The holomorphic couch theorem

We prove that if two conformal embeddings between Riemann surfaces with finite topology are homotopic, then they are isotopic through conformal embeddings. Furthermore, we show that the space of all conformal embeddings in a given homotopy class deformation retracts into a point, a circle, a torus, or the unit tangent bundle of the codomain, depending on the induced homomorphism on fundamental groups. Quadratic differentials play a central role in the proof.


Introduction
Loosely speaking, the 1-parametric h-principle is said to hold for a class of maps between manifolds if the only obstructions to connecting two maps in the class through maps in the same class are topological [10, p.60]. For example, the 1-parametric h-principle holds for immersions of S 2 in R 3 , so that the standard sphere can be turned inside out via immersions. This is known as Smale's paradox. Of course, there are situations where the 1-parametric h-principle fails due to geometric obstructions. A famous example is Gromov's symplectic camel theorem, which says that one cannot move a closed 4-dimensional ball through a hole in a wall in R 4 via symplectic embeddings if the ball is bigger than the hole.
In this paper, we prove that the 1-parametric h-principle holds for conformal embeddings between finite Riemann surfaces, where a finite Riemann surface is a Riemann surface with finitely generated fundamental group. Theorem 1.1 (The holomorphic couch theorem). If two conformal embeddings between finite Riemann surfaces are homotopic, then they are isotopic through conformal embeddings.
We think of the codomain as a house and the domain as a couch that we want to move around in the house without changing its holomorphic structure. Hence the name "holomorphic couch".
Given finite Riemann surfaces X and Y , and a topological embedding h : X → Y , we define CEmb(X, Y, h) to be the set of all conformal embeddings homotopic to h. We equip this set with the compact-open topology, which is the same as the topology of uniform convergence on compact sets with respect to any metric inducing the correct topology on Y . Theorem 1.1 is equivalent 1 to the statement that CEmb(X, Y, h) is path-connected whenever it is non-empty.
In fact, we prove a stronger result. Namely, we determine the homotopy type of the space CEmb(X, Y, h). The answer depends on the image of h at the level of fundamental groups. We say that h is trivial, cyclic, or generic if the image of the induced homomorphism π 1 (h) : π 1 (X, x) → π 1 (Y, h(x)) is trivial, infinite cyclic, or non-abelian, respectively. Theorem 1.2 (Strong holomorphic couch). Suppose that CEmb(X, Y, h) is non-empty. Then CEmb(X, Y, h) is homotopy equivalent to either the unit tangent bundle of Y , a circle, or a point, depending on whether h is trivial, cyclic, or generic. This is unless Y is a torus and h is non-trivial, in which case CEmb(X, Y, h) is homotopy equivalent to a torus.
If h is generic, then CEmb(X, Y, h) is contractible. The rest of the introduction gives an outline of the proof for this case.
Ioffe's theorem. A half-translation structure on a finite Riemann surface is an atlas of conformal charts (for the surface minus a finite set) whose transition functions are of the form z → ±z + c (with some extra condition on the finite set) and in which the surface has finite area. This data can be specified by an integrable holomorphic quadratic differential. The charts of the half-translation structure should also map the ideal boundary to curves which are piecewise horizontal or vertical.
A Teichmüller embedding is an injective map f : X → Y for which there exist half-translation structures on X and Y , and a constant K ≥ 1 such that: • f is locally of the form x+iy → Kx+iy (up to sign and translation); • the ideal boundary of X and Y is horizontal; • Y \ f (X) is a finite union of points and horizontal arcs. Our main tool is a theorem of Ioffe which says that extremal quasiconformal embeddings and Teichmüller embeddings are one and the same [20]. Theorem 1.3 (Ioffe). Let f : X → Y be a quasiconformal embedding which is not conformal. Then f has minimal dilatation (i.e. is closest to being conformal) in its homotopy class if and only if it is a Teichmüller embedding.
In the special case where X and Y are closed, we recover Teichmüller's celebrated theorem, since an embedding between closed surfaces is a homeomorphism. What is different from Teichmüller's theorem, however, is that Teichmüller embeddings are not necessarily unique in their homotopy class. This is an important issue which we discuss next.
Slit mappings. A slit mapping is a conformal Teichmüller embedding, i.e. one with K = 1. In this case, the half-translation structure on X is redundant, for it is the pull-back of the half-translation structure on Y by the slit mapping.
We show that if the space CEmb(X, Y, h) contains a slit mapping, then every element of CEmb(X, Y, h) is a slit mapping and CEmb(X, Y, h) is naturally homeomorphic to a point or a compact interval. Theorem 1.4 (Slit mappings are almost rigid). Suppose that CEmb(X, Y, h) contains a slit mapping with respect to a half-translation structure Ψ on Y . Then every g ∈ CEmb(X, Y, h) is a slit mapping with respect to Ψ. Moreover, for every x ∈ X, the evaluation map CEmb(X, Y, h) → Y sending g to g(x) is a homeomorphism onto its image, which is a (possibly degenerate) compact horizontal arc whose length does not depend on x.
A better way to say this is: any conformal embedding homotopic to a slit mapping differs from the latter by a horizontal translation which can be performed gradually.
Since an interval is contractible, Theorem 1.4 implies Theorem 1.2 whenever the space CEmb(X, Y, h) contains a slit mapping. The idea is then to reduce the general case to the above one by enlarging X until it barely fits in Y .
Modulus of extension. Suppose that CEmb(X, Y, h) does not contain any slit mapping. Then X has at least one ideal boundary component 2 . Thus, we may define a 1-parameter family of enlargements of X as follows. We first choose an analytic parametrization S 1 → C of each ideal boundary component C of X. Then, for every r ∈ (0, ∞], we let X r be the bordered surface X ∪ ∂X with a copy of the cylinder S 1 × [0, r) glued to each ideal boundary component along S 1 × {0} via the fixed parametrization. We also let X 0 = X.
For any f ∈ CEmb(X, Y, h), we define the modulus of extension m(f ) as the supremum of the set of r ∈ [0, ∞] such that f extends to a conformal embedding of X r into Y . Montel's theorem in complex analysis implies that: • for every f ∈ CEmb(X, Y, h) the supremum m(f ) is achieved by a unique conformal embedding f : X m(f ) → Y extending f ; • CEmb(X, Y, h) is compact; • m is upper semi-continuous. In particular, m achieves its maximum value. Using Ioffe's theorem, it is not too hard to show that if m attains its maximum at f , then f is a slit mapping. We prove that the same holds if f is a local maximum of m. 2 If not, then for any f ∈ CEmb(X, Y, h) the complement Y \f (X) is finite so that f is a slit mapping, as long as Y supports a half-translation structure. The only finite Riemann surface with non-abelian fundamental group which does not support any half-translation structure is the triply punctured sphere. But if each of X and Y is a punctured sphere and h is generic, then CEmb(X, Y, h) has at most one element, so there is nothing to show. Theorem 1.5 (Characterization of local maxima). Let f be a local maximum of m, and let f be the maximal conformal extension of f to X m(f ) . Then f is a slit mapping. Conversely, if g : X r → Y is a slit mapping, then g| X is a global maximum of m. The set M of all local maxima of m is homeomorphic to a point or a compact interval, and m is constant on M .
The initial motivation for studying m was to think of it as a Morse function for the space CEmb(X, Y, h). In an ideal world, flowing along the gradient of m would yield a deformation retraction of CEmb(X, Y, h) into M . This does not make sense though, since m is not even continuous unless it is constant equal to zero. However, the connectedness of CEmb(X, Y, h) is an easy consequence of Theorem 1.5. Theorem 1.6 (Weak holomorphic couch). CEmb(X, Y, h) is connected.
To improve upon this, we show that there are no local obstructions to contractibility.
Where can one point go? Given a point x ∈ X, we are interested in set of points in Y where x can be mapped by the elements of CEmb(X, Y, h). It is convenient to also keep track of how x gets mapped to a given y ∈ Y in the following sense. If f ∈ CEmb(X, Y, h), then by definition there exists a homotopy H : X × [0, 1] → Y from h to f . The homotopy class of the path t → H(x, t) from h(x) to f (x) does not depend on the particular choice of H. We thus let lift x (f ) be the homotopy class of this path. This is an element of the universal cover of Y , which can be identified with the unit disk D.
The map lift x : CEmb(X, Y, h) → D is continuous, and we let Blob(x) be its image. The blob is simpler than the image of the evaluation map in much the same way as Teichmüller space is simpler than moduli space. Indeed, the blob is as simple as can be.
Theorem 1.7 (The blob is a disk). If CEmb(X, Y, h) is non-empty and does not contain any slit mapping, then Blob(x) is homeomorphic to a closed disk.
The proof has four steps: (1) the blob is compact and connected; (2) the blob is semi-smooth; (3) every semi-smooth set is a 2-manifold with boundary; (4) there are no holes in the blob.
Step (1) follows because Blob(x) is the continuous image of CEmb(X, Y, h) which is compact and connected. The next two steps require some terminology. We say that a closed subset B of R 2 is semi-smooth if: • at each boundary point of B, there is at least one normal vector, i.e. a non-zero vector pointing orthogonally away from B; • the set of all normal vectors at a point is convex (and is thus a cone with total angle less than π); • any non-zero limit of normal vectors (not necessarily based at the same point) is a normal vector.
For example, 2-manifolds with C 1 boundary are semi-smooth, and so are closed convex sets with non-empty interior. To be clear, semi-smooth sets are allowed to have corners, but these should look convex as we zoom in on them. Corners pointing inwards-or dents-are forbidden since the first condition is not met. Similarly, outward pointing cusps are forbidden by the second condition. We now describe some of the ideas involved in step (2). Let π : D → Y be the universal covering map, let y = lift x (f ) be on the boundary of Blob(x), and let y = π( y) = f (x). It follows from Ioffe's theorem and a limiting argument that the restriction f : X \ x → Y \ y is a slit mapping. We say that f is a slit mapping rel x. We use the notion of extremal length, and in particular Gardiner's formula, to determine which tangent vectors are normal to Blob(x) at y. If F denotes the horizontal foliation of a halftranslation structure on X \ x with respect to which f is a slit mapping rel x, then the extremal length of F on X \ x is equal to the extremal length of f * F on Y \ y. Now the point is that extremal length does not increase under conformal embeddings. Therefore, whenever a point y close enough to y is such that the extremal length of f * F is bigger on Y \ y than on Y \ y, then the lift of y nearest to y is outside Blob(x). But Gardiner's formula tells us precisely how the extremal length of f * F on Y \ y varies as we move y around. The end result is that v is normal to Blob(x) at y if and only if π * (v) is vertical for some half-translation structure on Y \ y with respect to which f is a slit mapping rel x. We deduce that the blob is semi-smooth.
It is well-known that every closed convex set with non-empty interior in R 2 is a 2-manifold with boundary.
Step (3) is a generalization of this to semi-smooth sets.
Given (1), (2) and (3), we have that Blob(x) is a connected compact 2manifold with boundary in the plane. It only remains to show that Blob(x) does not have any holes. For this, we consider the family of blobs corresponding to the enlargements of X constructed earlier. In other words, we look at Blob(x, X r ) for r ∈ [0, R], where R is the maximal value of the modulus of extension m. These blobs are nested and are all 2-manifolds with boundary except for Blob(x, X R ), which is either a point or a compact interval. We show that the number of connected components of D \ Blob(x, X r ) does not depend on r. Since the complement of Blob(x, X R ) is connected, it follows that D \ Blob(x) is connected and hence that Blob(x) is a disk.
Moving one point at a time. Pick a countable dense set {x 1 , x 2 , ...} in X and let F ∈ CEmb(X, Y, h) be any conformal embedding. We now explain how to get a deformation retract of CEmb(X, Y, h) into {F } by moving one point at a time. Given f ∈ CEmb(X, Y, h), we join lift x 1 (f ) to lift x 1 (F ) by a path γ 1 in Blob(x 1 ). Such a path exists since Blob(x 1 ) is homeomorphic to a closed disk.
For every t ∈ [0, 1], we then look at where x 2 can go under maps g ∈ CEmb(X, Y, h) which satisfy lift x 1 (g) = γ 1 (t). This defines a new kind of blob, call it Blob t (x 2 ). We find that Blob t (x 2 ) moves continuously with t, which allows us to construct a second path γ 2 from lift x 2 (f ) to lift x 2 (F ) with the property that for every t ∈ [0, 1], the point γ 2 (t) belongs to Blob t (x 2 ).
Proceeding by induction, we obtain a sequence of paths γ 1 , γ 2 , . . . such that for every n ∈ N and every t ∈ [0, 1], there exists at least one map f n t ∈ CEmb(X, Y, h) such that lift x j (f n t ) = γ j (t) for every j ∈ {1, . . . , n}. If we fix t and pass to a subsequence, we get some limit f t ∈ CEmb(X, Y, h) for which lift x j (f t ) = γ j (t) for every j ∈ N. Since any two limits agree on the dense set {x 1 , x 2 , ...}, we actually have convergence f n t → f t for the whole sequence. By a similar argument, f t depends continuously on t. We thus found a path from f to F in CEmb(X, Y, h).
We construct the paths γ 1 , γ 2 , . . . carefully enough so that they depend continuously on the initial map f . The end result is a deformation retraction of CEmb(X, Y, h) into {F }.
Notes and references. The holomorphic couch problem arose in the context of renormalization in complex dynamics. Although our theorem does not have any direct applications to dynamics, some of the tools used heresuch as extremal length and Ioffe's theorem-do (see [39]).
The space CEmb(D, C), or rather its subspace S of conformal embeddings f : D → C satisfying f (0) = 0 and f (0) = 1, was the subject of much interest until the solution of the Bieberbach conjecture by de Branges in 1984. It is easy to see that S is contractible. On the other hand, S has isolated points when equipped with the topology of uniform convergence of the Schwarzian derivative instead of the compact-open topology [40]. The literature on the class S is quite vast. In comparison, not much has been written about conformal embeddings between general Riemann surfaces. Exceptions include [34], [20], [21], [6], and [35].
The holomorphic couch problem for embeddings of a multiply punctured disk in a multiply punctured sphere was first considered in [32]. However, the solution presented there relies in part on a rigidity claim [31] which is known to be false in general [22]. On a related note, Ioffe asserts that generic slit mappings are completely rigid in [20], but this is wrong. Indeed, we give examples where CEmb(X, Y, h) is an interval and not a singleton in Section 3. The conclusion of Theorem 1.4 is thus optimal.
There are various functionals similar to the modulus of extension m for which analogous versions of Theorem 1.5 hold. For example, instead of gluing the annuli to X, one can keep them disjoint from X [6]. More generally, there is a plethora of extremal problems on Riemann surfaces whose solutions involve quadratic differentials (see e.g. [23] and the references therein).
For the class S of normalized univalent functions from D to C, a suitable version of the blob is actually a round disk. More precisely, for every z ∈ D and every f ∈ S, the quantity w = log(f (z)/z) satisfies and every value w satisfying the inequality is achieved for some f ∈ S. This was proved by Grunsky in 1932 (see [5, p.323]). The blob for Kquasiconformal homeomorphisms was studied in [38] and [9]. Our approach for proving that the blob is homeomorphic to a closed disk seems similar to Strebel's, although the context is different.
The idea of moving one point at a time to get an isotopy is reminiscent of the finite "holomorphic axiom of choice" used by Slodkowski to extend holomorphic motions [36]. Our isotopies are holomorphic in the space variable and continuous in the time variable, whereas holomorphic motions are the other way around.
In closing, we note that our results can be extended to the situation where the domain X is a finite union of finite Riemann surfaces. This is the setup considered in Ioffe's paper [20]. We restrict ourselves to connected domains for simplicity.

Preliminaries
Ideal boundary and punctures. A Riemann surface is hyperbolic if its universal covering space is conformally isomorphic to the unit disk D. The only non-hyperbolic Riemann surfaces are C, C, C \ 0 and complex tori, where C is the Riemann sphere. A hyperbolic surface X can be regarded as the quotient of its universal covering space D by its group of deck transformations Γ. The limit set Λ Γ is the set of accumulation points in ∂D of the Γ-orbit of any point z ∈ D, and the set of discontinuity is Ω Γ = ∂D \ Λ Γ . The ideal boundary of X is ∂X = Ω Γ /Γ. The union X ∪ ∂X = (D ∪ Ω Γ )/Γ is naturally a bordered Riemann surface, since Γ acts properly discontinuously and analytically on D ∪ Ω Γ . If X is a finite hyperbolic surface, then ∂X has finitely many connected components, each homeomorphic to a circle.
A puncture in a Riemann surface X is an end for which there is a proper (preimages of compact sets are compact) conformal embedding D \ 0 → X. For example, C has one puncture at infinity and C \ 0 has two punctures. Every puncture can be filled, meaning that one can add the missing point and extend the complex structure there. The set of punctures of X is denoted byẊ and the Riemann surface obtained by filling the punctures is denoted by X ∪Ẋ. For hyperbolic surfaces, punctures are the same as a cusps, or ends with parabolic monodromy.
Given a finite hyperbolic surface X, we write X = X ∪ ∂X ∪Ẋ for the compact bordered Riemann surface obtained after adding the ideal boundary and filling the punctures. Suppose that ∂X is non-empty. Then if we take two copies of X-the second with reversed orientation-and glue them along ∂X with the identity, we get a closed Riemann surface called the double of X. Because of this construction, many theorems about closed Riemann surfaces with finitely many points removed are also true for finite Riemann surfaces. We will state and use such theorems without further remarks.
Montel's theorem. The simplest version of Montel's theorem says that the set of all holomorphic maps from D to D is compact. This implies a similar result for holomorphic maps between arbitrary hyperbolic surfaces. A sequence of maps f n : X → Y between Riemann surfaces diverges locally uniformly if for every compact sets K ⊂ X and L ⊂ Y , the sets f n (K) and L are disjoint for all large enough n. A set F of maps between two Riemann surfaces X and Y is normal if every sequence in F admits either a locally uniformly convergent subsequence or a locally uniformly divergent subsequence.
Theorem 2.1 (Montel). If X and Y are hyperbolic surfaces, then every set of holomorphic maps from X to Y is normal.
See [28, p.34]. Note that the limit of a convergent sequence of holomorphic maps is holomorphic. If every map in the sequence is injective, then the limit is either injective or constant. If every map in the sequence is locally injective, then the limit is either locally injective or constant.
Quasiconformal maps. A K-quasiconformal map between Riemann surfaces is a homeomorphism f such that in charts, its first partial derivatives in the distributional sense are locally in L 2 and the formal matrix df of partial derivatives satisfies the inequality df 2 ≤ K det(df ) almost everywhere. For each point z, the real linear map d z f sends circles in the tangent plane at z to ellipses of eccentricity d z f 2 / det(d z f ) in the tangent plane at f (z), and this ratio is called the pointwise dilatation of f at z. The dilatation of f , denoted Dil(f ), is the smallest K ≥ 1 for which f is K-quasiconformal. This is the same as the essential supremum of the pointwise dilatation of f .
A Beltrami form on a Riemann surface X is a map µ : T X → C such that µ(λv) = (λ/λ)µ(v) for every v ∈ T X and every λ ∈ C \ 0. In charts, the Wirtinger derivatives of a locally quasiconformal map f are The ratio ∂f /∂f is naturally a Beltrami form, and is sometimes called the Beltrami coefficient of f . The Beltrami coefficient of f encodes the field of ellipses in T X which df sends to circles. The measurable Riemann mapping theorem says that every measurable ellipse field with bounded eccentricity is the Beltrami coefficient of a quasiconformal map. Ahlfors-Bers). Let X be a Riemann surface and let µ be a measurable Beltrami form on X such that µ ∞ < 1. Then there exists a Riemann surface Y and a quasiconformal map f : X → Y such that ∂f /∂f = µ almost everywhere. The surface and the map are unique up to conformal isomorphism.
An important consequence is the following factorization principle. Suppose that f : X → Y is locally quasiconformal (or quasiregular). Then f = F • ϕ where ϕ : X → X is a quasiconformal homeomorphism and F : X → Y is holomorphic. Indeed, we can take ϕ to be the solution of the Beltrami equation with µ = ∂f /∂f and let F = f • ϕ −1 .
Montel's theorem generalizes to suitably normalized K-quasiconformal maps from the disk to itself. Theorem 2.3. For every K ≥ 1, the space of all K-quasiconformal maps from D to D fixing 0 is compact.
Lastly, we will use the fact that isolated points are removable singularities for quasiconformal maps. The reader may consult [2] for background on quasiconformal maps.
Quadratic differentials. A quadratic differential on a Riemann surface X is a map ϕ : T X → C such that ϕ(λv) = λ 2 ϕ(v) for every v ∈ T X and every λ ∈ C\0. A quadratic differential on X is holomorphic (resp. meromorphic) if for every open set U ⊂ X, and every holomorphic vector field v : U → T U , the function ϕ • v : U → C is holomorphic (resp. meromorphic). All quadratic differentials in this paper will be holomorphic or meromorphic.
The pull-back f * ϕ of a quadratic differential ϕ by a holomorphic map f is defined in the usual way by the formula is horizontal (resp. vertical) for every t ∈ I. The absolute value |ϕ| is an area form, and its integral ϕ = X |ϕ| is the norm of ϕ. For a finite Riemann surface X, we denote by Q(X) the set of all integrable holomorphic quadratic differentials ϕ on X which extend analytically to the ideal boundary of X, and such that ϕ(v) ∈ R for every vector v tangent to ∂X. Every ϕ ∈ Q(X) extends to a meromorphic quadratic differential on X with at most simple poles onẊ. The set Q + (X) is similarly defined, but with the additional requirements that ϕ ≥ 0 along ∂X and that ϕ is not identically zero. The set Q(X) is a real vector space inside of which Q + (X) forms a convex cone.
For every simply connected open set U ⊂ X where a quadratic differential ϕ does not have any zero or pole, there exists a locally injective holomorphic map z : U → C such that ϕ = dz 2 . The map z is unique up to translation and sign and is called a natural coordinate when it is injective. If ϕ ∈ Q(X), then the atlas of natural coordinates for ϕ is a half-translation structure on X in the sense of the introduction. Such a structure induces a flat geometry with cone points on X which we will discuss in Sections 3 and 4. The standard reference for this material is [37].
Teichmüller's theorem. A Teichmüller map between finite Riemann surfaces X and Y is a homeomorphism f : X → Y such that there exists a constant K > 1 and non-zero ϕ ∈ Q(X) and ψ ∈ Q(Y ) such that f is locally of the form x + iy → Kx + iy (up to sign and translation) in natural coordinates. Such a map is K-quasiconformal with constant pointwise dilatation.
The following theorems of Teichmüller are central in the theory of deformations of Riemann surfaces. Theorem 2.4 (Teichmüller's existence theorem). Let h be a quasiconformal map between finite Riemann surfaces. If there is no conformal isomorphism homotopic to h, then there is a Teichmüller map homotopic to h. Theorem 2.5 (Teichmüller's uniqueness theorem). Let f be a Teichmüller map of dilatation K between finite Riemann surfaces that are not annuli nor tori. If g is a K-quasiconformal map homotopic to f , then g = f .
For annuli and tori, Teichmüller maps are unique up to conformal automorphisms homotopic to the identity. Teichmüller's theorem is usually stated and proved for closed Riemann surfaces, but the general case follows from the closed case by doubling across the ideal boundary and by taking a branched cover of degree 2 or 4 ramified at the punctures [1].
Reduced Teichmüller spaces. Let S be a finite Riemann surface. The reduced Teichmüller space T # (S) is defined as the set of pairs (X, f ) where X is a finite Riemann surface and f : S → X is a quasiconformal map, modulo the equivalence relation (X, f ) ∼ (Y, g) if and only if g • f −1 is homotopic to a conformal isomorphism. The equivalence class of (X, f ) is denoted [X, f ], or just X when the marking f is implicit. The Teichmüller distance between two points of T # (S) is defined as where the infimum is taken over all quasiconformal maps h homotopic to g • f −1 . By Teichmüller's theorem, the infimum is realized by a (usually unique) quasiconformal map h which is either conformal or a Teichmüller map. The space T # (S) is a contractible real-analytic manifold of finite dimension. Let M(X) denote the space of essentially bounded Beltrami forms on X ∈ T # (S). By the measurable Riemann mapping theorem, the tangent space to T # (S) at X can be identified with the quotient of M(X) by its subspace M 0 (X) of infinitesimally trivial deformations. There is a natural pairing between M(X) and Q(X) given by and it turns out that M 0 (X) = Q(X) ⊥ with respect to this pairing. Therefore the tangent and cotangent spaces to T # (S) at X are isomorphic to M(X)/Q(X) ⊥ and Q(X) respectively. See [7] and [8] for more details.
Homotopies. If two maps are homotopic, then they induce the same homomorphism between fundamental groups, up to conjugation. The converse also holds under appropriate conditions [2, p.60] [3, §6]. Lemma 2.6 (Ahlfors). Let X be a space which has a universal cover, let Y be a metric space whose universal cover is a uniquely geodesic space in which geodesics depend continuously on endpoints, and let f 0 , f 1 : X → Y be continuous maps. Suppose that for some x ∈ X the induced homomorphisms π 1 (f j ) : π 1 (X, x) → π 1 (Y, f j (x)) agree up to conjugation by a path between f 0 (x) and f 1 (x). Then f 0 and f 1 are homotopic.
Proof. Let X and Y be the universal covers of X and Y , and let α be a path connecting f 0 (x) to f 1 (x) which conjugates the homomorphisms π 1 (f 0 ) and π 1 (f 1 ). Given a lift f 0 : X → Y , the path α allows us to lift f 1 in such a way that f 0 and f 1 are equivariant with respect to the same homomorphism of deck groups. The homotopy from f 0 to f 1 sending (x, t) ∈ X × [0, 1] to the point at proportion t along the geodesic from f 0 (x) to f 1 (x) in Y is continuous and equivariant, so it descends to a homotopy from f 0 to f 1 . This is also true if X is a CW-complex and Y is a K(π, 1) [18, p.90]. If X and Y are hyperbolic surfaces, then either hypotheses are satisfied. The most useful consequence for us is that homotopy classes of maps from a finite Riemann surface to a hyperbolic surface are closed.
Corollary 2.7. Let X and Y be hyperbolic surfaces, with X finite, and let f n , f : X → Y be continuous maps such that f n → f . Then f n is homotopic to f for all large enough n.
Proof. Let β 1 , . . . , β k be loops based at x ∈ X which generate π 1 (X, x) and let V be a simply connected neighborhood f (x) in Y .
Let n be large enough so that f n (x) ∈ V and f n (β j ) is freely homotopic to f (β j ) for every j ∈ {1, . . . , k}. If α is any path between f n (x) and f (x) in V , then α * f (β j ) * α is homotopic to f n (β j ) for every j. By the previous lemma, f n is homotopic to f .
In Teichmüller theory, one often goes back and forth between punctures and marked points as convenient. This passage is justified by the removability of isolated singularities for quasiconformal maps and the following elementary lemma whose proof is left to the reader.
Lemma 2.8. Let X and Y be Riemann surfaces, let f 0 , f 1 : X → Y be continuous maps, and let x ∈ X and y ∈ Y . Suppose that f −1 j (y) = {x} for j = 0, 1 and suppose the restrictions The analogous statement for finite sets of points holds as well.

Ioffe's theorem
The goal of this section is to characterize quasiconformal embeddings that have minimal dilatation in their homotopy class. We start with a compactness lemma that guarantees the existence of such extremal quasiconformal embeddings. From this section to Section 10, X and Y are assumed to be finite hyperbolic surfaces. Definition 3.1. An embedding between Riemann surfaces is parabolic if the image of the induced homomorphism on fundamental groups is cyclic and generated by a loop around a puncture.
Recall also that h is trivial or cyclic if the image of π 1 (h) is trivial or infinite cyclic respectively. Proof. Let π X : D → X and π Y : D → Y be universal covering maps with respective deck groups Deck(X) and Deck(Y ). The map h induces a homomorphism Θ : Deck(X) → Deck(Y ) once we fix basepoints. If f : X → Y is a K-quasiconformal embedding homotopic to h, then it lifts to a locally K-quasiconformal map f : D → D satisfying f (α · z) = Θ(α) · f (z) for every z ∈ D and every α ∈ Deck(X). The lift f is only defined up to composition by elements of the two deck groups. The condition that f be injective is equivalent to the condition π Y ( f (z)) = π Y ( f (w)) ⇒ π X (z) = π X (w).
There is a unique factorization f = F • ϕ where ϕ : D → D is a Kquasiconformal homeomorphism fixing 0 and 1, and where F : D → D is holomorphic and locally injective. As mentioned earlier, the space of Kquasiconformal homeomorphisms of D fixing the origin and the space of holomorphic maps from D to D are both compact. Thus, given a sequence of K-quasiconformal embeddings f n : X → Y and corresponding lifts f n = F n • ϕ n , we can pass to a subsequence such that F n → F and ϕ n → ϕ and hence f n → f = F • ϕ.
Suppose that each f n is homotopic to h and that the latter is neither trivial nor cyclic. Then the limit f is not constant. Indeed, if there is a constant w 0 ∈ D such that f (z) = w 0 for every z ∈ D, then for every α ∈ Deck(X). However, a Fuchsian group which fixes a point in D is cyclic, contradicting the assumption on h. In particular, the holomorphic function ϕ is not constant, thus has image in D, and by Hurwitz's theorem it is locally injective. Therefore f = F • ϕ is locally K-quasiconformal. Moreover, the equality f (α · z) = Θ(α) · f (z) for every z ∈ D and every α ∈ Deck(X) implies that f descends to a locally K-quasiconformal map f : X → Y .
We have to show that f is injective. If f (π X (z)) = f (π X (w)), then π Y ( f (z)) = π Y ( f (w)). Since f n → f and since these maps are open, we can find a sequence z n converging to z and a sequence w n converging to w such that f n (z n ) = f (z) and f n (w n ) = f (w) for all n large enough. Then π Y ( f n (z n )) = π Y ( f n (w n )), so that π X (z n ) = π X (w n ), and thus π X (z) = π X (w). Lastly, f is homotopic to h because it is a limit of maps which are.
Suppose now that the image of Θ is a cyclic group generated by a hyperbolic element β ∈ Deck(Y ). Let z 0 ∈ D, and let D ⊂ D be a fundamental domain for β whose closure is disjoint from the fixed points of β. By applying an appropriate power of β, we can assume that the lift f n is such that f n (z 0 ) ∈ D. After such a normalization, the limit f (z 0 ) belongs to the closure D, so it is not one of the fixed points of β. It follows that f is not constant. Then the above argument applies verbatim.
Conversely, if h is trivial then its image is contained in a disk in Y . In other words, h can be written as h = F • ϕ where F : D → Y is a conformal embedding and ϕ : X → D is a K-quasiconformal embedding.
Consider the sequence f n = F n • ϕ where F n (z) = F (z/n). Each f n is a K-quasiconformal embedding isotopic to h, but the sequence converges to a constant map. Similarly, if h is parabolic, then we can form a sequence of isotopic K-quasiconformal embeddings which diverges to the corresponding puncture.
The following corollary is immediate. Corollary 3.3. Let h : X → Y be a non-trivial and non-parabolic quasiconformal embedding. There exists a quasiconformal embedding f : X → Y homotopic to h with minimal dilatation.
We present Ioffe's theorem in two parts. The first part says that every extremal quasiconformal embedding is a Teichmüller embedding.
Definition 3.4. A Teichmüller embedding of dilatation K ≥ 1 is an injective continuous map f : X → Y for which there exist quadratic differentials ϕ ∈ Q + (X) and ψ ∈ Q + (Y ) such that f has the form x + iy → Kx + iy in natural coordinates and such that Y \ f (X) is an analytic graph all of whose edges are horizontal for ψ. We say that ϕ and ψ are initial and terminal quadratic differentials for f . A slit mapping is a conformal Teichmüller embedding, i.e. one with K = 1.
Proof. Suppose that f is not conformal and let Let F : Y → Y µ be the solution to the Beltrami equation ∂F/∂F = µ. By construction we have ∂(F • f )/∂(F • f ) = 0 so that F • f is a conformal embedding. Let G : Y µ → Y be a quasiconformal map homotopic to F −1 with minimal dilatation. By Teichmüller's theorem, either G is conformal or a Teichmüller map. The composition G • F • f : X → Y is a quasiconformal embedding homotopic to f , so that Thus all the terms in this chain are equal. The equality Dil(G) = Dil(F −1 ) implies that F has minimal dilatation in its homotopy class. Since F is not conformal, it is a Teichmüller map. This means that there is a nonzero ψ ∈ Q(Y ) and a constant k ∈ (0, 1) such that µ = −kψ/|ψ| almost everywhere. In particular, Y \ f (X) has measure zero and f has constant pointwise dilatation. Thought of as a homeomorphism from X to f (X), the map f has minimal dilatation in its homotopy class and is thus a Teichmüller map. Let ϕ and ω be its initial and terminal quadratic differentials. Since F • f is conformal, the directions of maximal stretching for F and f −1 must be perpendicular, which means that ψ = cω on f (X) for some positive constant c, which we may assume is equal to 1 by rescaling.
We have to show that ϕ ∈ Q + (X). If not, then ϕ < 0 along some segment I ⊂ ∂X. We will explicitly construct a quasiconformal embedding f from X to f (X) with pointwise dilatation smaller than f near I. We may work in a natural coordinate chart for ϕ in which I is equal to the vertical segment [−i, i] in the plane and X is to the right of I. There is also a natural chart for ω in which f takes the form x + iy → Kx + iy. which is affine on the upper and lower halves of ∆, fixes all three vertices of ∆, and sends the midpoint of I to (1 − 1/K)δ. Extend L to be the identity on X \ ∆ and let f = f • L. The linear part of f on the lower half of ∆ is equal to 1 (K − 1)δ 0 1 and the dilatation of this matrix tends to 1 as δ → 0. Therefore, if δ is small enough, then the embedding f has strictly smaller pointwise dilatation than f on ∆. Moreover the global dilatation of f is the same as f , so that f also has minimal dilatation in its homotopy class. By the first paragraph of the proof, the pointwise dilatation of f must be constant. This is a contradiction, and hence ϕ ∈ Q + (X).
It remains to show that f (X) is the complement of a graph which is horizontal with respect to ψ. Recall that X is the compactification of X obtained by adding its ideal boundary and filling its punctures. The path metrics induced by |ϕ| 1/2 and |ψ| 1/2 extend to complete metrics on X and Y . Since f : X → Y is K-Lipschitz with respect to those metrics, it extends to a K-Lipschitz map f : X → Y . Moreover, f is surjective since Y \ f (X) has measure zero and hence empty interior. Let I be the closure of a connected component of ∂X \{zeros of ϕ}. There is a sequence {I n } of arcs in X which are horizontal for ϕ and converge uniformly to I. Since the image arcs f (I n ) are all horizontal for ψ, they can only accumulate onto horizontal arcs, and thus f (I) is horizontal. Therefore, the complement Y \ f (X) = f (∂X ∪Ẋ) is a union of finitely many points and horizontal arcs for ψ. In particular, the ideal boundary ∂Y is horizontal for ψ so that ψ ∈ Q + (Y ).
In the last paragraph of the proof we actually showed the following useful criterion.
Lemma 3.6. Let ϕ ∈ Q + (X), let ψ ∈ Q(Y ), and let f : X → Y be an embedding which is locally of the form x + iy → Kx + iy in natural coordinates. If f (X) is dense in Y , then ψ ∈ Q + (Y ) and f is a Teichmüller embedding.
The second part of Ioffe's theorem says that every Teichmüller embedding is extremal. The proof is very similar to the proofs of Teichmüller's uniqueness theorem given in [3] and [11,Chapter 11].
Theorem 3.7 (Ioffe). Let f : X → Y be a Teichmüller embedding of dilatation K with terminal quadratic differential ψ, and let g : X → Y be a K-quasiconformal embedding homotopic to f . Then g is a Teichmüller embedding of dilatation K with terminal quadratic differential ψ, and the map is locally a translation in natural coordinates.
Proof. Let ϕ be the initial quadratic differential of f . By rescaling, we may assume that ϕ = 1. This implies that ψ = K since f multiplies area by a factor K and f (X) has full measure in Y . We equip X and Y with the complete conformal metrics |ϕ| 1/2 and |ψ| 1/2 . Since f is K-Lipschitz with respect to those metrics, it extends to a K-Lipschitz map f : X → Y .
We may assume that f mapsẊ intoẎ . Indeed, we can fill all the punctures of Y that are not poles of ψ, and then fill all the punctures of X that do not map to poles of ψ. The map f extended to those points is still a Teichmüller embedding.
Let M > 1. For every n ∈ N, let G n : X → X be a smooth Mquasiconformal and M -Lipschitz embedding such that • G n is homotopic to the identity relẊ; • G n ( X) is contained in X ∪Ẋ; • G n → id uniformly in the C 1 norm as n → ∞. Such maps are not difficult to construct. Given a smooth vector field on X pointing inwards on ∂X and vanishing onẊ, we can let G n be the corresponding flow at small enough time t n . Then let g n = g • G n : X → Y , which is a KM -quasiconformal embedding.
Let H : X × [0, 1] → Y be a homotopy from f to g n which is constant oṅ X. For every z ∈ X, let (z) = inf α |ψ| 1/2 where the infimum is taken over all piecewise smooth paths α : [0, 1] → Y that are homotopic to the path t → H(z, t) rel endpoints. IfẎ is empty, then the infimum (z) is realized by a unique geodesic. However, in general the infimum need not be realized since the restriction of |ψ| 1/2 to Y is not complete. In any case, it is easy to see that is continuous on X and extends continuously to X. Since X is compact, there exists a constant B such that (z) < B for every z ∈ X.
Let γ be a horizontal arc of length L in X. Since f is a Teichmüller embedding of dilatation K, it sends γ to a horizontal arc of length KL in Y . Let z 0 and z 1 be the endpoints of γ. We can obtain a path homotopic to f (γ) by taking the concatenation of a piecewise smooth path α 0 of length at most B homotopic to t → H(z 0 , t), the image g n (γ), and a piecewise smooth path α 1 of length at most B homotopic to t → H(z 1 , 1 − t). Since horizontal arcs minimize horizontal travel among all homotopic paths [37, p.76], we have Let dg n denote the matrix of partial derivatives of g n with respect to natural coordinates 3 and (dg n ) 1,1 its first entry. If z and ζ are natural coordinates for ϕ and ψ, then Remove from X all trajectories that go through a puncture of X or a zero of ϕ and denote the resulting full measure subset by U . For every z ∈ U , there is a unique (possibly closed) bi-infinite horizontal trajectory through z. For every L > 0 and every z ∈ U , let γ L z be the horizontal arc of length L centered at z. Since g n is quasiconformal, it is absolutely continuous on almost every horizontal trajectory. Upon applying Fubini's theorem, we find We claim that X |(dg) 1,1 | · |ϕ| = lim n→∞ X |(dg n ) 1,1 | · |ϕ| and hence that X |(dg) 1,1 | · |ϕ| ≥ K. This is a consequence of the Vitali convergence theorem [33, p.94]. In order to apply the theorem, we need to check that the functions |(dg n ) 1,1 | are uniformly integrable. First observe that X det(dg) · |ϕ| = g(X) |ψ| ≤ K, so that det(dg) is integrable. It follows that for every ε > 0, there exists a δ > 0 such that if A ⊂ X is measurable and A |ϕ| < δ, which shows uniform integrability and proves the claim. Applying Cauchy-Schwarz to the inequality K ≤ X |(dg) 1,1 | · |ϕ| yields Since the two ends of this chain of inequalities agree, each intermediate inequality is in fact an equality. Equality in (b) implies that |(dg) 1,1 | is equal to a constant almost everywhere on X, and that constant is equal to K by (a). The inequality (c) is based on Equality implies that dg 1 0 = ± K 0 . Moreover, since dg stretches maximally in the horizontal direction which is preserved, dg must be diagonal, i.e. dg = ± K 0 0 * with 0 < * ≤ K. Then the equality K 2 = dg 2 = K det(dg) determines that dg = ± K 0 0 1 almost everywhere on X.
almost everywhere on f (X), so that g • f −1 is conformal and in particular smooth. Since f is smooth except at the zeros of ϕ, the same holds for g. Therefore the equality dg = ± K 0 0 1 holds everywhere except at the zeros of ϕ, and g takes the form x + iy → ±(Kx + iy) + c in natural coordinates. Equality in (d) means that g(X) has full measure in Y . By Lemma 3.6, g is a Teichmüller embedding with terminal quadratic differential ψ. Finally, the equality d(g • f −1 ) = ± 1 0 0 1 holds everywhere, so that g • f −1 is a local translation.
In the case where X and Y are closed hyperbolic surfaces, we get that g • f −1 : Y → Y is a conformal isomorphism (hence an isometry) homotopic to the identity. It is well known [3, §12.A] that the identity is the only such map, so that g = f . This fact is also used in the proof of Hurwitz's theorem on automorphisms of closed hyperbolic surfaces [11, p.215]. We prove a more general statement for later use.
Lemma 3.8. Let X be a hyperbolic Riemann surface not isomorphic to D nor D \ 0, and let h : X → X be a holomorphic map homotopic to the identity. Then h is equal to the identity unless X is an annulus and h is a rotation.
Proof. By the Schwarz lemma, h is 1-Lipschitz with respect to the hyperbolic metric. Therefore, if α is a closed geodesic in X, then h(α) is at most as long as α. But geodesics minimize length in their homotopy class, so that h(α) = α. In particular, h is an isometry along α.
If X is an annulus, then it contains a unique simple closed geodesic α. We can post-compose h by a rotation r of X so that r • h is equal to the identity on α and hence on all of X by the identity principle.
If X is not an annulus, then it contains a closed geodesic α which selfintersects exactly once. Then h fixes this self-intersection point, thus all of α pointwise, and hence all of X pointwise by the identity principle.
As pointed out before the proof, this lemma implies the uniqueness of Teichmüller maps between closed hyperbolic surfaces. However, the above argument does not apply to non-surjective Teichmüller embeddings. Indeed, Teichmüller embeddings are not necessarily unique in their homotopy class.
If f and g are homotopic Teichmüller embeddings from X to Y , then the inclusion f (X) → Y and the map g • f −1 : f (X) → Y are homotopic slit mappings by Theorem 3.7. Therefore, it suffices to study the question of uniqueness for slit mappings. There are two obvious ways for uniqueness to fail: • if Y is a torus 4 , then we can post-compose f with any automorphism of Y isotopic to the identity; • if f (X) is contained in an annulus A ⊂ Y , then we can post-compose f with rotations of A. In [20], Ioffe claims these are the only exceptions, but this is wrong 5 . The next simplest example is as follows. Let Y be a round annulus in the plane with a concentric circular arc removed, and let X be the same annulus but with a slightly longer arc removed. Then we can obviously rotate X inside of Y by some amount. This gives a 1-parameter family of slit mappings between triply connected domains. If the annulus is centered at the origin, then the terminal quadratic differential is −dz 2 /z 2 . Figure 2 shows this example, but from a different perspective. The slit in Y was opened up and mapped on the outside. This is to illustrate a point which we have not explained yet. If Y is a finite Riemann surface, ψ ∈ Q + (Y ), and X is the complement of a collection of horizontal slits in Y , then ι * ψ belongs to Q + (X), where ι : X → Y is the inclusion map. The only thing to check is that ι * ψ extends analytically to the ideal boundary of X. Near the endpoint of a slit, the process of unfolding the slit to an ideal boundary component is the same as taking a square root. If we pull-back the quadratic differential dz 2 in C by the square root from C \ [0, ∞) to the upper half-plane, we get the quadratic differential 4z 2 dz 2 . In other words, unfolding a regular point of ψ yields half of a double zero of ι * ψ on the ideal boundary. More generally, if the slit ends at a zero of order k, unfolding yields half of a zero of order 2k + 2. It is perhaps more natural to count the number of prongs: an n-prong singularity transforms into half of a 2n-prong singularity (see Figure 3). Clearly ∂X is horizontal for ι * ψ. Here is a general method for constructing examples of non-unique slit mappings with arbitrary codomain Y . Let ω be a holomorphic 1-form on Y such that ω 2 ∈ Q + (Y ). If X ⊂ Y is the complement of enough horizontal slits for ω 2 , then X is not rigid. By "enough" we mean that for every point y ∈ Y which is either a zero of ω or a puncture of Y , and every trajectory γ of ω ending at y in forward time, we must remove a neighborhood of y in γ. Once this is done, then for all small enough t > 0 the time-t flow for the vector field 1/ω is conformal and injective on X, and is homotopic to the inclusion map yet different from it.
In the next section, we show that this kind of phenomenon is the worse that can happen: we can always get from any slit mapping to any homotopic one by flowing horizontally, although the quadratic differential does not have to be globally the square of an abelian differential.

Slit mappings are almost rigid
We prove that slit mappings are almost rigid in their isotopy class, in the sense that we can get from any one to any other by flowing horizontally.
Theorem 4.1. Let f 0 , f 1 : X → Y be homotopic slit mappings with terminal quadratic differential ψ. Then there is an isotopy f t from f 0 to f 1 through slit mappings with terminal quadratic differential ψ.
The statement is still true if we replace each occurrence of "slit mappings" with "Teichmüller embeddings of dilatation K". The general case follows from the case K = 1 by Theorem 3.7. The basic idea of the proof of theorem 4.1 is to use the flat metric |ψ| 1/2 on Y to modify the given homotopy H from f 0 to f 1 into a more geometric homotopy F . Based on the examples of non-rigid slit mappings we saw so far, we want t → F (x, t) to be the geodesic homotopic to t → H(x, t) on Y . If ψ has simple poles onẎ then geodesics are not unique, which is problematic. We use the following idea from [1, Section II.3] to get rid of the poles. Proof. Let k = #Ẏ . Suppose first that k is even. Then we can find disjoint simple smooth arcs in Y joining the points ofẎ in pairs. Take two copies of Y , cut each copy along the arcs, glue the left side of each arc in the first copy to the right side of the corresponding arc in the second copy and vice versa, let W be the resulting bordered Riemann surface, and let W be the interior of W minus the endpoints of the cuts. Then W = W and forgetting which copy of Y a point came from yields the desired degree 2 holomorphic branched cover c. The map c is locally injective except at the endpoints of the cuts where it has local degree 2. The restriction of c to W is thus a covering map.
If k is odd and larger than 1, then we can do the above construction with (k − 1) points. The point left over has two distinct lifts in in the resulting double branched cover. We can thus repeat the first construction with these two points, and get a degree 4 cover with degree 2 branched points overẎ .
Suppose that k = 1. Since Y is hyperbolic, either it has positive genus or non-empty ideal boundary. If Y has positive genus, then we can find a nonseparating simple smooth loop in Y , cut Y along that loop, and glue two copies along the loop. This gives an unbranched degree two cover in which Y has two distinct lifts, so we can perform the first construction with these two lifts. If Y has non-empty boundary, then we can cut Y along a simple smooth arc in Y joiningẎ to the boundary, and glue two copies along the arc. The forgetful map is then a degree two branched cover ramified only overẎ .
Remark. We do not use the orientation double cover of ψ because we do not want the map c to be branched over the zeros of ψ in Y .
Let W and c be as in the previous lemma. IfẎ is empty, then we let W = Y and take c to be the identity map. We then let U be the universal covering space of W and we let b : U → Y be the composition of U → W with the branched cover c : W → Y . Another way to say this is: consider Y as an orbifold with the group Z/2Z attached to each point ofẎ , and let b : U → Y be the corresponding universal covering orbifold.
To avoid unnecessarily heavy notation, we denote the pull-back b * ψ by the same letter ψ. The quadratic differential ψ is holomorphic on U since b is branched of degree 2 over every point ofẎ . The metric |ψ| 1/2 induces a distance on U defined by d(x, y) = inf α |ψ| 1/2 where the infimum is taken over all piecewise smooth paths α between x and y.
We recall some well-known facts about the geometry of (U, d). First, the metric d is complete since the corresponding metric on W is complete. A minimizing geodesic in (U, d) is a shortest path between two points, whereas a geodesic is a locally length minimizing path between two points. Since (U, d) is locally compact and complete, the Hopf-Rinow theorem implies that any two points in U can be joined by a minimizing geodesic, and that closed bounded sets in U are compact [4, p.35].
The metric induced by |ψ| 1/2 is simply the pull-back of the Euclidean metric by natural coordinates. Thus a geodesic in U travels in straight lines in natural coordinates. Near a k-pronged singularity the metric is isometric to a cone with angle kπ and a geodesic cannot turn inefficiently when it hits such a cone point. More precisely, a path γ : [0, 1] → U is geodesic if and only if • γ is smooth except perhaps at zeros of ψ; • the argument of ψ(γ (t)) is locally constant where γ is smooth; • at a zero of ψ, the cone angle on either side of γ is at least π.
The last point is called the angle condition.
A geodesic polygon in U is a concatenation of finitely many geodesics which closes up. Any geodesic polygon in U is compact and thus the region it bounds contains only finitely many zeros of ψ. Therefore, the Gauss-Bonnet theorem can be applied, and it takes a very simple form in this context.
Proof. The Gauss-Bonnet formula says that int(P ) The Gaussian curvature K of the metric |ψ| 1/2 is nil everywhere in the interior of P except at a zero z of ψ, where the curvature is equal to 2π minus the cone angle at z (which is the same as −π times the order of the zero). The geodesic curvature k is also zero everywhere along ∂P except at vertices of the polygon or at zeros of ψ. The geodesic curvature at such points is the "turning angle", or π minus the internal angle.
This implies that there is a unique geodesic between any two points of U. In particular, every geodesic is minimizing. Proof. Suppose that x, y ∈ U can be joined by two distinct geodesics α and β. Then α and β bound one (or more) simply connected polygon P . The boundary of the polygon is geodesic except for two points p and q where α and β intersect. Thus the turning angle of ∂P is non-positive everywhere except perhaps at p and q by the angle condition. Teichmüller's lemma gives which is a contradiction.
The Arzelà-Ascoli theorem and the uniqueness of geodesics together imply that geodesics depend continuously on endpoints [4, p.37]. The next theorem is a generalization of Strebel's "divergence principle": if two geodesic rays γ 0 and γ 1 are such that the angles they form with the geodesic between γ 0 (0) and γ 1 (0) sum to at least π, then the distance between γ 0 (t) and γ 1 (t) is non-decreasing. In the original statement [37, p.77], each angle is assumed to be at least π 2 . Our proof makes use of the following immediate corollaries of Teichmüller's lemma.
Lemma 4.6. The sum of the interior angles of a geodesic triangle in U is at most π, with equality only if the triangle is isometric to a Euclidean one. A geodesic triangle does not contain any zero of ψ in its interior.
Lemma 4.7. The sum of the interior angles of a geodesic quadrilateral in U is at most 2π, with equality only if the quadrilateral is isometric to a Euclidean one. If the angle sum is at least π, then there is no zero of ψ in the interior of the quadrilateral and the excess angle along the sides is no more than π.
Theorem 4.8. Let x 0 , x 1 , y 1 , and y 0 be the vertices of a geodesic quadri- Proof. We may assume that the sides [x 0 , y 0 ] and [x 1 , y 1 ] have no singularities in their interior, for otherwise we can partition them into pairwise congruent subintervals which have no singularities in their interior and connect the partition points in pairs, thereby creating a sequence of quadrilaterals satisfying the hypotheses of the theorem, stacked on top of each other. Note that [x 0 , y 0 ] and [x 1 , y 1 ] do not intersect, since this would create a triangle with angle sum at least π at the vertices x 0 and x 1 .
The quadrilateral Q can bound at most two region. Indeed, if it bounded more, then some of them would be bigons, which is impossible by uniqueness of geodesics. If Q bounds no open set, then it is isometric to a degenerate Euclidean parallelogram, with the pairs {x 0 , x 1 } and {y 0 , y 1 } linked.
Suppose that Q bounds two regions, hence two triangles ∆ 0 = x 0 z 0 y 0 and ∆ 1 = z 1 x 1 y 1 . The sides [x 0 , x 1 ] and [y 0 , y 1 ] possibly share an edge [z 0 , z 1 ] which we may safely disregard. We have to show that Since ψ does not vanish in the interior of Q, the developing map can be defined on each ∆ j . In general the developing map is only a local isometry, but in this case it turns out to be injective. The integral is only defined in the interior of ∆ j , but Ψ j extends continuously to the boundary. The image ∆ j of ∆ j by Ψ j is a "fake" triangle in the plane, meaning it can have bends (or fake vertices) along its sides. Since the bends are all outwards, each side of ∆ j is concave. Moreover, the sides of ∆ j only intersect at actual vertices. Indeed, if two sides intersect more than once, then they create a bigon. The winding number of the triangle around any point in such a bigon is equal to −1. However, since Ψ j is a holomorphic function defined in a simply connected domain, it maps ∂∆ j to a curve which winds a nonnegative number of times around any point in C. It follows that Ψ j (∂∆ j ) is a simple closed curve, which implies that Ψ j is injective by the argument principle.
There is no loss of generality in assuming that Ψ j ([y j , z j ]) is the straight line between Ψ j (y j ) and Ψ j (z j ), since this segment does not intersect the Figure 4. The two triangles bounded by Q and their superposition.
other two sides and is shortest possible. Thus ∆ j has two straight sides and a possibly bent one. Indeed, the side Ψ j ([x j , y j ]) is straight since [x j , y j ] has no singularities in its interior, hence no bends. To prove the required inequality, take the fake triangles ∆ 0 and ∆ 1 and superpose them along their side of common length with a rotation or a flip so that x 0 coincides with y 1 and y 0 with x 1 , and that the angles x 0 and x 1 are on the same side of the superposed edge. Then the concavity of the sides and the hypothesis x 0 + x 1 ≥ π forces the diagonals [y 0 , z 0 ] and [y 1 , z 1 ] to intersect, say at w. Moreover, the bent sides do not cross the diagonals. This is because x 0 ≥ π − x 1 ≥ y 1 and similarly x 1 ≥ y 0 . Now the bent side from x 0 to z 0 is strictly shorter than d(x 0 , w) + d(w, z 0 ). One way to see this is by a repeated use of the triangle inequality. If we replace a connection between consecutive bends by extending the adjacent sides until they meet, we make the bent side longer. We can repeat this until there is only one bent left, then extend the side of the bend closer to x 0 until it meets [w, z 0 ]. The result is shorter than d(x 0 , w) + d(w, z 0 ) by the triangle inequality and similarly if there is no bend at all. Thus for each j ∈ {0, 1} the length of the bent side Ψ j ([x j , z j ]) is bounded above by d(x j , w) + d(w, z j ). Adding these two inequalities together yields the desired result since in the superposed triangles we have d( . The inequality is strict in this case. Now suppose that Q bounds a single region. The developing map Ψ for Q is an injective local isometry for the same reasons as above. Thus we can think of Q as embedded in the plane, with the two sides [x 0 , y 0 ] and [x 1 , y 1 ] straight and the other two sides possibly bent but concave. There is no loss of generality in assuming that the side joining y 0 and y 1 is straight. Suppose that the remaining side is straight as well. Then Q is just an ordinary quadrilateral in the plane. Suppose that Q is not a parallelogram. Then π < x 0 + x 1 ≤ 2π and hence cos( x 0 )+cos( x 1 ) < 0. Now d(y 0 , y 1 ) is at least as large as the length of the orthogonal projection of [y 0 , y 1 ] onto the line through x 0 and x 1 . On the other hand, the length of the projection is equal to which is strictly bigger than d(x 0 , x 1 ). It remains only to treat the case where the side joining x 0 and x 1 has at least one bend. In this case we have y 0 + y 1 < π. If there is more than one bend, then we may make the side Ψ([x 0 , x 1 ]) longer by straightening consecutive bends, while making sure not to cross [y 0 , y 1 ]. After we get rid of all pairs of bends we are left with two possibilities. Either there is a single bend left which is disjoint from [y 0 , y 1 ], or Ψ([x 0 , x 1 ]) and [y 0 , y 1 ] share a subinterval or a point. In the latter case we can ignore the shared segment. This leaves us with two triangles, in which case we know that the strict inequality holds. Thus suppose there is only one bend, say at w. Then there is a unique straight line S which divides the sides [x 0 , y 0 ] and [x 1 , y 1 ] equally and passes through w. Then S divides Q into two triangles and one quadrilateral. The length of the bent side of Q is strictly less than the length of S by the case of two triangles, and the length of S is less than d(y 0 , y 1 ) by the case of an ordinary Euclidean quadrilateral.
Remark. In fact, Theorem 4.8 holds in any CAT(0) space (see [4, p.184] for the definition of angles in CAT(κ) spaces). The proof of this generalization is rather simple provided one knows the required technology. The idea is to split the quadrilateral into two triangles via a diagonal and to use comparison triangles. Since CAT(0) spaces won't be used in the sequel, we preferred to stick with the above tedious but elementary proof.
We can now return to proving Theorem 4.1. We have two slit mappings f 0 , f 1 : X → Y and a homotopy H : X × [0, 1] → Y between them. The maps f 0 and f 1 extend to continuous maps from X to Y since they are 1-Lipschitz with respect to the complete metrics |ϕ| 1/2 and The last equality holds because f 1 •f −1 0 is a local translation with respect to ψ by Theorem 3.7. We may assume that f 0 and f 1 both carryẊ intoẎ , since we can fill in any other punctures and treat them as points of X. Since f 0 is homotopic to f 1 on X, we must have f 0 = f 1 onẊ. Thus we may assume that the homotopy H : X × [0, 1] → Y extends to be constant at the punctures by Lemma 2.8.
Given x ∈ X, choose a lift of the path t → H(x, t) to U, let γ x be the unique geodesic between the endpoints of this lift, let L(x) be its length, and let F (x, t) = b • γ x (t). Note that L(x) and F (x, t) do not depend on the choice of lift, since the deck group of the branched cover b : U → Y acts by isometries on U.
The maps L and F are continuous on X by Lemma 4.5 and the fact that H is continuous. We extend L and F to X as follows. If x ∈ ∂X, then take a simply connected ball B containing x. The restriction H : (B ∩ X) × [0, 1] → W lifts to U and extends continuously to B × {0, 1}. For every y ∈ B ∩ X, we have a well-defined geodesic γ y . Let γ x be the limit of γ y as y → x, let L(x) be its length, and let F (x, t) = b • γ x (t) as before. Since H is constant onẊ, we have L(x) → 0 as x converges to any point inẊ and we extend is a local translation allows us to analyze the local maxima of L with the aid of Theorem 4.8.
Theorem 4.9. Suppose that f 0 = f 1 and that L has a local maximum at x ∈ X. Then L is locally constant near x. If γ x passes through a zero of ψ, then x ∈ ∂X and γ x is horizontal. If ϕ has a zero at x, then x ∈ ∂X and the zero is simple.
. We will write σ j = f j (σ) for every σ ∈ S. Since L has a local maximum at x, there is a simply connected ball B centered at x such that L(y) ≤ L(x) for every y ∈ B. If L(x) = 0, then L(y) = 0 for every y ∈ B, which means that f 0 = f 1 on B and hence on all of X by the identity principle. The hypothesis thus implies that L(x) > 0. We may therefore choose the radius of B to be less than L(x).
Case 1: Suppose that x ∈ X. Then neither f 0 (x) nor f 1 (x) is inẎ and by shrinking B we may assume that f 0 (B) and f 1 (B) are disjoint fromẎ . Let y ∈ B \ x and consider the quadrilateral Q formed by the vertices x 0 , x 1 , y 1 , and y 0 . If y 0 x 0 x 1 + x 0 x 1 y 1 ≥ π, then by Theorem 4.8. Since equality holds, Q is isometric to a Euclidean parallelogram and y 0 x 0 x 1 + x 0 x 1 y 1 = π. Now if y 0 x 0 x 1 + x 0 x 1 y 1 ≤ π, then we replace y by a point y on the other side of x to form another quadrilateral Q . More precisely, let y ∈ B \ x be such that the angle yxy is at least π on either side. Since f j is injective on B, we also have y j x j y j ≥ π on either side for j = 0, 1. We thus get By the above reasoning the quadrilateral Q with vertices x 0 , x 1 , y 1 , and y 0 is isometric to a Euclidean parallelogram. This shows that y 0 x 0 x 1 + x 0 x 1 y 1 = π and hence y 0 x 0 x 1 + x 0 x 1 y 1 = π. Thus the first situation holds and L(y) = L(x), i.e. L is constant on B. Suppose that Q is degenerate. This means that [x j , y j ] ⊂ γ x for j = 0 or j = 1. Since f j is injective, this can hold for at most one ray η ⊂ B. Let y, y ∈ B \ η be a pair of points such that yxy ≥ π on either side. The corresponding parallelograms Q and Q are non-degenerate and on different sides of γ x , which implies that there is no excess angle on either side of γ x . In other words, the interior of γ x does not contain any zero of ψ. Suppose however that ψ has zeros at the endpoints of γ x . Then ϕ has a zero at x and there exists a y ∈ B \ x such that the angle between [y 0 , x 0 ] and γ x is strictly greater than π. The corresponding quadrilateral Q is not isometric to a Euclidean parallelogram, which contradicts the above reasoning. Therefore, γ x is completely free of singularities.
If x ∈ ∂X, then the same basic idea works. However, we distinguish a few subcases to say a bit more when ϕ has a zero at x. Case 2: Suppose that x ∈ ∂X, that neither f 0 (x) nor f 1 (x) is inẎ , and that ϕ does not have a zero at x. Then let y, y be on opposite sides of x in the horizontal segment B ∩ ∂X. By the same argument as in Case 1, the corresponding quadrilaterals Q and Q are parallelograms. If Q and Q are degenerate, then for one of j = 0 or j = 1 we have [x j , y j ] ⊂ γ x and [x j+1 , y j+1 ] ⊂ γ x where the index is taken mod 2. Moreover, [x j , y j ] and [x j+1 , y j+1 ] are horizontal since f j is a local translation. It follows that for every y ∈ B, the angles y 0 x 0 x 1 and x 0 x 1 y 1 are complementary so that L(y ) = L(x). If Q or Q is non-degenerate, then neither is the other one, and the interior of γ x is free of zeros. Thus the slope of γ x is constant. This implies that y 0 x 0 x 1 + x 0 x 1 y 1 = π for every y ∈ B so that L(y ) = L(x). In either case L is constant on B.
Case 3: Suppose that x ∈ ∂X, that neither f 0 (x) nor f 1 (x) is inẎ , and that ϕ has a zero at x. Then the total angle at x is at least 2π. Thus for every y ∈ B \ x, there is a y ∈ B \ x such that yxy ≥ π on either side. Then we can argue in exactly the same way as in Case 1 and we conclude that L is constant on B. Suppose that the quadrilateral Q with vertices x 0 , x 1 , y 1 , and y 0 is degenerate for some y ∈ B. Then [x j , y j ] ⊂ γ x for j = 0 or j = 1. Since f j is injective on B ∩ X, this can hold for y in at most one ray η ⊂ B ∩ X. Thus we can still find two points y, y ∈ B ∩ X \ η such that yxy ≥ π on either side. The corresponding parallelograms Q and Q are non-degenerate and on different sides of γ x , so that the interior of γ x is free of singularities. Suppose that there is some y ∈ B \ x such that y 0 x 0 x 1 > π on either side. Then L(y) > L(x) by Theorem 4.8, which is a contradiction. It follows that the zero of ϕ at x is simple (i.e. half of a double zero) and that the initial portion of γ x is the horizontal trajectory bisecting f 0 (B ∩X).

Case 4:
Suppose that x ∈ ∂X and that f j (x) ∈Ẏ for some j ∈ {0, 1}. Since B ∩X is simply connected, the restriction f j | B∩X lifts to a map f j : B ∩X → U. Moreover the lift extends continuously to B ∩∂X since the corresponding part of the frontier of f j (B ∩ X) is horizontal, hence a piecewise analytic curve. Thus one of Case 2 or Case 3 applies without change.
We conclude that L is constant. Proof. Let M be the closed subset of X where L attains its maximum value. Every point of M is a local maximum of L. By Theorem 4.9, L is locally constant near every local maximum, so that M is open. Since X is connected, M = X.
Hence every point of X is a local maximum of L, so that the conclusions of Theorem 4.9 hold everywhere. Corollary 4.11. Suppose that f 0 = f 1 . ThenẊ is empty and ϕ has no zeros in X and at most simple zeros in ∂X. Moreover, γ x is horizontal for every x ∈ X.
Proof. Since L is constant, every x ∈ X is a local maximum of L. IfẊ is non-empty, then f 0 = f 1 . By Theorem 4.9, if ϕ has a zero at x ∈ X ∪ ∂X, then x ∈ ∂X and the zero is simple.
For every x ∈ X, the geodesic γ x is free of singularities. Moreover, for every y near x, the geodesic γ y forms a parallelogram with γ x . Thus the slope of γ x is locally constant as a function of x, hence constant on X. Since the map x → γ x is locally continuous on X ∪ ∂X, the slope is constant on all X ∪ ∂X.
Let y ∈ Y be a zero of ψ or a point ofẎ where ψ does not have a pole. Since the map f 0 : X → Y is surjective, there is some x ∈ X with f 0 (x) = y. The corresponding geodesic γ x has a zero at its starting point. By Theorem 4.9, x ∈ ∂X and γ x is horizontal. Thus all other geodesics are horizontal.
If ψ has no zeros in Y , then Y is either an annulus or a torus. Suppose that Y is an annulus. If the geodesics γ x are not horizontal, then they point away from Y on one of the two boundary components, which is of course impossible. If Y is a torus, thenẎ is non-empty since Y is hyperbolic. If there is a point ofẎ which is not a pole of ψ, then all the geodesics γ x are horizontal by the previous paragraph. On the other hand, if ψ has a pole inẎ then it has at least one zero in Y and once again all geodesics are horizontal.
We deduce that F is the desired isotopy from f 0 to f 1 . Proof. For every x ∈ X, the geodesic γ x is free of zeros and nearby geodesics are parallel of the same length, which shows that f t (x) = F (x, t) is a local translation. We have to show that f t is injective and that its image is contained in Y .
Since f t is a local isometry, the area of its image is at most the area of X, with equality only if f is injective. Thus to show that f is injective it suffices to show that it is almost surjective. Recall that there is a full measure subset U ⊂ X such that through every point of U passes a bi-infinite horizontal trajectory of ϕ. For every bi-infinite geodesic α ⊂ U , we have f t (α) = f 0 (α) since f 0 (α) is horizontal for ψ and F moves points horizontally. It follows that f t (U ) = f 0 (U ). But f 0 (U ) has full measure in Y since U has full measure in X and f 0 (X) has full measure in Y . Therefore f t (X) has full measure in Y and hence f t is injective. Since f t is a local translation and f t (X) is dense in Y , it follows that f t is a slit mapping with respect to ψ by Lemma 3.6.
It remains to show that f t (X) is disjoint fromẎ . Suppose that f t (x) ∈Ẏ for some x ∈ X. If f t (x) is not a pole of ψ in Y , then γ x (t) is a zero of ψ in U. By Theorem 4.9, this implies that x ∈ ∂X, contradiction. Thus f t (x) is a simple pole of ψ. Since f t is a local translation, ϕ has a simple pole at x. This is also impossible since Thus the open set f t (X) is contained in Y ∪ ∂Y , and hence in Y .
This completes the proof of Theorem 4.1. We record the following consequence for later reference.
Lemma 4.13. Suppose that CEmb(X, Y, h) contains a slit mapping. Then for every x ∈ X, the evaluation map which sends f ∈ CEmb(X, Y, h) to f (x) ∈ Y is a homeomorphism onto its image V (x) which is either a closed regular horizontal trajectory, a compact horizontal arc, or a point.
Proof. The evaluation map is continuous by definition of the compact-open topology. We now show that it is injective. Suppose that f 0 (x) = f 1 (x) for distinct f 0 , f 1 ∈ CEmb(X, Y, h) and some x ∈ X. Let F : X × [0, 1] → Y be the isotopy from f 0 to f 1 constructed in the proof of Theorem 4.1. Then t → α(t) = F (x, t) is a closed regular horizontal trajectory for ψ. Let A be an annular neighborhood of α foliated by closed trajectories which is symmetric about α. Since f 0 (x) is a fixed point of the local translation f 1 •f −1 0 , we have that f 1 • f −1 0 is either equal to the identity near f 0 (x) or a half-turn around f 0 (x). Suppose it is locally a half-turn. Let σ be the conformal involution of A which fixes f 0 (x) and permutes the two boundary components. By the identity principle f 1 • f −1 0 = σ on the connected component C of A ∩ f 0 (X) containing f 0 . Let β be any simple close curve contained in C and winding once around A and let β = f −1 0 (β ). Then f 1 (β) and f 0 (β) are not homotopic in Y , for they are inverses of one another. This is a contradiction. Otherwise, 0 is equal to the identity near f 0 (x) and hence on all of f 0 (X) by the identity principle. This contradicts the hypothesis that f 0 = f 1 and we conclude that the evaluation map is injective. Any injective continuous map from a compact space to a Hausdorff space is a homeomorphism onto its image.
Since the space CEmb(X, Y, h) is compact, its image V (x) by the evaluation map is compact. Moreover V (x) is path-connected and contained in a regular horizontal trajectory by Theorem 4.1. Thus V (x) is either a closed trajectory, a segment, or a point.

The modulus of extension
Let h : X → Y be a non-trivial and non-parabolic embedding.
Recall that the space CEmb(X, Y, h) is the set of all conformal embeddings homotopic to h equipped with the compact-open topology. This space is compact by Lemma 3.2. If X has empty ideal boundary, then CEmb(X, Y, h) is either empty or a singleton.
Lemma 5.1. If ∂X is empty, then CEmb(X, Y, h) has at most one element.
Proof. Every f ∈ CEmb(X, Y, h) extends to a conformal isomorphism f between the closed surfaces X and Y by Riemann's removable singularity theorem. If f and g are homotopic, then so are f and g. The composition g −1 • f is thus a conformal automorphism of X homotopic to the identity. If X has genus at least 2, then it is hyperbolic and f = g by Lemma 3.8. If X has genus 0, thenẊ must contain at least 3 points for X to be hyperbolic. Any conformal automorphism of the Riemann sphere fixing 3 points is the identity. If X has genus 1, thenẊ has at least 1 point. Any conformal automorphism of a torus homotopic to the identity is a translation, and a translation with a fixed point is the identity.
In light of this result, we assume that ∂X and CEmb(X, Y, h) are nonempty from now on. For each component C of ∂X, choose an analytic parametrization ζ C : S 1 → C. For every r ∈ (0, ∞] and every component C of ∂X, glue a copy of the cylinder S 1 × [0, r) to X ∪ ∂X along S 1 × {0} with the map ζ C (see Figure 6). We denote the resulting surface by X r . We also let X 0 = X. If ρ ≤ r, then the inclusion [0, ρ) ⊂ [0, r) induces a conformal inclusion X ρ ⊂ X r . Note that for every r ∈ [0, ∞], there is a homeomorphism H : X r → X which when followed by the inclusion X ⊂ X r is homotopic to the identity. We will abuse notation and write CEmb(X r , Y, h) instead of CEmb(X r , Y, h • H). X X r Figure 6. The surface X r is obtained by gluing a cylinder of modulus r to each ideal boundary component of X.
Given f ∈ CEmb(X, Y, h), we define the modulus of extension of f as It is easy to see that the supremum is realized and that the extension is unique.
Proof. Let r = m(f ), let r n r, and let f n : X rn → Y be a conformal embedding extending f . Every x ∈ X r is contained in X rn when n is large enough since X rn X r . Define f (x) = f n (x). This does not depend on n since f j = f j+1 on X r j by the identity principle. The function f is holomorphic and injective on X r because each f n is holomorphic and injective. The uniqueness of f follows from the identity principle.
Similarly, m is upper semi-continuous on CEmb(X, Y, h). Proof. Suppose that f n → f in CEmb(X, Y, h). Pass to a subsequence so that r n = m(f n ) converges to some r ∈ [0, ∞]. We have to show that m(f ) ≥ r. If r = 0, then there is nothing to prove, so we may assume that r > 0. Let ρ < r and let f n be the maximal injective holomorphic extension of f n . If n is large enough, then ρ ≤ r n , and we let g n be the restriction of f n to X ρ . Since g n belongs to the compact space CEmb(X ρ , Y, h), we may pass to a subsequence such that g n → g for some g ∈ CEmb(X ρ , Y, h). Now the restriction of g n to X is equal to f n , which by hypothesis converges to f . Thus the restriction of g to X is equal to f . In other words, g is a conformal embedding extending f , so that m(f ) ≥ ρ. Since ρ < r was arbitrary we have m(f ) ≥ r.
Since m is upper semi-continuous and CEmb(X, Y, h) is compact, m achieves its maximum at some f ∈ CEmb(X, Y, h). We will see that any such maximum is the restriction of a slit mapping. We first need to show that any limit of Teichmüller embeddings is itself a Teichmüller embedding.
Definition 5.4. Let X n ∈ T # (X) and Y n ∈ T # (Y ) be such that X n → X and Y n → Y as n → ∞, and let σ n : X n → X and τ n : Y n → Y be the Teichmüller maps that respect the markings. Let h n : X n → Y n and h : X → Y be any maps. We say that h n → h if τ n • h n • σ −1 n → h locally uniformly on X.
Lemma 5.5. Let h : X → Y be an embedding between finite Riemann surfaces. Suppose that X n → X in T # (X) and Y n → Y in T # (Y ), and let f n : X n → Y n be a sequence of Teichmüller embeddings homotopic to h. Then there exists a Teichmüller embedding f : X → Y homotopic to h such that f n → f after passing to a subsequence.
Proof. After passing to a subsequence, we may assume that Dil(f n ) → K for some K ≥ 1. By Lemma 3.2, we may also assume that f n → f for some K-quasiconformal embedding f : X → Y homotopic to h. We claim that f has minimal dilatation in its homotopy class. If not, let g : X → Y be a quasiconformal embedding homotopic to f with Dil(g) < Dil(f ). Then let σ n : X n → X and τ n : Y n → Y be the Teichmüller maps that respect the markings. By hypothesis we have Dil(σ n ) → 1 and Dil(τ n ) → 1. Thus if ε > 0 is small enough and n in large enough, then if n is large enough, which contradicts Theorem 3.7. Thus f has minimal dilatation in its homotopy class. If f is not conformal, then it is a Teichmüller embedding by Theorem 3.5, and we are done.
Suppose that f is conformal. Let ψ n be the terminal quadratic differential of f n , and let g n : Y n → Y n be the Teichmüller map of dilatation e 2 and initial quadratic differential ψ n . Since d(Y n , Y n ) = 1 for every n and Y n → Y , and since closed balls in T # (Y ) are compact, we may pass to a subsequence such that Y n → Y for some Y ∈ T # (Y ) with d(Y, Y ) = 1. Let g : Y → Y be the Teichmüller map that respects the marking. By a standard argument similar to the one above, g n → g. Indeed, every subsequence of {g n } has a converging subsequence. Any limit of any subsequence is a quasiconformal homeomorphism with minimal dilatation in its homotopy class, hence equal to g.
By construction, g n •f n : X n → Y n is a Teichmüller embedding. Moreover, g n • f n → g • f . As in the first paragraph of this proof, g • f has minimal dilatation in its homotopy class. This dilatation is equal to e 2 > 1, so that g • f is a Teichmüller embedding. Since f is conformal, the terminal quadratic differential of g • f is equal to the terminal quadratic differential of g. It follows that f = g −1 • (g • f ) is a slit mapping with respect to the initial quadratic differential of g.
We obtain the following characterization of the global maxima of m. Proof. Let R = m(f ). For every r > R, there is no conformal embedding g : X r → Y whose restriction to X is homotopic to h, for otherwise we would have m(g| X ) ≥ r > R = m(f ). By Ioffe's theorem, there exists a Teichmüller embedding g r : X r → Y whose restriction to X is homotopic to h. It is easy to see that X r moves continuously in T # (X) as a function of r ∈ [0, ∞). By Lemma 5.5, we can extract a limiting Teichmüller embedding g : X R → Y from some subsequence of g r as r → R. Since f is conformal and homotopic to g, Ioffe's theorem implies that f is itself a Teichmüller embedding. A conformal Teichmüller embedding is a slit mapping.
Observe that every end of the surface X ∞ is a puncture since the cylinder S 1 × [0, ∞) is conformally isomorphic to D \ 0. Thus if m(f ) = ∞, then the extension f extends further to a conformal isomorphism between X ∞ and Y . In particular, Y \ f (X ∞ ) is finite, so that f is a slit mapping with respect to any ψ ∈ Q + (Y ). This is unless Y is the triply punctured sphere in which case Q + (Y ) is empty. Thus the hypothesis m(f ) < ∞ in the above theorem is superfluous unless Y is the triply punctured sphere. In the latter case f is nevertheless unique in its homotopy class by Lemma 5.1.
We can in fact strengthen Theorem 5.6 by replacing the word "global" with "local". The proof requires another lemma similar to Lemma 5.5.
where the inclusion Y n ⊂ Y is homotopic to a homeomorphism, and let f n : X → Y n be a sequence of Teichmüller embeddings. Then there exists a Teichmüller embedding f : X → Y such that f n → f after passing to a subsequence.
Proof. Let K n be the dilatation of f n . The sequence K n is non-increasing and thus converges to a limit K ≥ 1. After passing to a subsequence, f n converges to a K-quasiconformal embedding f : X → Y .
We claim that f has minimal dilatation in its homotopy class. If not, let g : X → Y be a quasiconformal embedding homotopic to f with Dil(g) < Dil(f ). If r > 0 is small enough, then there is a quasiconformal homeomorphism σ : X r → X of dilatation strictly smaller than Dil(f )/ Dil(g). Thus the quasiconformal embedding f consisting of the inclusion X → X r followed by g • σ has dilatation strictly less than Dil(f ). Then f (X) ⊂ Y n whenever n is large enough. Indeed, the ends of f (X) of finite modulus are compactly contained in Y since they are surrounded by the union of collars g • σ(X r \ X). Thus the only way to go to infinity in the closure f (X) relative to Y is via punctures of f (X) that are also punctures of Y . For every such puncture p and every n ∈ N, a neighborhood of p in Y is contained in Y n . This is because f n is quasiconformal and as such it cannot map punctures of X to ends of Y n with finite modulus. Since Y n Y , the inclusion f (X) ⊂ Y n holds for n large enough. But the inequality Dil(f ) < Dil(f ) ≤ Dil(f n ) contradicts the hypothesis that f n is a Teichmüller embedding and hence of minimal dilatation in its homotopy class.
If f is not conformal, then it is a Teichmüller embedding by Theorem 3.5, and we are done. Thus suppose that f is conformal but is not a slit mapping. By Theorem 5.6, there exists an r > 0 and a conformal embedding g : X r → Y whose restriction to X is homotopic to f . By the above argument, we have g(X r/2 ) ⊂ Y n if n is large enough, and thus g(X) ⊂ Y n with complement having non-empty interior. On the other hand, the restriction g| X : X → Y n is conformal and homotopic to the Teichmüller embedding f n : X → Y n . By Theorem 3.7, g| X is a slit mapping so that Y n \ g(X) has empty interior, contradiction.
We come to the main result of this section, which is that every local maximum of m is the restriction of a slit mapping. Proof. Let R = m(f ). We first show that the complement Y \ f (X R ) is horizontal for some meromorphic quadratic differential on Y , and is in particular an analytic graph. Let {x 1 , x 2 , . . .} be a dense subset of X.
Proof of Claim. Suppose on the contrary that for every n ∈ N there exists an element g n of CEmb(X, Y, h) satisfying g n (x j ) = f (x j ) for every j ∈ {1, ..., n} such that m(g n ) > m(f ). As CEmb(X, Y, h) is compact, every subsequence of {g n } ∞ n=1 has a subsequence converging to some g ∈ CEmb(X, Y, h). Any limit g agrees with f on the dense set {x 1 , x 2 , ....}, and hence is equal to f . Thus g n → f with m(g n ) > m(f ). This contradicts the hypothesis that f is a local maximum of m.
Let n be as in the claim, and let P = {x 1 , x 2 , . . . , x n }. Then for every r > R, there is no conformal embedding g : X r → Y homotopic to f rel P . By Ioffe's theorem, there exists a Teichmüller embedding g r : X r \ P → Y \ f (P ) homotopic to f rel P .
Let g be any limit of any subsequence of g r as r R. Then g : X R \ P → Y \f (P ) is a Teichmüller embedding by Lemma 5.5. Since f is conformal and homotopic to g rel P , Ioffe's theorem implies that f is itself a slit mapping, considered as a map from X R \ P to Y \ f (P ). Therefore the complement Y \ f (X R ) is a finite union of horizontal arcs for some meromorphic quadratic differential on Y , possibly with simple poles on the set f (P ).
Let Γ = Y \ f (X R ), let {y 1 , y 2 , ...} be a dense subset of the graph Γ minus its vertices, and fix a Riemannian metric on Y , say the hyperbolic one.
Claim. There exists a k ∈ N such that for every r > R and every ε > 0, there is no conformal embedding g : X r → Y homotopic to h whose image is disjoint from the balls B(y 1 , ε), . . . , B ε (y k , ε).
Proof of Claim. Suppose that for every k ∈ N there exist an r k > R, an ε k > 0, and a conformal embedding g k : X r k → Y whose restriction to X is homotopic to h such that g k (X r k ) is disjoint from the balls B(y 1 , ε k ), . . . , B(y k , ε k ). We may assume that r k → R and ε k → 0. Let g be any limit of any subsequence of the sequence {g k }. Then g(X R ) is disjoint from the set {y 1 , y 2 , ...} and hence from its closure Γ, so that f −1 • g : X R → X R is a conformal embedding homotopic to the identity. If X R is not an annulus, then Lemma 3.8 implies that g = f and hence g k → f . If X R is an annulus, then we may pre-compose each g k by a rotation so that we still get g k → f . Since m(g k | X ) ≥ r k > R = m(f ), this contradicts the hypothesis that f is a local maximum of m.
Let k be as in the last claim, and let Q = {y 1 , . . . , y k }. For each ε > 0, let Y ε = Y \ (B(y 1 , ε) ∪ · · · ∪ B(y k , ε)). Let ι ε : Y ε → Y \ Q be a homeomorphism homotopic to the inclusion map, and let h ε = ι −1 ε • f . The embedding h ε : X R → Y ε followed by the inclusion Y ε → Y is homotopic to h. By the claim, for every r > R, there is no conformal embedding g : X r → Y ε homotopic to h ε . Therefore, there is a Teichmüller embedding g r ε : X r → Y ε homotopic to h ε . Letting r → R, we can extract a limiting Teichmüller embedding g ε : X R → Y ε by Lemma 5.5.
Since Y ε (Y \ Q) as ε 0, we can apply Lemma 5.7 and obtain a Teichmüller embedding g : X R → Y \ Q as a limit of a subsequence of {g ε }. Since f : X R → Y \ Q is homotopic to g, it is a slit mapping with respect to some ψ ∈ Q + (Y \ Q). Thus ψ is meromorphic on Y with at most simple poles on Q. Moreover, the graph Γ = Y \ f (X R ) is horizontal for ψ. Since every point of Q is contained in the interior of an edge of Γ, the quadratic differential ψ cannot have simple poles on Q. Indeed, there is only one horizontal ray emanating from any simple pole. Therefore, ψ is holomorphic on Y and f : X R → Y is an honest slit mapping.
In particular, every conformal embedding which is not a slit mapping can be approximated by a sequence of conformal embeddings, each extending by some amount. Corollary 5.9. Let g ∈ CEmb(X, Y, h). Then there exists a sequence g n converging to g in CEmb(X, Y, h) such that g n extends to a conformal embedding of X rn into Y for some r n > 0 unless g is a slit mapping.
Proof. If m(g) > 0, then we can take g n = g. If m(g) = 0 but g is not a local maximum of m, then there exists a sequence g n → g with m(g n ) > 0. If m(g) = 0 and g is a local maximum of m, then g is a slit mapping by the previous theorem.
A strong converse to Theorem 5.8 holds due to Ioffe's theorem.
Lemma 5.10. Suppose that g : X r → Y is a slit mapping such that g| X is homotopic to h. Then g| X is a global maximum of m.
Proof. First observe that m(g| X ) ≥ r. Suppose that m(f ) ≥ m(g| X ) for some element f of CEmb(X, Y, h) and let f be the maximal extension of f . Then f | Xr is homotopic to g. By Ioffe's theorem, f | Xr is a slit mapping. In particular, the complement of f (X r ) has empty interior in Y . Therefore X m(f ) \ X r is empty so that m(f ) ≤ r ≤ m(g| X ).
Furthermore, the almost rigidity of slit mappings implies that the set of local maxima of m is path-connected. Proof. Suppose first that there is some f ∈ M such that m(f ) < ∞. Then by Theorem 5.8, the maximal extension f is a slit mapping. By Lemma 5.10, f is a global maximum of m. In particular, m(g) < ∞ for every g ∈ M and thus every g ∈ M is a global maximum of m. In particular, m is constant on M , say equal to R. The map M → CEmb(X R , Y, h) defined by f → f is a homeomorphism with inverse g → g| X . By Lemma 4.13, the evaluation map ev x : CEmb(X R , Y, h) → Y is a homeomorphism onto its image for every x ∈ X R , and its image is either a point, a compact interval, or a circle.
Otherwise, m is constant equal to ∞ on M . In this case M is homeomorphic to CEmb(X ∞ , Y, h), which is a point by Lemma 5.1.
The fact that m has a connected plateau of local maxima easily implies that the space CEmb(X, Y, h) is connected.

The blob and its boundary
For Sections 6 to 10, we assume that the embedding h : X → Y is generic. By definition, this means that the induced homomorphism π 1 (h) : π 1 (X, x) → π 1 (Y, h(x)) has non-abelian image. Let Map(X, Y, h) be the space of all continuous maps f : X → Y homotopic to h. The following lemma shows that for every x ∈ X and every f ∈ Map(X, Y, h), there is a well-defined way to lift the image point f (x) to the universal covering space of Y . Proof. We use two standard facts about hyperbolic surfaces: • every abelian subgroup of π 1 (Y ) is cyclic; • every non-trivial element in π 1 (Y ) is the positive power of a unique primitive element. Let G be any homotopy from h to f . By composing H with G ran backwards, we get a homotopy from h to itself, hence a map F : X ×S 1 → Y . Suppose that the loop γ(t) = F (x, t) is not trivial in π 1 (Y, h(x)). Then it is equal to β k for some primitive element β and some k > 0.
Let α be any loop in X based at x. Then the map S 1 × S 1 → Y given by (s, t) → F (α(s), t) induces a homomorphism of Z 2 into π 1 (Y, h(x)). The image of this homomorphism is cyclic, and contains both [h•α] and [γ]. From the existence and uniqueness of primitive roots in π 1 (Y, h(x)), it follows that [h • α] = β j for some j ∈ Z.
Since α is arbitrary, we deduce that the image of the homomorphism π 1 (h) : π 1 (X, x) → π 1 (Y, h(x)) is contained in the cyclic group β . This contradicts the hypothesis that h is generic. We conclude that the loop γ(t) = F (x, t) is null-homotopic. Equivalently, the paths t → H(x, t) and t → G(x, t) are homotopic rel endpoints.
Fix some point x ∈ X and a universal covering map π Y : D → Y , and let b ∈ D be such that π Y (b) = h(x). Proof. Let f n , f ∈ Map(X, Y, h) be such that f n → f . Let K ⊂ X be a compact deformation retract of X containing x. Let ε > 0 be smaller than the injectivity radius of Y on f (K), and let n be large enough so that |f n − f | < ε on K. For every (ξ, t) ∈ K × [0, 1], let G (ξ, t) be the point at proportion t along the unique shortest length geodesic between f (ξ) and f n (ξ). This gives a continuous homotopy from f | K to f n |K. By composing the deformation retraction X → K with G , we get a homotopy G from f to f n moving points of K by distance at most ε.
Given any homotopy H from h to f , the concatenation H * G (this is H followed by G) is a homotopy from h to f n . Thus lift x (f n ) can be obtained by lifting the path t → G(x, t) starting at the point lift x (f ). The lift of t → G(x, t) is a geodesic of length at most ε, so | lift x (f n ) − lift x (f )| < ε.
We now get back to the space of conformal embeddings CEmb(X, Y, h) and look at where x can go under such maps.
We know that Blob(x) is at most 1-dimensional when CEmb(X, Y, h) contains a slit mapping. Moreover, the assumption that h is generic eliminates the possibility that Blob(x) be a circle. Lemma 6.5. If CEmb(X, Y, h) contains a slit mapping, then Blob(x) is homeomorphic to a point or a compact interval.
Proof. By Lemma 4.13, the evaluation map ev x : CEmb(X, Y, h) → Y is a homeomorphism onto its image V (x), and the latter is either a point, a compact interval, or a circle. Then π Y : Blob(x) → V (x) is a homeomorphism with inverse lift x • ev −1 x . Suppose that CEmb(X, Y, h) is homeomorphic to a circle. Then there are Z many homotopically distinct homotopies from any f ∈ CEmb(X, Y, h) to itself, which contradicts Lemma 6.1.
We assume henceforth that CEmb(X, Y, h) is non-empty and does not contain any slit mapping. To recapitulate the hypotheses thus far: X and Y are finite hyperbolic, ∂X is non-empty, h : X → Y is generic, and CEmb(X, Y, h) is non-empty and does not contain any slit mapping.
Since CEmb(X, Y, h) is compact and connected, and since lift x is continuous, Blob(x) is compact and connected as well. Our goal is to show that Blob(x) is homeomorphic to a closed disk. The strategy of the proof is to analyze the boundary of Blob(x). We will show that every point in ∂ Blob(x) is attained by a special kind of map in CEmb(X, Y, h) which we call a slit mapping rel x. Definition 6.6. Let f ∈ Map(X, Y, h). We say that f is a Teichmüller embedding rel x if the restriction f : X \ x → Y \ f (x) is a Teichmüller embedding. A slit mapping rel x is a Teichmüller embedding rel x which is conformal.
The distinction to make here is that the initial and terminal quadratic differentials of f are allowed to have simple poles at x and f (x) respectively. Here is a useful construction for moving a point around on a surface. Lemma 6.7. Given a smooth path γ : [0, 1] → D, there exists a quasiconformal diffeomorphism Π : Y → Y and a homotopy H from id Y to Π such that the lift of t → H(π Y (γ(0)), t) based at γ(0) is equal to γ.
Proof. One way to construct Π is to break up the path π Y • γ into finitely many simple subarcs γ j and define Π as the composition of as many diffeomorphisms Π j . Push ∂/∂x forward by γ j , extend this to a compactly supported vector field V j on Y , and define Π j to be the time t j -flow for V j where t j is the time required to travel between the two endpoints of γ j . The resulting diffeomorphism Π is the identity outside a compact set and is therefore quasiconformal. By construction, the concatenation of flows is a homotopy H from the identity to Π such that H(π Y (γ(0)), t) = π Y (γ(t)). The lift of π Y • γ based at γ(0) is of course equal to γ.
A diffeomorphism such as in the previous lemma is called a point-pushing diffeomorphism. If a point y is on the complement of Blob(x), then by definition y is not attained by any conformal embedding. Using Ioffe's theorem, we see that y is attained by a Teichmüller embedding rel x. Proof. Let γ be any smooth path from the basepoint b over h(x) to y and let Π be a point-pushing diffeomorphism along the path. By definition of Blob(x), there does not exist any g ∈ CEmb(X, Y, h) such that lift x (g) = y. In other words, the set is empty. On the other hand, for any F ∈ CEmb(X, Y, h) the map Π • F is quasiconformal. By Ioffe's theorem there exists a Teichmüller embedding f : X\x → Y \π Y (y) homotopic to Π•h. Then f extends to a quasiconformal embedding f : X → Y homotopic to Π • h and hence to h. Thus f is a Teichmüller embedding rel x. Moreover, If g is another Teichmüller embedding rel x homotopic to h such that lift x (g) = y, then g is homotopic to f rel g. By Theorem 3.7, f and g have the same dilatation and g • f −1 is a slit mapping. Since g • f −1 fixes the puncture π Y (y), it is the identity by Corollary 4.11.
We now show that boundary points are attained by slit mappings rel x. Lemma 6.9. Let y be on the boundary of Blob(x). There is a unique f ∈ CEmb(X, Y, h) such that lift x (f ) = y. The map f is a slit mapping rel x. If ϕ and ψ are initial and terminal quadratic differentials for f rel x, then ϕ has a simple pole at x and ψ has a simple pole at f (x).
Proof. Since Blob(x) is closed, y belongs to Blob(x), and hence there exists an element f of CEmb(X, Y, h) such that lift x (f ) = y. Let y n ∈ D \ Blob(x) be such that y n → y as n → ∞. Let γ n be the geodesic from y to y n and let Π n : Y → Y be the corresponding point-pushing diffeomorphism. We can construct Π n in such a way that it converges to the identity in the C 1 norm, so that its dilatation converges to 1 as n → ∞.
By the previous lemma, there exists a Teichmüller embedding f n rel x homotopic to h such that lift x (f n ) = y n . Moreover, the sequence of marked surfaces [Y \ π Y (y n ), Π n ] converges to [Y \ π Y (y), id] in T # (Y \ π Y (y)). Therefore, there exists a Teichmüller embedding g : X \ x → Y \ π(y) homotopic to f by Lemma 5.5. Since f is conformal, its dilatation is less than or equal to the dilatation of g. Hence f is a slit mapping rel x by Ioffe's theorem. If there is some g ∈ CEmb(X, Y, h) be such that lift x (g) = y, then g is homotopic to f rel x. Thus g is a slit mapping rel x by Ioffe's theorem and g • f −1 is the identity since it fixes a puncture.
Suppose that the terminal quadratic differential ψ does not have a pole at y. Then f is a slit mapping from X to Y (not rel x). But we assumed that CEmb(X, Y, h) does not contain any slit mapping. Therefore ψ has a simple pole at f (x) and ϕ = f * ψ has a simple pole at x. This motivates the following definition. Definition 6.10. If y, f , ϕ, and ψ are as in the previous lemma, then we say that f , ϕ, and ψ realize y.
We point out that although f is unique, ϕ and ψ need not be unique. Nevertheless, the set of quadratic differentials realizing a given point y in the boundary of Blob(x) is convex. Lemma 6.11. Suppose that ψ 0 , ψ 1 ∈ Q + (Y \ π Y (y)) realize y ∈ ∂ Blob(x). Then for every α, β > 0 the quadratic differential αψ 0 + βψ 1 belongs to Q + (Y \ π Y (y)), realizes y, and has a simple pole at π Y (y).
Proof. αψ 0 + βψ 1 ≥ 0 along any arc in Y \ f (X), where f is the slit mapping rel x realizing y. If αψ 0 + βψ 1 does not have a pole at π Y (y), then f is a slit mapping from X to Y , contradiction.
We will see that any ψ ∈ Q + (Y \π Y (y)) which realizes y tells us something about the shape of Blob(x) near y. In order to explain this, we first need to discuss measured foliations and extremal length.

Extremal length of partial measured foliations
There are several ways to define extremal length for measured foliations. We follow the approach developed in [16] and [15]. Throughout this section, we use the expression almost-smooth to mean continuous and continuously differentiable except perhaps at finitely many points.

Definition 7.1. A partial measured foliation on a Riemann surface X is a collection of open sets
there is a well-defined 1-form |dv| on j U j , called the transverse measure.
We extend |dv| to be zero on X \ j U j . The leaves of |dv| are the maximal connected subsets of j U j on which each v j is locally constant.
Remark. We could relax the regularity condition on the functions v j and only assume that they belong to the Sobolev space W 1,2 . For the sake of brevity and simplicity we will stick to the almost-smooth condition.
For example, if ϕ is a quadratic differential on X and U j ⊂ X is a simply connected domain on which ϕ does not have any singularities, then the function v j (z) = Im z z 0 √ ϕ is well-defined on U j up to an additive constant and sign. The leaves of the resulting partial measured foliation are the horizontal trajectories of ϕ. We write F[ϕ] for the resulting partial measured foliation |dv| = | Im √ ϕ|. where z = x + iy is any conformal coordinate.
For example, if ϕ is a holomorphic quadratic differential on X and F[ϕ] = |dv| = | Im √ ϕ| is its horizontal foliation, then as is easily seen by computing the Dirichlet energy in natural coordinates where ϕ = dz 2 and v(z) = ± Im z + c. for every homotopy class c of closed curves or cross-cuts in X.
Note that the extremal length of |dv| only depends on the measure equivalence class of |dv|. Moreover, extremal length rescales quadratically in the sense that EL(λ|dv|; X) = λ 2 EL(|dv|; X) for every λ > 0.
The fundamental result about extremal length is that the horizontal foliation of any integrable holomorphic quadratic differential minimizes Dirichlet energy in its equivalence class. This was first proved for measured foliations in [24], then generalized in [13]. See chapters 2 and 11 in [14] or chapter VII in [37] for detailed expositions. The version for partial measured foliations appears in [16] and [15], but the proof from [24] applies verbatim.
The next definition justifies our preference for partial measured foliations over measured foliations. The push-forward of a partial measured foliation by a smooth embedding is naturally a partial measured foliation, whereas this is not so clear for measured foliations.
Definition 7.8. If |dv| is a partial measured foliation on X and f : X → Y is an almost-smooth embedding, then the push-forward f * |dv| is given by the collection of open sets f (U j ) and the almost-smooth functions v j • f −1 : We now show that extremal length can increase by a factor at most K under K-quasiconformal embeddings. In particular, extremal length does not increase under conformal embeddings. Lemma 7.9. Let ϕ ∈ Q(X) and let f : X → Y be an almost-smooth Kquasiconformal embedding.
Proof. Let ζ = σ + iτ be a conformal coordinate on f (X) and let z = x + iy be a conformal coordinate at f −1 (ζ). Since f is K-quasiconformal, we have Let v(z) = Im We compute It follows that where the last equality holds by Theorem 7.7.
The inequality is sharp as the case of Teichmüller embeddings illustrates.
Lemma 7.10. Let f : X → Y be a Teichmüller embedding of dilatation K with initial and terminal quadratic differentials ϕ and ψ. Then f * F[ϕ] is measure equivalent to F[ψ] on Y and we have is a finite union horizontal arcs, the integral of F[ψ] is zero along any piece of curve contained in Y \ f (X). For any piecewise smooth curve α in Y , we thus have Therefore Lastly, we have ψ = K ϕ because f stretches horizontally by a factor K and f (X) has full measure in Y . Putting everything together, we get and hence equality holds.
We will need a sufficient condition for when the push-forwards of a partial measured foliation by two homotopic embeddings are measure equivalent. Definition 7.11. We say that an embedding f : X → Y is tame if it is almost-smooth and extends to a continuous map X → Y which is piecewise smooth along ∂X.
For example, every Teichmüller embedding is tame. Lemma 7.12. Let ϕ ∈ Q + (X) and let f, g : X → Y be homotopic tame embeddings. Then f * F[ϕ] and g * F[ϕ] are measure equivalent on Y .
Proof. By symmetry, it suffices to show that f * F[ϕ] ≥ g * F[ϕ]. Let α be a piecewise smooth closed curve or cross-cut in Y . Given ε > 0, we have to find a closed curve or cross-cut β homotopic to α such that Let n be the number of components of α ∩ f (X). Let δ = ε/4(n + 1) and let X δ be X minus a δ-neighborhood of its ideal boundary in the metric |ϕ| 1/2 .
We will take β to be a slightly modified version of H(α). We keep the part H(α) ∩ g(X δ ), which has at most n components, unchanged. Then H(α) \ g(X δ ) has at most n + 1 components, the +1 occuring if α is a crosscut. We homotope each component c of H(α) \ g(X δ ) within Y \ g(X δ ) to an arc γ j with at most three parts: the beginning γ − j and end γ + j contained in g(X \ X δ ), and perhaps a middle portion γ 0 j contained in Y \ g(X). The reason we can do this is that Y \ g(X) is a deformation retract of Y \ g(X δ ). Then we homotope each of γ ± j within g(X \ X δ ) to an arc η ± j such that .
This is possible since the height of each component of X \ X δ is at most δ = ε/4(n+1). Moreover, no matter how many times g −1 (γ ± j ) winds around the annular component of X \ X δ in which it is contained, we can push this winding part toward the boundary of X which is horizontal for ϕ. In doing so, the height of the winding part tends to zero and is thus eventually less than ε/4(n + 1), for a total height of at most ε/2(n + 1).
We then let β be the concatenation of H(α) ∩ g(X δ ) with the arcs η − j ∪ γ 0 j ∪ η + j . By construction β is homotopic to α and we compute which is what we wanted.

The blob is semi-smooth
In this section we show that the boundary of Blob(x) satisfies a regularity condition which we call semi-smoothness. We need the following formula due to Gardiner for the derivative of extremal length [13].
Theorem 8.1 (Gardiner's formula). Let Z 0 be a finite Riemann surface and let q ∈ Q(Z 0 ). If µ is a smooth Beltrami form on Z 0 with µ ∞ < 1 and f µ : Z 0 → Z µ is the solution to the Beltrami equation, then Remark. In his proof, Gardiner shows that for a universal constant c 1 . We will use this inequality instead.
We apply this formula in the special case where Z µ = Y \ γ(t) for some analytic path γ : I → Y . The pairing µ, q is then approximately proportional to the real part of the residue of q in the direction of γ (0). See [27] for a similar but more general calculation.
Lemma 8.2. Let γ be an analytic arc on Y and let Z t = Y \ γ(t). Then Z t = Z µ(t) for a smooth Beltrami form µ(t) such that µ(t) ∞ = O(t) and for every q ∈ Q(Z 0 ) we have Proof. Extend γ (t) to a holomorphic vector field v on a disk D centered at y = γ(0) and let φ be a smooth bump function which vanishes outside D and is equal to 1 on a disk D around y compactly contained in D.
Then φv is a smooth vector field defined on all of Y . Let Φ t be the time-t flow for φv and let µ(t) = ∂Φ t /∂Φ t . We have µ (0) = ∂(φv) and hence µ(t) = t∂(φv) + O(t 2 ). We compute The equality from the first line to the second is by Green's theorem and the change of sign in the next equality comes from reversing orientation on the circle ∂D . To conclude the proof, recall that ν, q = Re Z 0 νq by definition.
Remark. The constants in the big O notation are not universal, but depend continuously on the size of the tangent vector γ (0) and the injectivity radius of Y at γ(0). We set up some notation before stating the next result. Given a point y ∈ D and a tangent vector v ∈ T y D, let y = π Y ( y) and let v = dπ Y ( v). For small t ∈ R, we let y( v, t) = y + t v and y(v, t) = π Y ( y( v, t)). Finally, we let Π(v, t) : Y → Y be a diffeomorphism pushing y to y( v, t) in the universal cover. For any ψ ∈ Q(Y \ y), we compute the variation of the extremal length of F[ψ] along the path t → Y \ y(v, t). Corollary 8.3. Let y ∈ D, let v ∈ T y D, and let ψ ∈ Q(Y \ y) have area 1. Then Proof. The inequality follows immediately from Lemma 8.2 and the remark following Theorem 8.1.
If ψ ∈ Q(Y \ y) has a simple pole at y, then there exists a tangent vector v ∈ T y Y (unique up to rescaling) such that Res y (ψv) < 0. We say that v is vertical for ψ. For example, if ψ = 1 z dz 2 and v = − ∂ ∂z , then Res 0 (ψv) = − 1 2πi For every e iθ ∈ ∂D we have Res y (ψe iθ v) = e iθ Res y (ψv). If v is vertical for ψ, then Re[Res y (ψe iθ v)] = cos(θ) Res y (ψv) is negative precisely when θ ∈ (− π 2 , π 2 ). Thus we may improve the previous corollary as follows. Lemma 8.4. Let θ 0 ∈ (0, π 2 ) and let K be a compact set of pairs ( y, ψ) where y ∈ D and ψ ∈ Q(Y \ y) has area 1 and a simple pole at y. There exists δ > 0 such that for every ( y, ψ) ∈ K we have for every t ∈ (0, δ) and every θ ∈ [−θ 0 , θ 0 ] where v is the vertical vector for π * Y ψ at y rescaled to have norm 1 with respect to the hyperbolic metric. Proof. By Corollary 8.3, we have where c( y) is a positive constant depending continuously on y. The right hand side is positive provided that This upper bound is bounded below by some δ > 0 since it is positive and depends continuously on θ, y, and ψ, which vary inside a compact set.
To see how the above inequality might be useful in studying Blob(x), recall that conformal embeddings do not increase extremal length. Lemma 8.5. Let f : X → Y be a tame embedding homotopic to h, let y = f (x), and let ϕ ∈ Q + (X \ x). Suppose that Π : Y \ y → Y \ y is a diffeomorphism such that Then there is no conformal embedding homotopic to Π • f : Proof. Suppose there is a conformal embedding g homotopic to Π • f rel x. By Corollary 5.9 we may assume that g is tame. Then g * F[ϕ] is measure equivalent to (Π • f ) * F[ϕ] by Lemma 7.12. Since g is conformal we have by Lemma 7.9, which is a contradiction. Hence lift x (Π • f ) does not belong to Blob(x), as there is no conformal embedding homotopic to Π•f rel x.
Using the above results, we will show that if v is vertical for π * Y ψ at y ∈ ∂ Blob(x) where ψ realizes y, then v is normal to Blob(x) in the sense that it points orthogonally away from it. Definition 8.6. Given v ∈ T z C, θ ∈ (0, π), and δ > 0, we denote by (v, θ, δ) the open angular sector based at z with radius δ and angle θ on either side of v. In symbols, The closure of (v, θ, δ) is denoted [v, θ, δ]. . Suppose that f realizes y with respect to ψ ∈ Q + (Y \ y) and that v ∈ T y D is vertical for π * Y ψ. Then v is normal to Blob(x).
As a corollary, we obtain the converse of Lemma 6.9.
Proof. Since f is conformal, lift x (f ) ∈ Blob(x). Let ψ be a terminal quadratic differential for f rel x. If ψ does not have a simple pole at y = f (x) then f is a slit mapping, contradicting the assumption that CEmb(X, Y, h) does not contain any. Thus ψ has a simple pole at y so that there exists a tangent vector v ∈ T y Y such that Res(ψv) < 0. Let v be the lift of v based at lift x (f ). We can apply the same reasoning as in the proof of Theorem 8.8 to conclude that lift x (f ) + t v ∈ Blob(x) for all t > 0 small enough, and hence that lift x (f ) ∈ ∂ Blob(x).
which means that f is locally of Teichmüller form with respect to ϕ and ψ. Observe that ψ = K ϕ since ψ n = K n ϕ n for every n. Therefore f (X) has full measure in Y , which by Lemma 3.6 implies that Y \ f (X) is a finite union of points and horizontal arcs for ψ.
The following lemma shows that the set of vectors vertical for some realizing quadratic differential at a given point is convex.
We are now able to show that normal vectors are vertical.
Theorem 8.13. Let v be normal to Blob(x) at y. Then there exists a quadratic differential ψ ∈ Q + (Y \ π Y (y)) realizing y such that that v is vertical for π * Y ψ. Proof. Let V y denote the set of vectors which are vertical for some quadratic differential realizing y. By Lemmas 6.11 and 8.12, V y is convex. Moreover V y ∪ {0 y } is closed. Suppose that v is not in V y . Then there is an open half-plane H through y containing V y such that v is not in the closure H. Let y n be a sequence converging to y along the ray r which is normal to H at y. Since r makes an angle strictly less than π 2 with v and since v is normal to Blob(x), we may assume that y n is not in Blob(x) for any n. Let f n be the Teichmüller embedding rel x realizing y n provided by Lemma 6.8 and let ψ n be its terminal quadratic differential, normalized to have area 1.
We may assume that ψ n converges to some ψ ∈ Q + (Y \ π Y (y n )) and that f n converges to a slit mapping f rel x with terminal quadratic differential ψ. If ψ is holomorphic at π Y (y), then f is a slit mapping on X, contrary to the assumption that CEmb(X, Y, h) does not contain any slit mapping. Therefore ψ has a simple pole at π Y (y), which implies that ψ n has a simple pole at π Y (y n ) for all but finitely many indices. Let w n be the vertical direction for π * Y ψ n at y n . By rescaling, we may assume that w n converges to a non-zero tangent vector w vertical for π * Y ψ at y. We have w ∈ V y ⊂ H. We will see that this yields a contradiction.
Let φ ∈ (0, π 2 ] be the angle that w makes with the line ∂H and let θ = π 2 − φ 2 . By Lemma 8.4 there exists a δ > 0 such that for every n and every w ∈ (w n , θ, δ) we have On the other hand, since w n → w, the angle between w n and y − y n is eventually less than θ. Thus if n is large enough then y ∈ (w n , θ, δ) and hence EL(Π(y n → y) * F[ψ n ]; Y \ π Y (y)) > EL(F[ψ n ]; Y \ π Y (y n )). Let ϕ n be the initial quadratic differential of f n corresponding to ψ n and let K n be the dilatation of f n . By Lemma 7.10, F[ψ n ] is measure equivalent to (f n ) * F[ϕ n ] on Y \ π Y (y n ). If we let Π n = Π(y n → y), then By Lemma 8.5, it follows that y = lift(Π n • f n ) does not belong to Blob(x), which is a contradiction.
We finally come to the main result of this section, which is that Blob(x) is semi-smooth.
• for every z ∈ ∂B, the set of normal vectors to B at z is non-empty and convex; • any non-zero limit of normal vectors is normal.
Proof. For every y ∈ ∂ Blob(x) the set of normal vectors to Blob(x) at y coincides with the set V y of vertical vectors for quadratic differentials realizing y. The set V y is convex by Lemmas 6.11 and 8.12. Suppose that y n ∈ ∂ Blob(x), that y n → y, that v n is vertical for ψ n , and that v n → v = 0. Then we can rescale ψ n and pass to a subsequence such that it converges to some ψ realizing y. We have Res y (ψv) ≤ 0 since Res yn (ψ n v n ) < 0 for every n. Moreover, ψ must have a simple pole at y for otherwise CEmb(X, Y, h) would contain a slit mapping. This means that Res y (ψv) = 0. Therefore v is vertical for ψ hence normal to Blob(x) at y.

The blob is a disk
In this section, we show that the blob is homeomorphic to a closed disk. We first prove that every semi-smooth set is a manifold.
Theorem 9.1 (The aquatic theorem). Every closed semi-smooth subset of C is a 2-dimensional manifold with boundary. Proof. Let B be a closed semi-smooth set. Every interior point of B has a neighborhood homeorphic to an open subset of C, namely the interior of B. Thus we only have to show that every boundary point z ∈ B has a neighborhood homeomorphic to a half disk. By applying an isometry of the plane, we may assume that z = 0 and that the vector i bissects the cone N 0 of vectors normal to B at 0. Let φ be half the angle of N 0 , let α = φ + π 2 and let β = π − α. Since B is semi-smooth, we have φ < π 2 and hence β > 0. Moreover, for every θ ∈ (0, α) there exists a δ > 0 such that the open sector (i, θ, δ) is disjoint from B. We now show the existence of closed sectors pointing downwards contained in B.
Proof of Claim. Suppose not. Then there exists a θ ∈ (0, β) and a sequence δ n 0 for which the corresponding angular sector S n = [−i, θ, δ n ] intersects the complement of B for every n. Let D n be a closed disk in S n disjoint from B. Slide the center of D n in a straight line towards 0 until the disk first hits B, and let D * n be the resulting disk. The intersection points of D * n with B all lie on the half of ∂D * n which is closest to 0. Let z n be any point in this intersection. Then z n is on the boundary of B and the unit vector v n pointing from z n to the center of D * n is normal to B. Since S n shrinks to 0, we have z n → z. Each vector v n makes an angle at most θ + π 2 with the downward direction. Therefore, the vectors v n can only accumulate onto vectors forming an angle at least β − θ with the cone N 0 . This contradicts the hypothesis that every limit of normal vectors is normal. Let θ + ∈ (0, α), let S + = (i, θ + , δ + ) be disjoint from B, let θ − ∈ (0, β), and let S − = [−i, θ − , δ − ] be contained in B. Let I ⊂ S + be a compact horizontal segment symmetric about the vertical line through 0 and lying entirely above S − . We define a map p : I → ∂B as follows. For z ∈ I, let z fall straight down until it first hits B, and let p(z) be this first hitting point. Note that p(x + iy) = x + iq(x, y) for some function q so that p is injective. Proof of Claim. It is easy to see that p is continuous at the midpoint p −1 (0). This is because p keeps the x-coordinate unchanged and moreover, p(z) is below S + and above S − . Thus the y-coordinate of p(z) converges to 0 as z → p −1 (0). Figure 10. The vertical projection p is akin to rain falling on the surface of B. The projection is continuous (in fact Lipschitz) by the squeeze theorem.
Let 0 < ε < β/2. By semi-smoothness, there exists a δ > 0 such that if w ∈ ∂B is within distance δ of 0, then every vector in N w is within angle φ + ε of the upward direction. For every w ∈ ∂B with |w| < δ, let v w be the bisector of N w and let φ w be half the angle of N w . For every θ + w ∈ (0, φ w + π 2 ) there is an open sector S + w = (v w , θ + w , δ + w ) disjoint from B by definition of we may choose θ + w so that S + w contains the vertical direction in its span. By Claim 9.2, there is also a closed sector The angle that −v w makes with the downward direction is equal to the angle that v w makes with the vertical direction, which is at most φ + ε − φ w hence strictly less than π 2 − φ w . Thus we may choose θ − w so that S − w contains the downward direction in its interior. By continuity of p at p −1 (0), there is a closed interval J ⊂ I centered at p −1 (0) such that p(J) is contained in the ball of radius δ about 0. Let z ∈ J, let w = p(z), and let S + w and S − w be angular sectors as described in the previous paragraph. Also let K ⊂ S + w be a compact horizontal segment crossing the vertical line through w and lying entirely above S − w . By construction, the vertical segment from z to w intersects B only at w. Since B is closed, we may assume that the rectangle with bottom edge K and upper edge L ⊂ I is disjoint from B, by making K shorter if necessary. For every ζ ∈ L, the image p(ζ) is thus squeezed between S + w and S − w , so that p is continuous at z. Proof of Claim. If δ is small enough, then the component C + of U δ \ p(J) above p(J) lies below J itself. By definition of p, for every z ∈ J the open vertical segment between z and p(z) is disjoint from B, so that C + is disjoint from B.
For the component lying below p(J), we use the same idea as in the proof of Claim 9.2. Suppose that the result does not hold. Then there is a sequence δ n 0 such that for every n, there is a closed disk D n contained in the component of U δn \ p(J) below p(J). Slide the center of D n upwards until the disk first hits B, and let D * n this hitting disk. Every intersection point of D * n with B is on the upper half of ∂D * n . Let z n be any point in that intersection. Then z n is on the boundary of B and the unit vector v n pointing from z n towards the center of D * n is normal to B. As n → ∞, we have z n → 0. Moreover, the vectors v n only accumulate onto vectors forming an angle at least π 2 with the upwards direction at 0, hence outside N 0 . This contradicts the semi-smoothness of B at 0.
By the claim, if δ is small enough then U δ ∩ B is equal to the union of U δ ∩ p(J) with the component of U δ \ p(J) below p(J). This neighborhood of z in B is clearly homeomorphic to the rectangle (−δ, δ)×(0, δ], which in turn is homeomorphic to a half disk. Thus B is a 2-manifold with boundary. Since Blob(x) is semi-smooth, it is a 2-manifold with boundary.
Corollary 9.5. Suppose that CEmb(X, Y, h) does not contain any slit mapping. Then Blob(x) is a compact connected 2-manifold with boundary.
In particular, the blob is the closure of its interior. We use this to show that the blob depends continuously on parameters. We first need to define a topology on closed subsets of a space.
Let S be a topological space and let CL(S) be the hyperspace of closed subsets of S. The Fell topology on CL(S) is the topology generated by neighborhoods of the form N (K, U), where K ⊂ S is compact, U is a finite collection of open subsets of S, and N (K, U) is the set of A ∈ CL(S) such that A ∩ K = ∅ and A ∩ U = ∅ for every U ∈ U.
Theorem 9.6 (Fell). For any topological space S, the hyperspace CL(S) is compact. If S is locally compact, then CL(S) is Hausdorff.
See [12] for a proof. If S is first-countable and Hausdorff, then a sequence {A n } ⊂ CL(S) converges to A ∈ CL(S) if and only if • for every a ∈ A, there exist a n ∈ A n such that a n → a; • for every sequence {a n } with a n ∈ A n , if {a n } accumulates onto a ∈ S, then a ∈ A. We use the Fell topology on closed subsets of D. To prove convergence, we mostly rely on Fell's compactness theorem and the above criterion for sequences.
It is fairly clear that the blob depends upper semi-continuously on parameters. The same holds for its boundary.
Lemma 9.7. Blob(x, X) and ∂ Blob(x, X) depend upper semi-continuously on the pair (X, x). More precisely, suppose that X n \x n → X\x in T # (X\x), that Blob(x n , X n ) → A in CL(D), and that ∂ Blob(x n , X n ) → B in CL(D). Then A ⊂ Blob(x, X) and B ⊂ ∂ Blob(x, X).
Proof. Let y ∈ A. By hypothesis there exist y n ∈ Blob(x n , X n ) such that y n → y. Let f n ∈ CEmb(X n , Y, h) be such that lift xn (f n ) = y n . Let σ n : X \ x → X n \ x n be the Teichmüller map in the right homotopy class and let K n be its dilatation (which converges to 1 by hypothesis). For any K > 1, if n is large enough then f n • σ n : X → Y is a K-quasiconformal embedding homotopic to h. The space of all K-quasiconformal embeddings from X to Y homotopic to h is compact by Lemma 3.2. After passing to a subsequence, we may thus assume that f n • σ n converges to some Kquasiconformal embedding f : X \x → Y in the same homotopy class. Since f is K-quasiconformal for every K > 1, it is conformal. Then lift xn (f n ) = lim n→∞ y n = y, so that y ∈ Blob(x, X). Now let y ∈ B and let y n ∈ ∂ Blob(x n , X n ) be such that y n → y. By Lemma 6.9, there exists a slit mapping f n rel x n from X n to Y homotopic to h such that lift xn (f n ) = y n . By Lemma 5.5, we can pass to a subsequence such that f n converges to some slit mapping f rel x from X to Y .
We do not know if the blob moves continuously in general, but it does when there are no slit mappings at the limiting parameters.
Lemma 9.8. Suppose that CEmb(X, Y, h) does not contain any slit mapping. If X n \ x n → X \ x in T # (X \ x), then Blob(x n , X n ) → Blob(x, X) and ∂ Blob(x n , X n ) → ∂ Blob(x, X) in CL(D).
Proof. By compactness of CL(D), it suffices to prove that if Blob(x n , X n ) converges to A and ∂ Blob(x n , X n ) converges to B, then A = Blob(x, X) and B = ∂ Blob(x, X).
Let us prove convergence of the blobs first. By Lemma 9.7, we have A ⊂ Blob(x, X). We claim that the interior of Blob(x, X) is contained in A. Let y be in the interior of Blob(x, X) and suppose that there is an infinite set J ⊂ N such that y is not contained in Blob(x n , X n ) for every n ∈ J. Then for every n ∈ J, there exists a Teichmüller embedding f n rel x n with lift xn (f n ) = y. After passing to a subsequence in J, we get that f n → f for some Teichmüller embedding f rel x by Lemma 5.5. We have lift x (f ) = y by continuity of lift x . By Corollary 8.9, y is in the complement of the interior of Blob(x, X). This is a contradiction, which means that y is contained in Blob(x n , X n ) for all but finitely many indices, and hence y ∈ A. Since A is closed and Blob(x, X) is the closure of its interior, we have Blob(X, x) ⊂ A and hence A = Blob(x, X).
By Lemma 9.7, we have B ⊂ ∂ Blob(x, X). Let y ∈ ∂ Blob(x, X). Let U be any connected neighborhood of y. We claim that if n is large enough, then U intersects both the complement of Blob(x n , X n ) and the interior of Blob(x n , X n ). Suppose on the contrary that U is contained in Blob(x n , X n ) for every n in an infinite set J ⊂ N. Then U ⊂ A = Blob(x, X), which is nonsense since y is on the boundary of Blob(x, X). Similarly, suppose that U is contained in the complement of Blob(x n , X n ) for every n in an infinite set J ⊂ N. Then for every z ∈ U and every n ∈ J there is a Teichmüller embedding f n : X n → Y rel x n homotopic to h such that lift xn (f ) = z. By Lemma 5.5, f n converges to a Teichmüller embedding f rel x after passing to a subsequence. Then lift x (f ) = z so that z ∈ ∂ Blob(x) by Corollary 8.9. This is a contradiction, which proves the claim. Let n be large enough so that U intersects both the interior and the complement of Blob(x n , X n ). Since U is connected, it also intersects ∂ Blob(x n , X n ). Since U can be chosen arbitrarily small, this shows that y ∈ B.
Similarly, nested families of blobs move continuously. In what follows, the surface X r is obtained from X by gluing a cylinder of modulus r to each ideal boundary component as in Section 5.
Proof. If r ∈ [0, R) and ρ → r, then Blob(x, X ρ ) → Blob(x, X r ) and ∂ Blob(x, X ρ ) → ∂ Blob(x, X r ) by Lemma 9.8, since CEmb(X r , Y, h) does not contain any slit mapping. It remains to prove continuity at r = R. By compactness of CL(D) and Lemma 9.7, it suffices to show that if r n R, if Blob(x, X rn ) → A, and if ∂ Blob(x, X rn ) → B, then A ⊃ Blob(x, X R ) and B ⊃ ∂ Blob(x, X R ). Let y ∈ Blob(x, X R ) = ∂ Blob(x, X R ). Then y ∈ Blob(x, X rn ) for every n. Indeed, r n ≤ R means that X rn ⊂ X R and hence CEmb(X R , Y, h) ⊂ CEmb(X rn , Y, h). It follows that y ∈ A. Let U be a connected neighborhood of y. Then U intersects Blob(x, X rn ) since y ∈ Blob(x, X rn ). Thus U intersects the interior of Blob(x, X rn ) because Blob(x, X rn ) is the closure of its interior. Suppose that U is contained in Blob(x, X rn ) for every n in an infinite set J ⊂ N. Then U is contained in A and hence in Blob(x, X R ). This is absurd since Blob(x, X R ) has empty interior. Thus U intersects ∂ Blob(x, X rn ) for all large enough n and hence y ∈ B.
We use continuity to show that the blob has no holes and is thus homeomorphic to a closed disk.
Theorem 9.10. Blob(x) is homeomorphic to a closed disk.
Proof. We will show that the complement of Blob(x) is connected, which is sufficient. Let z 1 , z 2 ∈ D \ Blob(x). Note that z 1 and z 2 are contained in D \ Blob(x, X ρ ) for every ρ ∈ [0, R] as the blobs are nested. Let r be the infimum of the set of ρ ∈ [0, R] such that z 1 and z 2 are in the same component of D \ Blob(x, X ρ ). The set of such ρ is non-empty since Blob(x, X R ) is a point or a compact interval, and hence has connected complement.
Suppose that z 1 and z 2 belong to different components of D \ Blob(x, X r ). Then r < R. In particular, Blob(x, X r ) is a 2-manifold and each boundary component of Blob(x, X r ) is a simple closed curve. Let C 1 be the component of ∂ Blob(x, X r ) surrounding z 1 , let C 2 be the one surrounding z 2 , and let γ be a simple closed curve in the interior of Blob(x, X r ) separating C 1 from C 2 . For all ρ close enough to r we have that ∂ Blob(x, X ρ ) is disjoint from γ. On the other hand, there is a sequence ρ n r such that z 1 and z 2 belong to the same component of D \ Blob(x, X ρn ). Let γ n be a path in D \ Blob(x, X ρn ) connecting z 1 and z 2 . For every n, γ n intersects γ, say at w n . Since γ is compact, we may pass so a subsequence so that w n → w for some w ∈ γ. Now w is in the interior of Blob(x, X r ). Let U be an open disk centered at w whose closure is contained in the interior of Blob(x, X r ). Since Blob(x, X ρn ) → Blob(x, X r ), the open set U must intersect Blob(x, X ρn ) for all large enough n. Since w n ∈ γ \ Blob(x, X ρn ) and since γ ∪ U is connected, the intersection of γ ∪ U with ∂ Blob(x, X ρn ) is non-empty. Let ζ n be in the intersection. After passing to a subsequence, ζ n converges to some point ζ in γ ∪ U . This is a contradiction since ∂ Blob(x, X ρn ) → ∂(x, X r ) but γ ∪ U is disjoint from ∂(x, X r ). Therefore z 1 and z 2 belong to the same component of D \ Blob(x, X r ).
Suppose that r > 0. Let γ be a path joining z 1 to z 2 in D \ Blob(x, X r ). Since γ is compact and Blob(x, X ρ ) depends continuously on ρ, the two are disjoint for all ρ sufficiently close to r. Then z 1 and z 2 belong to the same component of D \ Blob(x, X ρ ) for all ρ < r sufficiently close to r, which contradicts the minimality of r. We conclude that r = 0 and that z 1 and z 2 belong to the same component of D \ Blob(x, X 0 ). Since z 1 and z 2 were arbitrary, the complement of Blob(x) is connected and thus Blob(x) is homeomorphic to a closed disk.

The deformation retraction
Fix once and for all a countable dense subset {x 1 , x 2 , ...} ⊂ X. For each n ∈ N choose a lift b n of h(x n ) to D and define the map lift n = lift xn as in Section 6. Also fix a conformal embedding F ∈ CEmb(X, Y, h) which maximizes m. In this section, we construct a (strong) deformation retraction of CEmb(X, Y, h) into {F }.
Given any f ∈ CEmb(X, Y, h), we define a sequence of paths γ n : In words, γ 1 stays at lift 1 (f ) for half the time and then moves at constant speed along the conformal ray towards the "center" lift 1 (F ) of Blob(x 1 ). In particular, γ 1 (t) belongs to Blob(x 1 ) for every t ∈ [0, 1] so that there exists some g ∈ CEmb(X, Y, h) such that lift 1 (g) = γ 1 (t).
Lemma 10.2. For every t ∈ [0, 1], there exists a unique f t ∈ CEmb(X, Y, h) such that lift n (f t ) = γ n (t) for every n ∈ N. The map t → f t is continuous and satisfies f 0 = f and f 1 = F .

Proof.
Observe that E[n, t] is a non-empty closed subset of CEmb(X, Y, h) and is thus compact. Therefore, for each t ∈ [0, 1], the nested intersection ∞ n=1 E[n, t] is non-empty. Any two functions in the intersection agree on the dense set {x 1 , x 2 , ...} and hence on all of X. Therefore, there is a unique function f t in the intersection. Moreover, f t varies continuously with t. Indeed, if g is any limit of any subsequence of f t as t → s, then for every n ∈ N, we have so that g = f s . It follows that f t → f s as t → s. By construction we have lift n (f ) = γ n (0) and lift n (F ) = γ n (1) for every n ∈ N.
We thus have a map H : is continuous for every f ∈ CEmb(X, Y, h); • H(f, 0) = f and H(f, 1) = F for every f ∈ CEmb(X, Y, h); • H(F, t) = F for every t ∈ [0, 1]. The last point holds because if f = F , then every path γ n is constant and hence f t = F for every t. It remains to prove that H is continuous in both variables.
Lemma 10.3. Suppose that for every n ∈ N, the map f → γ n is continuous. Then H is continuous.
Proof. If for every n ∈ N the map (f, t) → H(f, t)(x n ) is continuous, then H is continuous. This is because of the compactness of CEmb(X, Y, h) and the fact that {x 1 , x 2 , . . .} is dense in X (see the proof of Lemma 10.2). Since H(f, t)(x n ) = f t (x n ) = π Y (γ n (t)), it thus suffices to show that the map (f, t) → γ n (t) is continuous. This condition is equivalent to the requirement that f → γ n be continuous, where the space of continuous maps [0, 1] → D is equipped with the compact open topology (which is the same as the topology of uniform convergence). This is because the interval [0, 1] is locally compact Hausdorff (see [29, p.287]).
Since the map f → lift 1 (f ) is continuous and the Riemann map G[1] is continuous, it is easy to see that f → γ 1 is continuous. We proceed by induction for the rest. Let n ≥ 2 and suppose that the maps f → γ j are all continuous for j = 1, ..., n − 1. We will prove that the map f → G[n] is continuous, which obviously implies that f → γ n is continuous. We use the following theorem of Radó, a proof of which is given in [30, p.26].
Theorem 10.4 (Radó). Let (D k , w k ) and (D, w) be topological closed disks in C, each with a marked point in the interior. Suppose that w k → w and that D k → D in the Fell topology. Suppose also that there are parametrizations c k : S 1 → ∂D k and c : S 1 → ∂D such that c k → c uniformly. Then the normalized Riemann map (D, 0) → (D k , w k ) converges uniformly on D to the normalized Riemann map (D, 0) → (D, w).
By a slight generalization 6 of Lemma 9.7, the maps (f, t) → Blob[n, t] and (f, t) → ∂ Blob[n, t] are upper semi-continuous in t. Moreover, they are continuous at every (f, t) for which E[n, t] does not contain any slit mapping by Lemma 9.8. But if E[n, t] contains a slit mapping, then Blob[n, t] = ∂ Blob[n, t] is a single point and thus upper semi-continuity at (f, t) implies continuity. By Lemma 10.1, the conformal center lift n (g[n, t]) of Blob[n, t] also depends continuously on (f, t). The only thing that remains to be checked is that the boundary ∂ Blob[n, t] can be parametrized as to converge uniformly.
Definition 10.5. Let {c k } ∞ k=1 be a sequence of simple closed curves in C. We say that {c k } ∞ k=1 has a collapsing finger if after passing to a subsequence, there exist x k , y k , z k , w k ∈ S 1 in cyclic order and x, y ∈ C with x = y such that c k (x k ) → x, c k (y k ) → y, c k (z k ) → x, and c k (w k ) → y.
x y Figure 11. A piece of a curve with a finger about to collapse.
We now show that collapsing fingers are the only obstructions to uniform convergence of simple closed curves.
Theorem 10.6. Let c k and c be simple closed curves in C such that c k (S 1 ) converges to c(S 1 ) in the Fell topology. If {c k } ∞ k=1 does not have any collapsing finger, then we can reparametrize c k such that c k → c uniformly.
Proof. By the Jordan-Schoenflies Theorem, c can be extended to a homeomorphismĉ : C → C. Thenĉ −1 • c k (S 1 ) → S 1 in the Fell topology and the sequence {ĉ −1 • c k } ∞ k=1 does not have any collapsing finger. Moreover, if σ k : S 1 → S 1 is a homeomorphism such thatĉ −1 •c k •σ k converges uniformly to the inclusion map S 1 → C, then c k • σ k converges uniformly to c. We may thus assume that c is the inclusion map S 1 → C.
If k is large enough, then c k (S 1 ) is disjoint from 0 and ∞. We claim that if k is large enough, then the winding number of c k around the origin is ±1. Since c k is simple, its winding number is either −1, 0, or 1. Suppose the claim is false. Then after passing to a subsequence, the winding number of c k is 0 for every k. Let arg(c k ) = c k /|c k | and let Arg(c k ) : S 1 → R be a lift of arg(c k ) under the universal covering map R → S 1 . This lift exists because the winding number is zero. Let [a k , b k ] be the image of Arg(c k ). Since c k (S 1 ) converges to S 1 , it follows that the image arg(c k )(S 1 ) converges to S 1 as well, and hence lim inf n→∞ b k − a k ≥ 2π. Thus if k is large enough, then b k − a k > π . Let x k and z k in S 1 be such that Arg(c k )(x k ) = a k and Arg(c k )(z k ) = min(b k , a k + 2π). Also let y k ∈ x k z k and w k ∈ z k x k be such that Arg(c k )(y k ) = Arg(c k )(w k ) = a k + π. Since c k (S 1 ) → S 1 , we may pass to a subsequence so that c k (x k ), c k (z k ), c k (y k ), and c k (w k ) converge to some x, y, z, and w in S 1 . Then x = z, y = w, and x = y, i.e. {c k } ∞ n=1 has a collapsing finger. This is a contradiction, which proves the claim.
If the winding number of c k around the origin is −1, then we reverse the parametrization so that it becomes +1. Let ζ k 1 < ζ k 2 < . . . < ζ k k be a partition of S 1 into k congruent arcs. Since c k has winding number 1, we can find ξ k 1 < ξ k 2 < . . . < ξ k k in S 1 such that arg(c k )(ξ k j ) = ζ k j for every j ∈ {1, . . . , k}. Let σ k : S 1 → S 1 be any homeomorphism such that σ k (ζ k j ) = ξ k j for every j ∈ {1, . . . , k}. We claim that c k • σ k converges uniformly to the inclusion map c : S 1 → C.
To simplify notation, we assume that c k was parametrized correctly from the start, i.e. we assume that for every k 0 and every j ∈ {1, . . . , k}, we have arg(c k )(ζ k j ) = ζ k j . If c k does not converge uniformly to c, then there exists an ε > 0 and an infinite set J ⊂ N such that for every k ∈ J, there exists a y k ∈ S 1 such that |c k (y k ) − y k | ≥ ε. Since S 1 is compact and c k (S 1 ) → S 1 , we can pass to a subsequence such that y k → x and c k (y k ) → y for some x and y in S 1 . Note that |y − x| ≥ ε and in particular y = x. Let j ∈ {1, . . . , k} be such that ζ k j ≤ y k < ζ k j+1 , where we define ζ k k+1 = ζ k 1 . Then let x k = ζ k j and z k = ζ k j+1 . Also let w k ∈ {ζ k 1 , . . . , ζ k k } be the closest point to y which comes after z k but before x k in the cyclic order on S 1 . We have c k (x k ) = x k → x, c k (y k ) → y, c k (z k ) = z k → x, and c k (w k ) = w k → y. In other words, the sequence {c k } ∞ k=1 has a collapsing finger, which is a contradiction. Therefore, c k converges uniformly to c.
To conclude the proof, we show that ∂ Blob[n, t] does not have any collapsing fingers. The reason for this is that the blobs Blob[n, t] are uniformly semi-smooth, meaning that any non-zero limit of a sequence of vectors normal to some blob is normal to the limiting blob 7 . Now if there was a collapsing finger somewhere, then we would see two normal vectors pointing opposite to each other in the limit, which is forbidden by the definition of semi-smoothness. Proof. Let B k = Blob[n, t k ], B = Blob[n, t], c k = ∂B k , and c = ∂B. Suppose that after passing to a subsequence we can find x k , y k , z k , w k ∈ c k in cyclic 7 The proof of this is a straightforward generalization of Theorem 8.15 order and x, y ∈ c with x = y such that x k , z k → x and y k , w k → y. Rotate and translate the picture in such a way that x = 0 and that the upward direction i bisects the cone N 0 of vectors normal to B at 0.
By the proof of Theorem 9.1, there exists a rectangle Q centered at 0 with sides parallel to the coordinate axes such that Q ∩ c is the graph of a continuous function. Since y = 0, we can shrink Q so that it does not contain y. Let δ > 0 be such that the vertical δ neighborhood U δ of Q ∩ c is contained in Q. Then Q \ U δ is compact and disjoint from c. Let k be large enough so that x k and z k are in Q, y k and w k are not in Q, and c k is disjoint from Q \ U δ . Then of the three subarcs x k y k , y k z k , and z k w k of c k , at least two must cross the same vertical side S of Q. This implies that S \ B k is disconnected. Hence there is an open subinterval I of S \ B k whose highest point is contained in B k . Let D be a closed round disk centered on I and contained in C \ B k . Move the center of D upwards until the boundary of the translated disk D * first hits B k . Any intersection point p k of D * with B k is on the top half of ∂D * . Moreover, the unit vector v k based at p k and pointing in the direction of the center of D * is normal to B k .  Figure 12. If the sequence of blobs has a collapsing finger, then we can find a sequence of normal vectors which accumulate to a vector which is not normal to the limiting blob.
Since we can choose Q to be arbitrarily small, we can arrange so that p k → 0. Then the normal vectors v k accumulate onto vectors pointing towards the lower half-plane at 0. This is a contradiction since the cone of normal vectors N 0 is contained in the upper half-plane.
Thus by Theorem 10.6, the boundary of Blob[n, t] can be parametrized in a way that depends uniformly continuously on (f, t). By Theorem 10.4, this implies that the Riemann map G[n] : D → Blob[n, t] depends uniformly continuously on (f, t). Therefore the path γ n depends uniformly continuously on f , and hence H is continuous by Lemma 10.3. This shows that CEmb(X, Y, h) is contractible whenever it is non-empty and h is generic.

The remaining cases
In this section, we describe the homotopy type of CEmb(X, Y, h) when h is not generic. As before, we let {x 1 , x 2 , . . .} be an arbitrary countable dense subset of X.
Y is the sphere. For any triple (y 1 , y 2 , y 3 ) of distinct points in C, there exists a unique Möbius transformation M [y 1 , y 2 , y 3 ] sending (y 1 , y 2 , y 3 ) to (0, 1, ∞). Thus there is a homeomorphism CEmb(X, C) → Aut( C) × CEmb(X \ {x 1 , x 2 , x 3 }, C \ {0, 1, ∞}) The factor CEmb(X \ {x 1 , x 2 , x 3 }, C \ {0, 1, ∞}) deformation retracts into a point since the implicit underlying embedding is generic. As for the factor Aut( C), it is homeomorphic to the set of triples (a, v, b) where a, b ∈ C are distinct and v ∈ T a C is non-zero. The homeomorphism is given by f → (f (0), f (0), f (∞)). This set of triples deformation retracts into T C by moving the point b along the spherical geodesic to the antipode of a. Lastly, T C clearly deformation retracts into the unit tangent bundle T 1 C.
The second factor in this product is contractible since the corresponding embedding is generic. As for the first factor, it is homeomorphic to T C via the map f → (f (0), f (0)). The tangent bundle T C deformation retracts into the unit tangent bundle T 1 C, which deformation retracts further into a circle.
X is the disk. We assume that the codomain Y is not the sphere or the plane. Then we equip Y with a metric of constant curvature (either Euclidean or hyperbolic). We first define a map from the unit tangent bundle T 1 Y to CEmb(D, Y ) as follows. Given v ∈ T 1 y Y , we let D v be the largest embedded geometric disk in Y centered at y, and we let F v : D → D v be the Riemann map with F v (0) = y and F v (0) = λv for some λ > 0. The map v → F v is an embedding. We construct a deformation retraction of CEmb(D, Y ) into the image of that map.
Given f ∈ CEmb(D, Y ), let v ∈ T 1 Y be the unique vector such that f (0) = λv for some λ > 0. Then let r ∈ (0, 1] be the largest such that f (rD) ⊂ F v (D) and let f † (z) = f (rz). Then F −1 v • f † : D → D is a conformal embedding with a positive derivative at the origin which it fixes.
We define a deformation retraction of CEmb(D, Y ) into the unit tangent bundle T 1 Y by We leave it to the reader to check that this map is continuous.
h is trivial. Assume that Y is not the sphere or the plane, and that X is not the disk. Fix a non-zero tangent vector v ∈ T x X. Given f ∈ CEmb(X, Y, h), we can define a disk D f ⊂ Y by filling the holes of f (X). Then we define defined by f → (F, F −1 • f ). The first factor is contractible whereas the second factor deformation retracts into a circle. We can compose the two deformation retractions to obtain a deformation retraction of CEmb(X, Y ) into a circle.
Y is a torus. Let Y be a torus and suppose that h : X → Y is non-trivial. Let x 0 ∈ X and let y 0 = h(x 0 ). For every y ∈ Y , there is a unique conformal automorphism M y : Y → Y homotopic to the identity such that M y (y) = y 0 . We thus have a homeomorphism CEmb(X, Y, h) ≈ Y × CEmb(X \ x 0 , Y \ y 0 , h).
given by f → (f (x 0 ), M f (x 0 ) • f ). Since the restriction h : X \ x 0 → Y \ y 0 is generic, the factor CEmb(X \ x 0 , Y \ y 0 , h) is contractible. It follows that CEmb(X, Y, h) is homotopy equivalent to Y .