Energy minimizing N-covering maps in two dimensions

We show that the N-covering map, which in complex coordinates is given by uN(z):=z↦zN/N|z|N-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u_{_{\scriptscriptstyle {N}}}(z):=z \mapsto z^{N}/\sqrt{N}|z|^{N-1}$$\end{document} and where N is a natural number, is a global minimizer of the Dirichlet energy D(v)=∫B|∇v(x)|2dx\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {D}(v)=\int _B |\nabla v(x)|^2 \, dx$$\end{document} with respect to so-called inner and outer variations. An inner variation of uN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u_{_{\scriptscriptstyle {N}}}$$\end{document} is a map of the form uN∘φ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u_{_{\scriptscriptstyle {N}}}\circ \varphi $$\end{document}, where φ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi $$\end{document} belongs to the class A(B):={φ∈H1(B;R2):det∇φ=1a.e.,φ|∂B(x)=x}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {A}(B):=\{\varphi \in H^1(B;\mathbb {R}^2): \ \det \nabla \varphi = 1 \ \text {a.e.}, \ \varphi \arrowvert _{\partial B}(x) = x\}$$\end{document} and B denotes the unit ball in R2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^2$$\end{document}, while an outer variation of uN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u_{_{\scriptscriptstyle {N}}}$$\end{document} is a map of the form ϕ∘uN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi \circ u_{_{\scriptscriptstyle {N}}}$$\end{document}, where ϕ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi $$\end{document} belongs to the class A(B(0,1/N))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {A}(B(0,1/\sqrt{N}))$$\end{document}. The novelty of our approach to inner variations is to write the Dirichlet energy of uN∘φ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u_{_{\scriptscriptstyle {N}}}\circ \varphi $$\end{document} in terms of the functional I(ψ;N):=∫BN|ψR|2+1N|ψτ|2dy\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I(\psi ;N):= \int _{B} N |\psi _{_R}|^2 + \frac{1}{N} |\psi _\tau |^2 \, dy$$\end{document}, where ψ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi $$\end{document} is a suitably defined inverse of φ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi $$\end{document}, and ψR\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi _{_R}$$\end{document} and ψτ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi _{\tau }$$\end{document} are, respectively, the radial and angular weak derivatives of ψ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi $$\end{document}, and then to minimise I(ψ;N)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I(\psi ;N)$$\end{document} by considering a series of auxiliary variational problems of isoperimetric type. This approach extends to include p-growth functionals (p>1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p>1$$\end{document}) provided the class A(B)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {A}(B)$$\end{document} is suitably adapted. When 1<p<2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1<p<2$$\end{document}, this adaptation is delicate and relies on the deep results of Barchiesi et al. on the space they refer to in Barchiesi et al. (Arch Ration Mech Anal 224(2):743–816, 2017) as Ap\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {A}_p$$\end{document}. A technique due to Sivaloganthan and Spector (Arch Ration Mech Anal 196:363–394, 2010) can be applied to outer variations. We also show that there is a large class of variations of the form v=h∘u2∘g\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v=h\,\circ \,u_2\,\circ \,g$$\end{document}, where h and g are suitable measure-preserving maps, in which u2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u_2$$\end{document} is a local minimizer of the Dirichlet energy . The proof of this fact requires a careful calculation of the second variation of D(v(·,δ))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {D}(v(\cdot ,\delta ))$$\end{document}, which quantity turns out to be non-negative in general and zero only when D(v(·,δ))=D(u2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {D}(v(\cdot ,\delta ))=\mathbb {D}(u_2)$$\end{document}.

be the Dirichlet energy of u. Ball showed in [1] that there is a minimizer of D(·) among functions in the class Y := {y ∈ H 1 (B; R 2 ) : det ∇ y = 1 a.e. in B, y| ∂ B = u N }. (1.2) Here, u N is the N -covering map given by where 0 ≤ R ≤ 1 and 0 ≤ α < 2π. The prefactor 1/ √ N ensures that u N satisfies det ∇u N = 1 a.e., which, together with the observation that ∇u N is essentially bounded, implies that u N belongs to Y. This paper examines and finds evidence in support of the conjecture that u N is itself a singular (i.e. non-smooth), global minimizer of D in Y by proving that u N is the unique global minimizer of D with respect to generalized inner and outer variations. A generalized inner variation of u N in this case is formed by varying the independent variable only, i.e. by forming a map u N • ϕ where ϕ belongs to the class We also study the corresponding problem in the case of energies with subquadratic growth, a typical example of which would be a functional such as the p-Dirichlet energy. In the subquadratic setting the construction of generalised inner variations is much more delicate, so we postpone its description until Sect. 5 of the paper, where, it should be noted, we shall rely on the ideas of Barchiesi et al. [4].
The problem of minimizing D in Y is of interest for several reasons, the first of which is that this problem automatically delivers a singular minimizer whenever N ≥ 2. An argument of Ball in [3,Section 2.3] can be adapted to establish that, when N ≥ 2, no member of Y is C 1 , and so, in particular, the minimizer of D in Y cannot be C 1 . The constraint det ∇ y = 1 a.e. in B is instrumental in limiting the regularity in this way: without it, it is easy to show that the unique global minimizer of D among maps y agreeing with u N on ∂ B is the smooth harmonic extension of u N | ∂ B . Nor can topology be ignored: it turns out that any y in Y is continuous and hence, by topological degree theory, is such that, roughly speaking, y(B) covers the ball B N N times. When N = 1, the minimizer of D in Y is trivially the identity map, which coincides with u 1 in the notation introduced above. Thus we focus on non-trivial changes in topology by choosing N ≥ 2.
The second reason to study D in detail is that, formally at least, u N is a critical point of an appropriately perturbed version of D in Y. To be precise, it can be checked directly that u N solves the Euler-Lagrange equation associated with the energy D(u) + B 2 p N ln R det ∇u dx, (1.4) where p N := N − 1/N is constant. (See [6,Proposition 3.4] in the case N = 2; the case for general N follows similarly.) The prefactor of det ∇u in (1.4) is a Lagrange multiplier traditionally used to encode the requirement that functions in Y obey the constraint det ∇u = 1 a.e.. We note that the theory which underpins Lagrange multipliers in the case of nonlinear elasticity has been established in [8,14,15,[21][22][23]. The main difference between that setting and ours lies in the lack of invertibility of maps belonging to Y which, as indicated above, can be thought of as being N -to-1, whilst those considered by the works cited above, for example, are assumed to be invertible, with sufficiently regular inverses. Thus their results do not necessarily provide for the existence of a Lagrange multiplier for our problem in the case of the Dirichlet energy D. Nevertheless, the direct calculation alluded to above conforms to the conclusions of both [15,Theorem 3.1] and [8,Theorem 4.1], for example, without necessarily obeying all their hypotheses. This observation may be of independent interest. It is straightforward to check that, when ϕ ∈ A(B), u N • ϕ has finite Dirichlet energy, agrees with u N on ∂ B, and satisfies det ∇(u N • ϕ) = 1 a.e. in B, and that the same assertions are true of outer variations φ • u N , where φ ∈ A(B(0, 1/ √ N )). We remark that u N • ϕ represents a quite general form of inner variation that is appropriate to the constraint det ∇v = 1 a.e. imposed on members of A(B). In the setting of nonlinear elasticity more generally, where the constraint det ∇v > 0 a.e. is enforced, inner variations of admissible maps v often take the form v(x + φ(x)), where φ is a smooth, compactly supported function and is chosen sufficiently small that det ∇(x + φ(x)) is bounded away from 0. See [2] together with [5,Theorem A.1]; see also [3,Section 2.4]. In the incompressible setting we consider in this paper maps of the form ϕ(x) = x + φ(x), with φ smooth and compactly supported in B, belong to A(B) only if φ(x) = 0 for all x. Thus this particular form of inner variation appears to be too restrictive.
The first main result of this paper, Theorem 2.2, is the following.
Theorem Let u N be the N-covering map and let ϕ be a map in H 1 id (B; R 2 ) which satisfies det ∇ϕ = 1 a.e. in B. Then (1.5) with equality if and only if ϕ is the identity map. In particular, u N is the unique global minimizer with respect to inner variations of the Dirichlet energy.
The corresponding result for outer variations, Theorem 4.3, demonstrates that, within the class of outer variations, u N is the unique global minimizer of the Dirichlet energy among maps agreeing with u N on ∂ B, i.e. D(φ • u N ) ≥ D(u N ) for all φ in the class A(B(0, 1/ √ N )), with equality if and only if φ is the identity map.
It has to be checked that inner and outer variations in this setting are genuinely different, and this is done rigorously in Proposition 6.1. It is therefore quite natural that the approaches needed to prove Theorems 2.2 and 4.3 are necessarily different too. Sections 2 and 3 describe our approach to the inner variation problem, which proceeds by writing D(u N • ϕ) as a functional in ψ, a suitably defined inverse of ϕ, and by solving a series of problems of isoperimetric type. The approach needed in the case of outer variations is adapted from [30], and, although isoperimetry is again involved, the method does not apply to inner variations. Taken together, these two results clearly do not settle the question of whether u N minimizes D in Y. Nonetheless, we believe that Theorems 2.2 and 4.3 are interesting intermediate steps.
Our approach to inner variations can be generalized to functionals with subquadratic pgrowth. In this setting, the extension of A(B) is a delicate matter, and we rely heavily on the results of [4] concerning the class which they refer to as A p . The details can be found in Sect. 5, which deals with functionals of the form where f is convex, of class C 1 (R + ), and such that f (t) ≥ 0 for t > 0. We suppose that, for some 1 < p < 2, there is a constant C > 0 such that 1 C t p ≤ f (t) ≤ C(1 + t p ) for all t > 0. The corresponding result to Theorem 2.2 is Theorem 5.7, which we record here for completeness.
Theorem Let the functional E be given by (1.6) and let u N • ϕ be a generalized inner variation of u N . Then E(u N • ϕ) ≥ E(u N ), with equality if and only if ϕ = id.
We remark that since u N is a non-affine, positively one-homogeneous function, ∇u N (x) is discontinuous at the point x = 0, a feature it has in common with many of the examples of singular minimizers in the higher dimensional calculus of variations. In brief, [17,24,27,32] all construct one-homogeneous minimizers of smooth, strongly convex functionals, the last of these being in the smallest possible, and hence optimal, dimensions m = 3, n = 2. Here, m and n are, respectively, the dimensions of the domain and codomain of the energy minimizing map. The works [24,32] also contain cogent summaries of the wider theory, which the reader should consult for further details. To relate some of this to the problem considered in our paper, we note that in the dimensions m = n = 2, and without a constraint on the determinant, minimizers should be smooth. This follows from standard results about harmonic functions (or indeed from the classical result [25,Theorem 1.10.4 (iii)] in the more general case of strongly convex, quadratic growth integrands), and from [28] in the case that the candidate minimizer is supposed to be one-homogeneous. In the presence of the determinant constraint, however, these results cannot apply to members of Y when N ≥ 2.
The paper concludes by focusing in detail on the case N = 2, where we examine a rich subclass of Y comprised of maps of the form v = h • u 2 • g, where h and g are self-maps of the balls B(0, 1/ √ 2) and B respectively. In fact, the maps g and h are generated by flows in a way that is made precise in Sect. 6.2, and because of this there is a natural parameter in v, which we call δ, on which both h = h(z; δ) and g = g(y; δ) depend smoothly, and for which it holds that g(y; 0) = y for all y in B and h(z; 0) = z for all z in B(0, 1/ √ 2). This enables us to write v(y; δ) = h(u 2 (g(y; δ)); δ), (1.7) which is such that v(·; 0) = u 2 , and so D(v(·; δ)) depends naturally on δ and obeys D(v(·; 0)) = D(u 2 ). The final main result of the paper, Theorem 6.3, concerns maps of the type v(·, δ) and can be described as follows. Note that J stands for the 2 × 2 matrix representing a rotation of π/2 radians anticlockwise.
Theorem Let v(·, δ) be given by (1.7) and let = J ∇ξ and = J ∇σ be smooth, divergence-free maps with compact support in B(0, 1/ √ 2)\{0} and B\{0} respectively. Suppose that h and g are solutions of the following equations In particular, the map u 2 is a local minimizer of the Dirichlet energy with respect to all variations of the form v(·, δ).

Notation
For non-zero x, we will write x = x/|x|. Note then that for any weakly differentiable map ϕ we may write ∇ϕ(x) = ϕ R ⊗ x + ϕ τ ⊗ J x, where J is the 2 × 2 matrix representing a rotation of π/2 radians anticlockwise. For any two vectors a, b in R 2 , the 2× 2 matrix a ⊗ b is defined by its action a ⊗ b v := (b · v)a on any v in R 2 . In plane polar coordinates (R, α), the angular derivative ψ τ is (ψ α )/R and the radial derivative is ψ R . The inner product of two matrices A 1 and A 2 will be denoted by A 1 · A 2 = tr (A T 1 A 2 ) and the norm by |A 1 | 2 = tr (A T 1 A 1 ). The two-dimensional Lebesgue measure of a (measurable) set E ⊂ R 2 will be written |E| provided it is unambiguous to do so, and otherwise as L 2 (E). Throughout the paper, we will use C r to represent the circle of radius r and centre 0. The rest of the notation is either standard or is explained where it is introduced.

The Dirichlet energy of inner variations of u N
Let ϕ belong to A(B) and consider the composed map u N • ϕ. A short calculation shows that The right-hand side of (2.1) can be rewritten in terms of ∇ϕ T ϕ and ∇ϕ T J ϕ to give For later use, we note that for N ≥ 1 Note that, from (2.2) and (2.4), In the following we shall need to refer to the inverse, ψ, say, of a map ϕ belonging to A(B). The existence and regularity of a continuous map ψ satisfying ψ(ϕ(x)) = x for a.e. x in B was established in [31, Lemma 6 and Theorem 8], as was the validity of the expression ∇ψ(ϕ(x)) = (∇ϕ(x)) −1 . Bearing the constraint det ∇ϕ = 1 a.e. in mind, this immediately implies that ∇ψ(ϕ(x)) = adj ∇ϕ(x) a.e., and hence, via [31, Theorem 2 (ii)], that ∇ψ ∈ L 2 (B; R 2 ). where Proof Apply [31,Theorem 8] together with the constraint det ∇ϕ(x) = 1 a.e. to obtain ∇ψ(ϕ(x)) = adj ∇ϕ(x) a.e. in B. Letting y = ϕ(x) and using the identity adj which proves (2.6). Integrating the expression above with respect to x and than applying the change of variables formula [31, Theorem 2 (ii)] leads to (2.7).
Using Proposition 2.1, we can now study the Dirichlet energy of the map u N • ϕ by considering I (ψ, N ). The latter can be expressed very simply in polar coordinates as and hence, by convexity, the inequality must hold. By a direct calculation using the boundary condition ψ(y) = y for y ∈ ∂ B, the two integrals combine to give where dν corresponds to the measure dL 2 (y)/|y|. Defining we claim that if ψ belongs to A(B) then F(ψ) ≤ π. Supposing for now that this is established, we immediately obtain from (2.11) that I (ψ, N ) ≥ I (id, N ), which, thanks to (2.7), implies that Our goal is now to prove the following result.
Theorem 2.2 Let u N be the N-covering map and let ϕ be a map in H 1 id (B; R 2 ) which satisfies det ∇ϕ = 1 a.e. in B. Then with equality if and only if ϕ is the identity map. In particular, u N is the unique global minimizer with respect to inner variations of the Dirichlet energy.
A short proof of Theorem 2.2 which draws together all its supporting results will be given at the end of Sect. 3.

Bounds on F(Ã)
It is enough to establish the stronger result that, for any admissible ψ and almost every r ∈ (0, 1), the functional when subject to the constraint obeys F(ψ; r ) ≤ 2πr . Integrating (3.1) over r ∈ (0, 1) will then yield the claim that F(ψ) ≤ π.
Proof (a) This is [31,Theorem 5]   The result for general measurable G follows by approximation. Finally, to prove (3.2) it is clearly enough to show that ( * ) B r det ∇ψ dx = 1 2 C r J ψ · ψ τ dH 1 for smooth maps ψ and then apply an approximation argument. The latter expression is a natural consequence of the fact that det ∇ψ is a null Lagrangian, and is hence a divergence. In this case, det ∇ψ = (1/2)div (ψ 2 J ∇ψ 1 − ψ 1 J ∇ψ 2 ), which can be integrated using Green's theorem to give ( * ). Equation (3.2) is now immediate from ( * ) and det ∇ψ = 1.  We first deal with the question of the existence of a maximiser of the functional F(·; r ) among suitable functions in the set C(r ) described by (3.5) below. Let r ∈ (0, 1) be such that Proposition 3.1(b)-(c) apply to ψ| C r . Define and let We can assume that for almost every r in (0, 1), sequences (ψ (n) C r ) n∈N in C(r ) are bounded in H 1 -norm. To see this, consider the full sequence (ψ (n) ) n∈N in A. Without loss of generality ||ψ (n) || H 1 (B) is uniformly bounded, and hence, in particular, so must ||ψ (n) τ C r || L 2 (C r ) be for a.e. r in (0, 1). Note that ∂ τ and restriction to C r commute, so we can write (ψ| C r ) τ = ψ τ C r for any function ψ| C r in C(r ). Next, since F(·, r ) is clearly bounded above, we can choose (ψ (n) C r ) n∈N such that as n → ∞. By the foregoing argument, we can extract a subsequence (not relabelled) such that ψ (n) . It is then immediate, by Sobolev embedding, that F(ψ (n) C r ; r ) → F(v; r ), and that by Sobolev embedding together with the weak convergence ψ . Hence F(v; r ) = sup C(r ) F(ψ| C r ; r ) and A(v) = πr 2 . Further, by again considering the full sequence (ψ (n) ) n∈N in A, which we may assume, without loss of generality, converges weakly to ψ in A(B), it follows that v = ψ| C r . Hence, for almost every r ∈ (0, 1), a maximiser ψ| C r of F(·; r ) exists in C(r ).
In terms of ρ and σ , and By identifying ψ| C r with the pair of functions (ρ, σ ) as described above, it is convenient to alter the notation slightly so that A(ρ, σ ) := A(ψ| C r ) and F(ρ, σ ) := F(ψ| C r ; r ). We now determine conditions necessary for ψ| C r = (ρ, σ ) to be a stationary point of F amongst maps which obey the constraint A(ρ, σ ) = πr 2 .
It remains to identify the solutions to (3.10) and (3.11), and thereby calculate the extreme values of F(ψ| C r ; r ) among ψ| C r in C(r ).
Step 1. Let k = 0 be a constant to be determined shortly, and let provided cos = 0 and where, to start with, we suppose that cos σ 0 = 0. The motivation for transforming variables in this way is explained in Remark 3.4 below. The aim is to show that with ρ, and Y so arranged, both (3.10) and (3.11) automatically hold. To that end, note that by first differentiating (3.14) and then using Y +Y = (λk) −1 , Bearing (3.13) in mind, the latter expression rearranges to give (3.10). To see that (3.11) holds, differentiate (3.13) and use (3.14) and its derivative to obtain (It may help to notice that the bracketed term in the penultimate line also appears in (3.15), and is therefore (λkY ) −1 .) Step 2. Eliminating from (3.13) and (3.14) gives When cos σ 0 = 0, we impose ρ(0) = r by setting k = −r cos σ 0 in (3.13) (and by recalling that Y (0) = 1). This gives rise to two possible sets of equations expressing cos and sin purely in terms of Y as follows: the ± sign to be interpreted as sgn (cos σ 0 ). Suppose for now that cos σ 0 > 0. Note that from (3.16), (3.17) and (3.18), we can express the components of ψ| C r as these simplify to give The same equation results when cos σ 0 < 0 is assumed at the outset.
Step 4. We now deal with the case cos σ 0 = 0. Firstly, note that by sending cos σ 0 → 0 in (3.22) we obtain four distinct maps (corresponding to λ −1 = ±r , σ 0 = π/2 or 3π/2) which, by a direct calculation, obey (3.10), (3.11) and the relevant boundary conditions. The point of this final step is to verify that these are the only such solutions.
Thus we see that the extremals of F(·; r ) on C(r ) are in the form of two families of maps which take circles to circles, each family being parametrized by the change in polar angle σ 0 . Note also that the condition that re 1 maps to re(σ 0 ) is achieved by effectively 'pivoting' the original circle C r about a suitably chosen point.
none of which is smooth in a neighbourhood of = 0. By contrast, the quantities ρ cos = −r (1 − cos ) and ρ sin = −r sin , which are the building blocks of ψ| C r , are clearly smooth in , and hence in α.
As promised at the end of Sect. 2, we now give the proof of Theorem 2.2.
Proof Let ϕ be as described in the statement of the theorem. By Proposition 2.1, Eq.(2.9) and inequality (2.11), it follows that where ψ = ϕ −1 and the functional F is given by (3.1). By the argument preceding Lemma 3.2, the results of Lemma 3.3 apply to F(ψ; r ) for almost every r in (0, 1). In particular, F(ψ; r ) ≤ 2πr a.e., from which it clearly follows that the second term in (3.25) is nonnegative for N ≥ 1. Finally, since the term π(N + 1/N ) in (3.25) is exactly the Dirichlet energy of u N on the unit ball, (2.13) follows. To prove that u N is the unique global minimizer of the Dirichlet energy with respect to inner variations, we note that if (2.13) holds with equality then, in particular, (2.11) must also hold with equality. Since the functional I (ψ, N ) in (2.9) is strictly convex in ψ, (2.10) holds with equality if and only if ψ is the identity map, whence ϕ must be the identity map too.

Remark 3.6
Note that equality in (2.13) also implies that F(ψ; r ) = 2πr for almost every r , so, for such r , it must be that ψ = ψ| C r for an appropriate choice of σ 0 and with λ −1 = r , as described in (3.22) (and in the notation introduced in (3.6) and (3.7)). Supposing this to be the case, it follows that for a suitable, weakly differentiable mapσ satisfyingσ (1) = 0. A short calculation then reveals that det ∇ψ = 1 a.e. if and only ifσ is a.e. zero, and hence that ψ is equivalent to the identity map. In this way the uniqueness part of the proof of Theorem 2.2 can be deduced without appealing to strict convexity.

u N is a minimizer within the class of outer variations
We now apply a technique of Sivaloganathan and Spector [30] to demonstrate that u N is a minimizer of the Dirichlet energy within the class of outer variations. In the rest of this section we will write B N := B(0, 1/ √ N ) for brevity.
By Proposition 3.1, we may assume that φ is continuous and measure-preserving, and that for a.e. r ∈ (0, 1/ √ N ), φ| C r belongs to H 1 (C r ; R 2 ) and is H 1 -a.e. 1 − 1. We have need of the following technical result.
(iii) By Proposition 3.1(b), we can assume without loss of generality that φ| C r belongs to We now state and prove our result concerning outer variations. Proof A short calculation shows that to which, by [13,Theorem 5.35], we can apply the change of variables Here  (v N , B, z) directly. Now let f N : R + → R + be given by and notice that f N is strictly convex on R + and strictly increasing on the interval (1/ √ N , ∞). It follows by the calculation above and (4.1) that where Jensen's inequality has been applied to pass from the third to the fourth lines above. By Proposition 4.2(i) and (iii), the argument of f N in the last line above is at least 1, and But, by inspection, D(u N ) = π N f N (1), which proves the inequality stated in (4.2). If D(φ • u N ) = D(u N ) then all the inequalities derived above become equalities, implying that (a) Equation (4.1) must hold with equality a.e. in B N , and hence, by the condition for equality in the Cauchy-Schwarz inequality, there is a scalar-valued function k(z), say, such that J φ R (z) = k(z)φ τ (z) for a.e. z in B N ; (b) C r |φ τ (z)| dH 1 (z) = 2πr must also hold for a.e. r ∈ (0, 1/ √ N ), which when integrated over r in that range gives Hence (c) and (4.3) give Since f N is strictly convex, Jensen's inequality tells us that with equality if and only if |φ τ (z)| = − B N |φ τ (z)| dz for a.e. z. But, by (4.3), this implies that |φ τ (z)| = 1 for a.e. z, and hence from (a) together with the constraint J φ R (z) · φ τ (z) = 1 a.e., we obtain that k(z) = 1 a.e. Thus we can write which has the property that cof ∇φ(z) = ∇φ(z) a.e. Thus, by Piola's identity, φ is a weak solution of Laplace's equation which agrees with the identity on ∂ B N , so it must be that φ = id in B N .

Remark 4.4
From the proof above, we see that Now compare this with the corresponding expression for an inner variation, (2.9), which we reprint here for convenience, The technique used in Theorem 4.3 does not apply to functional D(u N •ψ), and neither do the methods used to prove Theorem 2.2 apply to the functional D(φ • u N ). In each case, it seems that the weighting of the radial and angular derivatives determines the approach required.

Extension to functionals with p-growth for 1 < p < 2
It is natural to ask whether the techniques of the preceding section carry over to functionals besides the Dirichlet energy. The functionals we have in mind are of the form where f is convex, of class C 1 (R + ), and such that f (t) ≥ 0 for t > 0. In addition, so that the setting of the problem is W 1, p (B; R 2 ), we suppose that there is a constant C > 0 such that 1 C t p ≤ f (t) ≤ C(1 + t p ) for all t > 0. When p > 2, the analysis of E(u N • ϕ) carries over from that given for D(u N • ϕ) with only minor changes, so we do not address that question here. We focus on the case 1 < p < 2, where one has to construct carefully a suitable analogue to the function space A(B) introduced in (1.3). The chief difficulties are that, in contrast to members of A(B), a typical map ϕ ∈ W 1, p (B; R 2 ), with p < 2, obeying det ∇ϕ = 1 a.e. in B and ϕ| ∂ B = id need not be continuous, Eq. (3.2) need not hold, nor need ϕ be invertible in the sense described just before Proposition 2.1. Recall that the continuity and invertibility of ϕ ∈ A(B) were needed to apply results depending on the topological degree and to transform the energy functional B W (φ, ∇ϕ, N ) dx into a more tractable form involving ψ := ϕ −1 (see e.g. (2.7) and (2.8)), while Eq. (3.2) led to the area constraint in the study of the functional F(ψ) described in Sect. 3.
Fortunately, thanks to works of Müller and Spector [26], Henao and Mora-Corral [18-20] and Barchiesi et al. [4], there is a substantial framework which provides a suitable candidate for A(B). In short, the required invertibility and other properties (such as the validity of Eq. (3.2), for example) can be found in the class which Barchiesi, Henao and Mora-Corral refer to in [4] as A p . We shall recall the definition of A p from [4] below, describe some of its properties, and note how a supplementary condition, given later, ensures that the local inverse of [4] is, in this setting at least, effectively an inverse on the entire image domain.
For later use, we recall that if ϕ ∈ W 1, p (B, R 2 ) then its distributional determinant Det ∇ϕ obeys We also recall the definition of Müller and Spector's condition (INV), as given in [26,Definition 3.2] in terms of a bounded, open domain ⊂ R n with Lipschitz boundary: Definition 5. 1 We say that u : → R n satisfies (INV) provided that for every a ∈ there exists an L 1 null set N a such that, for all r ∈ (0, r a )\N a , u| ∂ B(a,r ) is continuous, B(a, r )) ∪ u (∂ B(a, r )) for L 2 -a.e. x ∈ B(a, r ), and (ii) u(x) ∈ R 2 \im T (u, B(a, r )) for L 2 -a.e. x ∈ \B(a, r ).
The topological image im T (u, B(a, r )) is given by im T (u, B(a, r )) := {y ∈ R n \u (∂ B(a, r )) : d(u, ∂ B(a, r ), y) = 0} whenever u| ∂ B(a,r ) is continuous: see [26,Section 3] or [4, Section 3]. We will need to apply condition (INV) to the precise representative of a function in W 1, p . One can either follow [ Proposition 5.2 Let 1 ≤ p < n and u ∈ W 1, p ( ; R n ). Let p * := np/(n − p) be the Sobolev conjugate exponent. Denote by P the set of points x 0 ∈ where the following property fails: there exists u * (x 0 ) ∈ R n such that Then cap p (P) = 0.

Definition 5.3 Let ϕ ∈Ã p . Then a generalized inner variation of u N is a function of the form
SinceÃ p is weakly closed, it follows from Sobolev embedding that the class of generalized inner variations is also weakly closed. With this in mind, an application of the direct method of the calculus of variations yields the following.

Proposition 5.4 Let E be given by (5.1). Then there is a minimizer ϕ ∈Ã p of E(u N • ϕ).
Proof Note thatÃ p contains the identity map, so, in particular, the set of generalized inner variations is nonempty. Take a sequence (ϕ ( j) ) j∈N such that E(u N •ϕ ( j) ) inf{E(u N •ϕ ( j) ) : ϕ ∈Ã p }. By (2.3) and the assumed p-growth of f , it follows that, for a subsequence, ϕ ( j) ϕ in W 1, p (B, R 2 ), and, by the argument above, that ϕ ∈Ã p . The convexity of E finishes the proof.
We are now in a position to apply the techniques of Sects. 2 and 3 of this paper to prove the following result. Proof Let ϕ ∈Ã p . Recall (2.5) and note that, by (5.5), E(u N • ϕ) can be written where ψ is the inverse of ϕ, as described above. By the convexity of f , A direct calculation shows that W

is constant, and in polar coordinates
1 2 for any a, b ∈ R 2 and note that g is convex. In particular,

(5.8)
Integrating and applying the boundary condition ψ| ∂ B = id, it follows that (5.9) where F(ψ) is given by (2.12). The constraint (3.2) is in force, and the arguments of Sect. 3 continue to hold with W 1, p in place of H 1 throughout (with the exception of part (a) of Proposition 3.1; this is not needed, now that (3.2) has been established above by different means). In particular, by Lemma 3.3, we see that F(ψ) ≤ π, and hence the rightmost term in (5.9) is nonnegative. Since f ≥ 0, it follows that the second line in (5.7) is nonnegative, from which the inequality To prove that the identity map is the unique minimizer, first suppose that E(u N • ϕ) = E(u N ). Then, in particular, (5.8) holds with equality for a.e. y in B. The same calculation which proves the convexity of g also shows that if x := (a, b) ∈ R 4 then D 2 g(x)[ξ, ξ ] = 0 if and only if x and ξ are proportional. Equality in (5.8) therefore implies that D 2 g(ȳ, Jȳ)[ξ, ξ ] = 0 with ξ := (ψ R −ȳ, ψ τ − Jȳ), and hence, by the previous remark, that (ψ R , ψ τ ) = k(y)(ȳ, Jȳ) for some function k(y) and a.e. y in B. Since det ∇ψ = 1 a.e., it follows that k 2 (y) = 1 a.e., and hence that ∇ψ(y) ∈ SO(2) for a.e. y. By a version of Liouville's theorem (see e.g. [29]), ∇ψ is smooth and everywhere equal to a constant matrix, and hence, via the boundary condition, ψ = id. Thus ϕ = id.

A class of variations in which u 2 is a local minimizer
We now focus on the double-covering map, u 2 , with the twin goals in mind of (a) establishing that there are maps in the admissible class Y that are not inner variations of u 2 and (b) demonstrating that u 2 is a local minimizer in a large subclass of A whose description we give in Sect. 6.2. The proofs of both facts rely to differing extents on being able to generate self-maps on the balls B or B using flows, the ideas for which come from [10] and the references therein.

Admissible maps that are not inner variations of u 2
The following class of counterexamples apply to the case N = 2, i.e. to the double-covering map, but the principle can be extended to N -covering maps if we wish. To be clear, we seek maps v belonging to H 1 (B; R 2 ) which obey both det ∇v = 1 a.e. in B and v| ∂ B = u 2 , but which cannot be expressed as inner variations u 2 • ϕ where ϕ belongs to A(B). To clarify the following argument we introduce the following notation. Given sets X , Y , if and only if x = a. By definition of u 2 and ψ we therefore have 0 ↔ p 0 . Now let y 1 and y 2 be the local inverses of u 2 defined on B , so that u 2 (y i ( p 0 )) = p 0 for i = 1, 2. Note that because p 0 = 0, neither of y 1 ( p 0 ), y 2 ( p 0 ) is zero and y 1 ( p 0 ) = y 2 ( p 0 ). Let ϕ ∈ A, where the class A is defined in (1.3). Since d(ϕ, B, y) = 1 for all y in B, it must in particular be that there are points x 1 , x 2 in B such that ϕ(x i ) = y i ( p 0 ) for i = 1, 2. Moreover, x 1 = x 2 because y 1 ( p 0 ) = y 2 ( p 0 ). Hence, and with obvious notation, Fig. 1.) If we now suppose for a contradiction that v = u 2 • ϕ then we must have x i v → p 0 for i = 1, 2. But 0 v ↔ p 0 , which implies x 1 = x 2 = 0 and thereby contradicts x 1 = x 2 . Since ϕ was arbitrary, we conclude that v is not expressible as u 2 • ϕ.
The key to the result above is the use of 'outer variations' of u 2 of the form ψ • u 2 , and it naturally leads us to consider compositions of the form ψ • u 2 • ϕ where ψ and ϕ are measure-preserving maps of the balls B and B respectively. By further requiring that tr (ψ) = id and tr (ϕ) = id, it should be clear that ψ • u 2 • ϕ belongs to Y. In this way, we can generate a rich subclass of Y which can be analysed in a neighbourhood of u 2 . This is the topic of the next subsection.

Local minimality of u 2 in the class S • u 2 • S −1
In this section we focus on admissible maps where h ∈ S , g = G −1 with G ∈ S, and where the classes S and S are defined in terms of flows, as follows. Firstly, let with a corresponding description for T (B ). Then we define S and S respectively by and S := G : B → B, ∃ ∈ T (B) and Now, by [9,Lemma 14.11], maps belonging to S are self-maps of B which obey (i) det ∇ z h(z, δ) = 1 for all z ∈ B , |δ| < δ 0 and (ii) h(z, δ) = z for all z ∈ ∂ B , |δ| < δ 0 . Similarly, maps in S are smooth self-maps of B with unit Jacobian and which agree with the identity on ∂ B. In particular, by the inverse function theorem, any map G belonging to S is smooth, invertible and agrees with the identity on ∂ B, so that g = G −1 is well-defined. Letting the set S −1 be the notation in the title of the section is now self-explanatory, and, by inspection, the map v given by (6.1) belongs to Y. Now recall that When v is expressed using (6.1), a short calculation shows that |∇h(u 2 (y))∇u 2 (y)adj ∇G(y)| 2 dy, (6.2) where G = g −1 . Indeed, it is clear from (6.1) and the definition of D(v) that By making the substitution y = g(x) and using the fact that ∇g(x) = adj ∇G(y) when det ∇g(x) = 1 and G = g −1 , the expression (6.2) results.

Lemma 6.2 Let D(v) be as in (6.2) with h a smooth self-map of B and G a smooth self-map of B. Then D(v) can be written as
Futhermore, on letting
In particular, the map u 2 is a local minimizer of the Dirichlet energy with respect to all variations of the form v(y, δ).
The proof of Theorem 6.3 relies on a number of auxiliary results, which we now present.
Proof (a) Since G(y, 0) = y, the relation G R (y, 0) = y, whose left-hand side is b 1 (y, 0) by definition, is immediate. Similarly, h s (z, 0) = z for any z = 0, so take z = u 2 (y) and recall the definition of a 1 (y, 0). The proof of part (b) follows similarly.
(c) Consider (6.8) and take the derivative with respect to the radial variable R. Since the derivatives in δ and R commute, we have b 1 (y, δ) = ∇ (G(y, δ))b 1 (y, δ) y ∈ B, |δ| < δ 0 . (6.16) By taking δ = 0, applying (a) and the fact that G(y, 0) = y for y ∈ B, it follows that The right-hand side of the equation above is R (y), as claimed. Similarly, h s (z, 0) = s (z)z, in which we set z = u 2 (y) to obtain a 1 (y, 0) = s (u 2 (y)). Notice that the condition y = 0 is precisely what is needed to ensure z = u 2 (y) = 0. The proof of part (d) follows similarly.
The next three lemmas gather together and simplify various quantities which will shortly be of use and which exploit the divergence-free nature of and appearing in (6.7) and (6.8). The first of the two results is an identity that is easily deduced, so we present only a sketch of the proof. The second and third results are more involved.

Lemma 6.5 Let σ belong to the class C
Proof Equation (6.17) follows from the facts that each of which can be established through a relatively straightforward sequence of integrations by parts. Lemma 6.6 Let b 1 (y, δ) be given by (6.13), where the corresponding map G(y, δ) evolves according to (6.8) and where = J ∇σ is a smooth, divergence-free map with compact support in the punctured ball B\0. Similarly, let a 1 (y, δ) be given by (6.12), where the corresponding map h(z, δ) evolves according to (6.7) and where = J ∇ξ is a smooth, divergence-free map with compact support in the punctured ball B \0. In addition, let X (y) := ξ(u 2 (y)) y ∈ B, (6.20) and and (c) the following identities hold: Proof (a) We have = σ R J y − σ τ y, so R = σ R R J y − σ τ R y, and part (a) is immediate.
We now replace X and σ in (6.33) with their Fourier series representations where y = (R cos θ, R sin θ) and where, a 0 , A 0 and, for j = 1, 2, . . ., a j , b j and A j , are smooth functions of R ∈ (0, 1). Note that, by (6.20), X is an even function of y, and so its Fourier series must be as stated. The following result records the effect of this substitution into (6.33), but in a new set of variables which arise naturally from a j , b j and A j .
and whereḟ denotes differentiation with respect to the variable t. Then, in these terms, the expression (6.33) takes the form Proof Firstly, it is straightforward to check using elementary properties of Fourier series that when X and σ are expressed in terms of (6.37) and (6.38), (6.33) gives B L (0) dy = 2π 1 0 3 4 Now we use (6.39), (6.40) and (6.41) to write a j = z j +ż j , a j = (ż j +z j )/R and (a j /R) =ż j /R, with similar conversions for A j and b j . Using also the facts that where u, v, are any of w j , z j , Z j , we find, after simplifying and re-arranging terms, that It is now clear that the integrand of the first term in (6.44) is equal to 1 4 O(z 0 − Z 0 ; 0), and that the remaining integrand equates to O(z j − Z j ; j) + O(w j ; j), which leads directly to Eq. (6.43) as stated. Corollary 6.9 Let the functions σ and X belong to C ∞ c (B\0) and be related to B L (0) dy through (6.33). Then Proof To show (6.45), begin by noting the lower bounds the last two of which hold for all j ∈ N. A short calculation using the Fourier decompositions (6.37) and (6.38), and the changes of variables (6.39), (6.40) and (6.41), shows that Inserting inequalities (6.46), (6.47) and (6.48) into (6.43), and using (6.49) and (6.50), yields (6.45).
It may help at this point to take stock of the results obtained so far, which have established the lower bound (1.8), i.e.
for variations v(·, δ) given by (6.10). The functions σ and X = ξ • u 2 are connected to the evolution of v(·, δ) for |δ| < δ 0 through the systems (6.7) and (6.8). In proving Theorem 6.3 we are faced with two possibilities, which are that either the right-hand side of (1.8) is strictly positive or it is not. The strictly positive case is easily dealt with, while the degenerate case that requires further analysis. This is the purpose of the next three results, after which we will finally turn to the proof of Theorem 6.3.
In the following, we abbreviate H (y, δ) to H , and, later, G(y, δ) to G; we also make use of the notation z = z/|z| for z = 0. By definition of H (y, δ), H (y, δ) = ∇u 2 (G(y, δ))G (y, δ) where, in order to pass from the fifth to the sixth line of the calculation, we have applied Lemma 6.10 with G(y, δ) in place of y. The conclusion now follows.
The consequences of Lemma 6.11 are quite strong, as we now show.
Proof Part (a) of Theorem 6.3 follows from Eq. (6.32) in Lemma 6.7 and the fact that B X τ R − σ τ R dy = 0 for any smooth function with compact support in the set B\0.
To prove part (b), we twice apply a suitable dominated convergence theorem to the expression (6.28) for D(v(·, δ)) to deduce that after (6.45) of Corollary 6.9 has been applied. If the lower bound in (6.52) is positive then a standard argument implies that, for sufficiently small δ, the inequality D(v(·, δ)) > D(v(·, 0)) = D(u 2 ) must hold. Note that we have implicitly used part (a) of Theorem 6.3 here.

Fig. 2 Left
The arrangement of ([0, α * )) and ([α * , 2π)) is illustrated. Both generalised exterior normals ν andν point into the region U 0 := {w ∈ R 2 \ψ(C r ) : d(ψ, B r , w) = 0}, and the degree changes by 1 as each curve is crossed in the direction of its normal. By Proposition 3.1 part (d), the degree can take only the values 0 or 1. Thus there is a non-trivial region in a neighbourhood of (α * ) where the degree is both 0 and 1, which is impossible. The point z is depicted here as the origin for convenience. Right The curve ψ(C r ) with σ (2π) − σ (0) = −2π is illustrated. According to Proposition 3.1 part (e), the exterior normal ν is obtained by rotating the tangent vector, here written as τ for brevity, π/2 radians clockwise. Since ν points into the region U 0 defined above, it follows that ψ(B r ) lies entirely outside ψ(C r ) and is, in particular, unbounded. This is impossible By rotating the coordinate system in the image domain, we may also assume that (0) ∈ {z + t(1, 0) : t ≥ 0} =: R(0), say. Introduce for each α ≥ 0 the shorthand notation Since ψ(C r ) is a closed loop, there must exist a least α 1 > 0, say, such that (α 1 ) ∈ R(0), and so we may define the closed loop α 1 by which connects (0) to itself. For such a loop there are two possibilities: either α 1 is contractible in the set R 2 \{z} to the point (0) or it is not. Note that, since 2π = ψ(C r ) encircles z, it is certainly the case that 2π is not contractible to (0) in R 2 \{z}, so the set E := {α ∈ (0, 2π] : (α) ∈ R(0), α is not contractible in R 2 \{z} to (0)} contains 2π and is bounded below. Thus α * := inf E is well defined. Suppose that α j → α * and α j ∈ E for all j. Then, since ψ| C r belongs to H 1 (C r , R 2 ), we have H 1 ( ((α j , α * ))) → 0 as j → ∞, so the non-contractibility in R 2 \{z} of α * to (0) follows from that of α j for sufficiently large j. Thus E is closed and α * ∈ E is a minimum. Note that 0 = { (0)}, which is clearly contractible to (0) in R 2 \{z}. Hence 0 / ∈ E, and so, in particular, α * > 0.