On the analytic systole of Riemannian surfaces of finite type

In our previous work we introduced, for a Riemannian surface $S$, the quantity $ \Lambda(S):=\inf_F\lambda_0(F)$, where $\lambda_0(F)$ denotes the first Dirichlet eigenvalue of $F$ and the infimum is taken over all compact subsurfaces $F$ of $S$ with smooth boundary and abelian fundamental group. A result of Brooks implies $\Lambda(S)\ge\lambda_0(\tilde{S})$, the bottom of the spectrum of the universal cover $\tilde{S}$. In this paper, we discuss the strictness of the inequality. Moreover, in the case of curvature bounds, we relate $\Lambda(S)$ with the systole, improving a result by the last named author.


Introduction
Small eigenvalues of Riemannian surfaces, in particular of hyperbolic surfaces, have been of interest in different mathematical fields for a long time. Buser and Schmutz conjectured that a hyperbolic metric on the closed surface S = S g of genus g ≥ 2 has at most 2g − 2 eigenvalues below 1/4 [Bus10,Sch91]. In [OR09], Otal and Rosas proved a generalized version of this conjecture. They showed that a real analytic Riemannian metric on S g with negative curvature has at most 2g − 2 eigenvalues ≤ λ 0 (S), whereS denotes the universal covering surface of S, endowed with the lifted Riemannian metric, and where λ 0 (S) denotes the bottom of the spectrum ofS. Recall here that, for a Riemannian surface F (possibly not complete) with piecewise smooth boundary ∂F (possibly empty), the bottom of the spectrum of F is defined to be λ 0 = λ 0 (F ) = inf R(ϕ), (1.1) GAFA ON THE ANALYTIC SYSTOLE OF RIEMANNIAN SURFACES 1071 multiplicity of λ 0 (F ) as an eigenvalue of F is one. We then call the corresponding positive eigenfunction of F with L 2 -norm one the ground state of F . For a Riemannian surface S, with or without boundary, we define the analytic systole to be the quantity Λ(S) = inf where the infimum is taken over all subsurfaces F inS with smooth boundary which are diffeomorphic to a closed disc, annulus, or cross cap. (A cross cap is frequently also called a Möbius strip.) Note that the fundamental groups of disc, annulus, and cross cap are cyclic, hence amenable. By the work of Brooks, we therefore have for all complete and connected Riemannian surfaces S, see [Bro85, Theorem 1] and also Theorem A.1 below. The strictness of this inequalilty and other estimates of Λ(S) are the topics of this article. To clarify our terminology, a surface is a smooth manifold of dimension two. A Riemann surface is a surface together with a conformal structure. They are not the topic of this article. We study Riemannian surfaces, that is, surfaces together with a Riemannian metric.
We say that a surface S is of finite type if its Euler characteristic χ(S) is finite and its boundary is compact (possibly empty). It is well known that a connected surface S is of finite type if and only if S can be obtained from a closed surface by deleting a finite number of pairwise disjoint points and open discs.
After first extensions of the results of Otal and Rosas in [Mat13] and [Mon14], we showed in [BMM16] and [BMM17] that any complete Riemannian metric on a connected surface S of finite type with χ(S) < 0 has at most −χ(S) eigenvalues ≤ Λ(S), where the eigenvalues are understood to be Dirichlet eigenvalues if ∂S = ∅. This result explains the significance of the analytic systole and the interest in establishing strictness in (1.3).

GAFA ON THE ANALYTIC SYSTOLE OF RIEMANNIAN SURFACES 1073
2) If S is non-compact and the fundamental group of S is not cyclic, then S carries complete Riemannian metrics such that λ ess (S) > Λ(S). Moreover, if χ(S) < 0, then such metrics may be chosen to have curvature K ≤ −1 and finite or infinite area. 3) If λ 0 (S,g) > 0 for some Riemannian metric g on S, then a generic complete Riemannian metric g on S in any neighborhood of g in the uniform C ∞ topology satisfies the strict inequality λ ess (S, g ) > Λ(S, g ).
Here |S| denotes the area of S and sys(S), the systole of S, is defined to be the minimal possible length of an essential closed curve in S. Theorem 1.6. For a closed Riemannian surface S with curvature K ≤ κ ≤ 0, we have Remarks 1.7. 1) For closed Riemannian surfaces S with curvature K ≤ κ < 0, we know in general only that λ 0 (S) ≥ −κ/4. Therefore Theorem 1.6 may not imply the strict inequality Λ(S) > λ 0 (S) for such S. In fact, the relation between λ 0 (S) and the right hand side in Theorem 1.6 is not clear. Our method of proof, involving isoperimetric inequalities and Cheeger's inequality, does not seem to be sophisticated enough to capture the difference between them.
2) The proof of Theorem 1.6 also applies to non-compact surfaces of finite type.
In this case one needs to define the systole as the infimum over all homotopically non-trivial curves, not only the essential (not homotopic to a boundary component or a puncture) ones. For this reason, the corresponding statement is not really interesting anymore. If |S| < ∞, then sys(S) = 0 (by a refinement of the isosystolic inequality), and if |S| = ∞, then sys(S)/|S| = 0. 3) The difference Λ(S) − λ 0 (S) can not be estimated from below by a positive constant, which only depends on the topology and the area of S. In fact, given any ε > 0 and natural number n, if the metric on S is hyperbolic with sufficiently small systole, then λ n (S) < 1/4 + ε, by [Bus77, Satz 2] or the proof of Theorem 8.1.2 in [Bus10].
One may view Theorem 1.6 also as an upper bound on the systole in terms of a curvature bound and Λ(S). Together with our next result, this explains the name analytic systole. For a closed Riemannian surface S, we say that a closed geodesic c of S is a systolic geodesic if it is essential with length L(c) = sys(S). Clearly, systolic geodesics are simple.
Here we say that a simple closed curve in S is two-sided or one-sided if it has a tubular neigborhood which is diffeomorphic to an annulus or a cross cap, respectively.
Combining Theorems 1.6 and 1.8, we get that, for hyperbolic metrics, Λ(S) is squeezed between two functions of the systole. Corollary 1.9. For closed hyperbolic surfaces, we have with w = w(sys(S)) as in Theorem 1.8.
We conclude that the analytic systole of hyperbolic metrics on closed surfaces tends to 1/4 if and only if their systole tends to 0.

Main problems and arguments.
The only surfaces S in Theorems 1.1 and 1.2 with Euler characteristic χ(S) ≥ 0 are torus and Klein bottle. For these, the proof of the inequality Λ(S) > λ 0 (S) is quite elementary. The proof of the hard direction of Theorem 1.2, namely establishing the strict inequality Λ(S) > λ 0 (S) under the condition λ ess (S) > λ 0 (S), is rather involved in the case χ(S) < 0. The domain monotonicity of the first Dirichlet eigenvalue implies that Λ(S) can not be realized by any compact subsurface F ⊆ S diffeomorphic to a disc, an annulus or a cross cap. Keeping this in mind, our general strategy for the proof of Theorem 1.2 is to show that the equality Λ(S) = λ 0 (S) would imply the existence of a non-trivial λ 0 (S)-eigenfunctionφ onS or an appropriate cyclic quotientŜ ofS that vanishes on an open set.
The condition λ ess (S) > λ 0 (S) forces a subsurface F with λ 0 (F ) close to λ 0 (S) to stay almost completely in a large compact set in a weighted sense, the weight being GAFA ON THE ANALYTIC SYSTOLE OF RIEMANNIAN SURFACES 1075 the ground state. One then works essentially within a fixed compact subsurface of S. Two main problems that we still have to overcome in establishing the existence ofφ as above are 1) a priori non-existence of a fixed quotient ofS along a sequence of subsurfaces approximating λ 0 (S) and 2) the absence of the compact Sobolev embedding H 1 → L 2 on these covering spaces.
As for the first problem, the case of cross caps can be reduced to the case of annuli by considering the two-sheeted orientation covering of the original surface. The case of annuli is tackled by showing that only finitely many isotopy types of annuli have the bottom of their spectrum close to λ 0 (S). A keystone of the argument is Lemma 5.3 which relates the bottom of the spectrum of compact surfaces F with the sum of the lengths of shortest curves in the free homotopy classes of the boundary circles of F .
To tackle the second problem, we establish, in Lemma 6.4, an inradius estimate for superlevel sets of suitably truncated ground states of a sequence of subsurfaces F n approximating λ 0 (S). The inradius estimate is proved by means of isoperimetric inequalities, extending arguments from the proof of the Cheeger inequality.
1.3 Structure of the article. In Sect. 2, we collect the relevant facts about isoperimetric inequalities on Riemannian surfaces. In Sect. 3, we extend Osserman's refined version of the Cheeger inequality [Oss77, Lemma 1] for plane domains to compact Riemannian surfaces with boundary. We also recall Osserman's elegant proof since we will need consequences and extensions of his arguments. The isoperimetric inequalities from Corollary 2.2 and the Cheeger inequality are then used in Sect. 4 to obtain a generalized version of Theorem 1.6. The arguments here are very much in the spirit of Osserman [Oss77] and Croke [Cro81]. As an application of our discussion, we obtain Theorem 1.2 for the case where S is a torus or a Klein bottle. This section closes with the proof of Theorem 1.8, which involves methods which are different from those of the rest of the article. Sections 5 and 6 are concerned with properties of the ground states of compact Riemannian surfaces with boundary. The main objectives are Lemma 5.3 on the relation of the bottom of the spectrum to other geometric quantities and Lemma 6.4 on the inradius of superlevel sets of ground states. In Sect. 7, we complete the proof of Theorem 1.2. Section 8 contains the proof of Proposition 1.5 and some remarks and questions. In particular, we draw attention to problems in optimal design which are related to optimal estimates of the analytic systol. In "Appendix A", we discuss an extension of the result of Brooks quoted in connection with (1.3).

Isoperimetric Inequalities
The content of the present section is related to and extends Lemma 1 of [Oss77] in the way we will need it.
Let F be a compact and connected surface with piecewise smooth boundary ∂F = ∅ and interiorF = F \ ∂F . The components of ∂F are piecewise smooth circles. Denote by χ = χ(F ) the Euler characteristic of F.
Assume that F is endowed with a Riemannian metric and denote by K the Gauss curvature of F . Let |F | and |∂F | be the area of F and the length of ∂F , respectively, and For a function f : F → R, write f + = max(f, 0) for its positive part. We recall the following isoperimetric inequalities.
Theorem 2.1. For any F as above and κ ∈ R, we have (1) (2) If F is not a disc and κ < 0, then where denotes the sum of the lengths of the shortest loops in the free homotopy classes (in F ) of the boundary circles of F .
Corollary 2.2. If K ≤ κ, then we have: Note that we always have |∂F | 2 ≥ 2 , by the definition of .

Cheeger Inequality Revisited
In Lemma 2 of [Oss77], Osserman discusses a refinement of the Cheeger inequality for compact planar domains, endowed with Riemannian metrics. We will need an extension of Osserman's Lemma 2. As above, we let F be a compact and connected Riemannian surface F with piecewise smooth boundary ∂F = ∅. The Cheeger constant of F is defined to be the number where the infimum is taken over all compact subsurfaces F ofF with smooth boundary. Note that closed surfaces cannot occur as subsurfaces F of F since F is connected with non-empty boundary.
This shows that the infimum h can be taken over compact and connected subsurfaces F ofF with smooth boundary. Let now C be a component of F \ F . Suppose first that the boundary of C does not contain a boundary circle of F . Then F = F ∪ C is a compact and connected subsurface ofF with area |F | > |F | and length of boundary |∂F | < |∂F |. It follows that the infimum h is attained by compact and connected subsurfaces F ofF with smooth boundary such that the boundary of each component of F \ F contains a boundary circle of F .
If the boundary of C would not have a boundary circle inF , then C would have to coincide with F since F is connected. But then F would be empty, a contradiction.
By a slight variation of the standard terminology, we say that a subsurface S of a surface T is incompressible in T if the induced maps of fundamental groups are injective, for all connected component C of S. In particular, embedded discs are always incompressible.

Proposition 3.2. The Cheeger constant is given by
where the infimum is taken over all incompressible compact and connected subsurfaces F ofF with smooth boundary such that no component of F \ F is a disc or a cross cap. Any such F satisfies χ(F ) ≥ χ(F ) with equality if and only if F \ F is a collared neighborhood of ∂F , consisting of annuli about the boundary circles of F .
For example, if F is an annulus, then we only need to consider discs and incompressible annuli F in F ; if F is a cross cap, then only discs and incompressible annuli and cross caps F . Proof of Proposition 3.2. By Lemma 3.1, the Cheeger constant h is realized by compact and connected subsurfaces F ofF with smooth boundary such that each component of F \ F has at least two boundary circles. This excludes discs and cross caps as components of F \ F . We have since the intersection of F with F \F consists of circles and since no component of F \F is a disc. Furthermore, equality can only occur if χ(F \F ) = 0. By what we already know, this can only happen if the components of F \F are annuli. By Lemma 3.1 and since F ⊆F , they constitute a collared neighborhood of ∂F .
It remains to show the incompressibility of F . If this would not hold, F would contain a Jordan loop c which is not contractible in F , but is contractible inF . Then c would be the boundary of an embedded disc D inF which is not contained in F . Since ∂D ⊆ F , D \ F would consist of components of F \ F . Their boundary would be in D ⊆F in contradiction to Lemma 3.1.
Recall the classical Cheeger inequality.
In the proofs of Lemmas 5.3 and 6.4, we will need arguments and consequences of the proof of Theorem 3.3 and, therefore, recall the elegant arguments from the proof of the corresponding Lemma 2 in [Oss77].
Recalling the proof of the Cheeger inequality. Since F is compact with piecewise smooth boundary, λ 0 = λ 0 (F ) is the first Dirichlet eigenvalue of F . Let ϕ be the corresponding ground state and set ψ = ϕ 2 . By the Schwarz inequality, we have For regular values t > 0 of ψ, let F t = {ψ ≥ t} and denote by A(t) and L(t) the area and the length of F t and ∂F t = {ψ = t}, respectively. For the null set of singular values of ψ, set A(t) = L(t) = 0. The coarea formula gives On the other hand, since F ψ computes the volume of the domain Combining (3.2) and (3.5), we get λ 0 ≥ h 2 /4 as asserted.
We start with a version of Theorem 1.6 for surfaces with (possibly empty) boundary.
2) If κ > 0 and S is orientable, then 3) If κ > 0 and S is non-orientable, then Proof. For a closed disc D in S, Corollary 2.3.1 implies that (4.1) Suppose now that A is a closed annulus in S. Suppose first that the boundary circles of A are null-homotopic in S. Then by the Schoenflies theorem (see also [BMM16, Then |∂F | ≤ |∂A| and |F | ≥ |A|. Using Corollary 2.3.1 again, we get that (4.1) also holds for A in place of D.
Assume now that the boundary circles of A are not null-homotopic in S. By Corollary 2.3.3 and the statement after it, we have where l(A) denotes the length of a shortest curve in the free homotopy class in A of the two boundary circles of A. Since the boundary circles of A are not homotopic to zero in S, we have l(A) ≥ sys(S). Hence If C is a cross cap in S, then S is not orientable. Now the soul of C is not homotopic to zero in S and the fundamental group of S is torsion free. Since the boundary circle ∂C of C is freely homotopic to the soul of C, run twice, we get that ∂C is not homotopic to zero in S. In particular, we always have |∂C| ≥ sys(S). If κ ≤ 0, then a shortest curve in S in the free homotopy class of the soul of C, run twice, is a shortest curve in S in the free homotopy class of the boundary circle of C. Hence |∂C| ≥ 2 sys(S) if κ ≤ 0. We conclude that (4.2) also holds for C in place of A if κ ≤ 0. In the general case, |∂C| ≥ sys(S) implies a modified version of (4.2) with C in place of A, where the factor 4 on the right hand side is replaced by 1. Now the assertions of Theorem 4.1 follows from the Cheeger inequality (Theorem 3.3) in combination with Proposition 3.2, (4.1), and (4.2) or the modified version of (4.2), respectively.
Proof of Theorem 1.6. It remains to show that sys(S) 2 /|S| ≤ π if S is closed with curvature K ≤ 0. In fact, in that case, the injectivity radius of S is sys(S)/2. Then the exponential map exp p at any point p ∈ S is a diffeomorphism from the disc of radius sys(S)/2 in T p S to its image, the metric ball B = B(p, sys(S)/2) about p in S. By comparison with the flat case, we get |B| ≥ π sys(S) 2 /4 and therefore Remarks 4.2.
1) If S is a compact and connected surface with non-empty boundary, then S contains a finite graph G in its interior which is a deformation retract of S. Given a Riemannian metric on S, a sufficiently small tubular neighborhood T of G in S is a Riemannian surface diffeomorphic to S with sys(T ) ≥ sys(S) and with arbitrarily small area. Moreover, any upper bound on the curvature persists. In other words, we cannot expect to remove the minimum on the right hand side of the estimates in Theorem 4.1. Note also that the right hand side of the inequalities in 2) and 3) of Theorem 4.1 is positive if and only if |S| < 4π/κ, that is, if and only if |S| is smaller than the area of the sphere of constant curvature κ > 0. 2) In [Gro83, Corollary 5.2.B], Gromov shows that sys(S) 2 /|S| 2 ≤ 4/3 for any closed Riemannian surface. The point is, of course, that his estimate is curvature free. His work in [Gro83] also implies that Proof of Theorem 1.2 in the case χ(S) ≥ 0. In view of Proposition 1.5, it remains to show that Λ(S) > λ 0 (S) in the case where S is a torus or a Klein bottle. Then S admits a flat background metric h which is conformal to the given metric g of S. Theorem 1.6 applies to h and shows that where (S, h) denotes S, endowed with the metric h. Furthermore, since we are in the case of surfaces, the Dirichlet integral of smooth functions is invariant under conformal changes; that is, we have GAFA Since S is compact, there is a constant α ≥ 1 such that for all tangent vectors v of S. Using (4.3) and (4.4), we obtain On the other hand, the fundamental group of S is amenable and hence λ 0 (S) = λ 0 (S) by [Bro85, Theorem 1]. Now S is a torus or a Klein bottle, hence λ 0 (S) = 0.
Proof of Theorem 1.8. Suppose first that S is orientable, that is, that S = S g for some g ≥ 2, and let c be a systolic geodesic on S. Then by [Bus10, Theorem 4.3.2], the tubular neighborhood T of c of width is an open annulus. Since c is essential, T is incompressible. Note also that T can be exhausted by incompressible compact annuli with smooth boundary. In particular, for any r < w 2 , the closed metric ballB(p, r) of radius r about a point p on c is contained in an incompressible compact annulus A r ⊆ T with smooth boundary.
Since B(p, r) ⊆ A r , we may use Theorem 1.1 and the first displayed formula on page 294 of [Che75] to conclude that By the definition of Λ(S), we have Λ(S) ≤ λ 0 (A r ) for any r as above. Hence the claim of Theorem 1.8 follows in the case S = S g .
Suppose now that S is not orientable. Let Or(S) → S be the orientation covering of S and c be a systolic geodesic on S. There are two cases: In both cases, 1) and 2), we can now conclude the proof of the claim of Theorem 1.8 as in the case S = S g .
Remark 4.3. The arguments in the proof of Theorem 1.8 also show that diam S ≥ w with w = w 2 and w = w 1 , respectively. Hence we get [Che75]. In view of Λ(S) < λ −χ(S) , this gives another, but weaker upper bound for Λ(S).

On the Ground State
Throughout this section, we let F be a compact Riemannian surface with smooth boundary ∂F = ∅ and ϕ be the ground state of F . We also set ψ = ϕ 2 and let F t = {ψ ≥ t}. Note that ψ = 1.
By the Hopf boundary lemma [GT83, Lemma 3.4], ϕ does not have critical points on ∂F . Moreover, since ϕ > 0 in the interiorF of F , a point inF is critical for ψ if and only if it is critical for ϕ. All points of ∂F are critical for ψ.
In our first result, we elaborate on the argument from the middle of page 549 in [Oss77].
Lemma 5.1. Let 0 < t < max ψ be a regular value of ψ. Then F t is a compact subsurface ofF such that the boundary of each component of F \ F t has at least one boundary circle inF and contains at least one boundary circle of F .
Proof. Since t is a regular value of ψ with 0 < t < max ψ, F t is a compact subsurface ofF with smooth boundary. If the boundary of a component C of F \ F t would not contain a boundary circle of F , then ϕ would be a non-constant superharmonic function on C which attains its maximum √ t along ∂F t , a contradiction. Clearly, the boundary of C must have at least one boundary circle in ∂F t ⊆F . Proof. Suppose that a component D of F \ C would be a disc or a cross cap. Then D \ F t would consist of components of F \ F t with boundary inF , a contradiction to Lemma 5.1. Substituting C for F , the rest of the proof of Proposition 5.2 is now more or less the same as that of Proposition 3.2.
As in Theorem 2.1, we denote by the sum of the lengths of the shortest loops in the free homotopy classes (in F ) of the boundary circles of F . Furthermore, we let Λ (F ) = inf λ 0 (F ), where the infimum is taken over all incompressible compact and connected subsurfaces F ofF with smooth boundary and Euler characteristic χ(F ) > χ(F ).
Proof. Since the quantities involved in the lemma vary continuously with respect to variations of the metric (in the C 0 -topology), we may assume, by Theorem 8 in [Uhl76], that ϕ is a Morse function. Then the critical points of ψ inF are nondegenerate. Moreover, since ϕ does not have critical points on ∂F , F \F t is a collared neighborhood of ∂F , consisting of annuli about the boundary circles of F , for all sufficiently small t > 0. On the other hand, for t < max ψ suffciently close to max ψ, F t is a union of embedded discs, one for each maximum point of ψ. Hence the topology of F t undergoes changes as t increases from 0 to max ψ.
Since ϕ is a Morse function, ψ has only finitely many critical points inF . By Lemma 5.1, ψ does not have local minima inF . Hence critical points of ψ inF are saddle points and local maxima.
Let 0 = β 0 < · · · < β m = max ψ be the finite sequence of critical values of ψ and choose ε > 0 with ε < min{β i+1 − β i }. In a first step, we select now a critical value β = β i according to specific requirements.
By Proposition 5.2, each component C of F β1+ε has Euler characteristic χ(C) ≥ χ(F ). Therefore there are two cases. Either each component C of F β1+ε has Euler characteristic χ(C) > χ(F ). Then we set β = β 1 . Or else there is a component C with χ(C) = χ(F ). Then F \C is a collared neighborhood of ∂F , consisting of annuli about the boundary circles of F , by Proposition 5.2. In that case, by Lemma 5.1, the other components of F β1+ε are discs contained in these annuli.
We assume that we are in the second case and consider the second critical value β 2 . By Proposition 5.2, there are again two cases. Either each component C of F β2+ε has Euler characteristic χ(C) > χ(F ); then we set β = β 2 . Or else there is a component C of F β2+ε with χ(C) = χ(F ). Then F \ C is a collared neighborhood of ∂F consisting of annuli about the boundary circles of F . In the latter case, we pass on to the next critical value β 3 . Since χ(F ) ≤ 0, we will eventually arrive at a first critical value β = β i with the property that the complement of a component of F β−ε is a collared neighborhood of ∂F consisting of annuli about the boundary circles of F and such that each component C of F β+ε has Euler characteristic χ(C) > χ(F ). Note that this property then holds for all sufficiently small ε > 0 since β is the only critical value of ϕ in (β i−1 , β i+1 ). It follows that for any regular value 0 < t < β of ψ, F t has a component C such that F \ C is a collared neighborhood of ∂F consisting of annuli about the boundary circles of F . In particular, |∂F t | ≥ for all such t. Using (3.2) and (3.3), we obtain where L(t) denotes the length of ∂F t .

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For ε > 0 as above, the smooth function ϕ ε = ϕ − √ β + ε is smooth on F β+ε , vanishes on ∂F β+ε and satisfies Now the first term on the right hand side of (5.2) satisfies For the second term on the right hand side of (5.2), we have by the Schwarz inequality and the Peter and Paul principle. Combining (5.2), (5.3), and (5.4) and using that the L 2 -norm of ϕ is one and that 2 − 1/δ < 0, we obtain For the Rayleigh quotient of ϕ ε , we get Since F β+ε is a disjoint union of incompressible compact and connected subsurfaces F with smooth boundary and χ(F ) > χ(F ), we also have R(ϕ ε ) ≥ inf Λ (F ).

On the Ground State (Continued)
Let S be a complete and connected Riemannian surface of finite type with χ(S) < 0 and λ 0 (S) < λ ess (S). Since χ(S) < 0, S carries complete hyperbolic metrics. Using a decomposition of S into a finite number of pairs of pants, it is clear that we may choose such a metric h such that the connected components of a neighborhood of the ends of S is a finite union of hyperbolic funnels, that is, cylinders of the form (−1, ∞) × R/Z with metric dr 2 + cosh(r) 2 dϑ 2 .
Then the curves r = 0 are closed h-geodesics of length 1. The original metric of S will be denoted by g.
We fix a smooth and proper function from S to [0, ∞), which agrees outside a compact set with the coordinates r in each of the ends. By abuse of notation, we denote this function by r. Choose an increasing sequence 0 < r 0 < r 1 < r 2 < · · · → ∞. Then the subsurfaces is an exhaustion of S. By choosing the sequence of r i suitably, we may assume that a) there exist cutoff functions η i : S → [0, 1] with η i = 1 on K i , η i = 0 outside K i+1 , and |∇η i | 2 ≤ 1/i, b) λ 0 (S) < λ 0 (S \ K 0 ), where we note that λ 0 (S \ K i ) < λ 0 (S \ K i+1 ) · · · → λ ess (S).
In the case where S is compact, we have K i = S for all i and part of the following discussion becomes trivial.
We now let F be a compact subsurface of S with smooth boundary ∂F = ∅. As in Sect. 5, we denote by ϕ the ground state of F and let F t = {ϕ 2 ≥ t}.
Lemma 6.1. For a subset R ⊆ (0, max ϕ 2 ) of full measure, F t is a smooth subsurface ofF such that ∂F t = {ϕ = t} and ∂K i intersect transversally for all i.
Proof. Since ϕ is smooth up to the boundary of F and has no critical points on ∂F , there is a smooth extensionφ of ϕ to S such thatφ is strictly negative on S \ F . The restrictionφ 0 ofφ to the union of the curves {r = r i } is then smooth, and hence there is a set R 0 ⊆ R of full measure such that any t ∈ R 0 is a regular value ofφ 0 . Note that ∇φ is not perpendicular to the curve {r = r i } at points p ∈ {r = r i } with ϕ(p) ∈ R 0 . On the other hand, ∇φ is perpendicular to ∂F t for any regular value t of ϕ 2 in (0, max ϕ 2 ). Therefore the intersection R of R 0 with the set of regular values of ϕ 2 in (0, max ϕ 2 ) satisfies the required assertions.

Lemma 6.2. For any t ∈ R, the intersection F t ∩ K i is a subsurface of F with piecewise smooth boundary and any connected component C of F t ∩ K i is incompressible in F . In particular, we have χ(C) ≥ χ(F ).
Proof. For any t ∈ R, ∂F t = {ϕ = t} and ∂K i intersect transversally for all i, and then F t ∩ K i is a subsurface of F with piecewise smooth boundary. Since S \ K i is a cylindrical neighborhood of the ends of S, a disc in S has to be contained in K i if its boundary is in K i . Hence the components of Since 0 < θ < 1, we conclude that Now for i 0 sufficiently large, the right hand side is smaller than ε for all i ≥ i 0 − 1. For any i ≥ i 0 , we then have There is a sequence of constants 1 ≤ α 0 ≤ α 1 ≤ · · · such that for all tangent vectors v of S with foot point in K i , where no index and index h indicate measurement with respect to g and h, respectively. Over K i , the area elements da of g and da h of h are then estimated by with corresponding inequalities for the areas of measurable subsets and for integrals of non-negative measurable functions. Let now again ϕ be the ground state of F , t ∈ R, and F t = {ϕ 2 ≥ t}. In our next result, we estimate the inradius of F t ∩ K i for sufficiently large i.
Lemma 6.4. Let F be a disc, an annulus, or a cross cap. Assume that λ 0 (F ) ≤ θλ 0 (S \ K 0 ) for some 0 < θ < 1 and let δ > 0. Then there is an integer Proof. In a first step, we estimate the Rayleigh quotient of η i ϕ. Computing as in the proof of Lemma 6.3, we have where i 0 is taken from Lemma 6.3. Therefore where we may assume that i 1 (θ, δ) ≥ i 0 (θ, δ). In a second step, we follow the proof of Cheeger's inequality, Theorem 3.3. Computing as in (3.1), we get

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By the coarea formula and (6.3) and since supp η i ⊆ K i+1 and η i = 1 on K i , we have Here we note that, for the integration, it suffices to consider s ∈ R which are also regular for η 2 i ϕ 2 . Then {ϕ 2 = s} meets ∂K i transversally and, therefore, {η 2 i ϕ 2 = s} ∩ (F \ K i ) consists of arcs a j connecting their corresponding end points on {ϕ 2 = s} ∩ ∂K i . Replacing the a j by the corresponding segments b j on ∂K i , we obtain the boundary of the subsurface F s ∩K i . Now |b j | h ≤ |a j | h by the choice of the hyperbolic metric h on S and since the numbers r i defining the K i are positive. Hence we have for any s ∈ R. This implies Furthermore, by Lemma 6.2, the connected components of F s ∩ K i are subsurfaces of F with piecewise smooth boundary and have non-negative Euler characteristic for any s ∈ R. Therefore we get (6.10) Lemma 6.4 follows now from combining (6.5) -(6.10).
Lemmas 5.3 and 6.4 will lead to the apriori estimates in Lemma 7.1 and in the proof of Lemma 7.4, which are essential in the proof of Theorem 1.1.

Qualititative Estimates of Λ(S)
In this section, we prove Theorem 1.2 in the case χ(S) < 0. Throughout, we let S be a complete and connected Riemannian surface of finite type and set where the infimum is taken over all embedded closed discs D, incompressible annuli A, and cross caps C inS with smooth boundary, respectively. As we will explain in Sect. 8.1, we have Λ D (S) ≥ Λ A (S) and Λ(S) = inf{Λ A (S), Λ C (S)} if the fundamental group of S is infinite. Nevertheless, since the case of discs reveals an essential idea of the proof and since we will need the estimate anyway, we include the discussion of Λ D (S). We fix a hyperbolic metric h on S as in Sect. 6 and denote by g the original Riemannian metric of S. If not otherwise mentioned, statements refer to g and not to h.
We will use the setup and notation from the previous section. The following assertion is an immediate consequence of Lemma 6.4.
We now discuss the cases of discs and annuli separately. Proof. Suppose that there is a sequence of discs D n in S with smooth boundary such that λ 0 (D n ) → λ 0 (S). Let ϕ n : D n → R be the positive λ 0 (D n )-Dirichlet eigenfunction with ||ϕ n || 2 = 1. By passing to a subsequence if necessary, we may assume that λ 0 (D n ) ≤ θλ 0 (S \ K 0 ) for some 0 < θ < 1. By Lemma 7.1 and up to passing to a subsequence, there are positive constants ε 0 and ρ 0 and a point ,1/2) . Fix a pointx 0 ∈S above x 0 . Then there is a unique liftD n of D n toS containing x 0 such thatD n → D n is a diffeomorphism (including the boundary). Thus we may also lift ϕ n toφ n onD n and extendφ n to a functionφ n onS n by settingφ n = 0 onS \D n . Since the boundary ofD n is smooth andφ n is smooth onD n , it follows thatφ n ∈ H 1 0 (S) with H 1 -norm ||φ n || H 1 = ||ϕ n || 2 H 1 = λ 0 (D n ) + 1, where we use Green's formula for the second equality. In particular, up to extracting a subsequence, we have weak convergencẽ ϕ n φ ∈ H 1 0 (S) with ||φ|| H 1 ≤ lim inf ||ϕ n || H 1 . Up to extracting a further subsequence, the sequence ofφ n converges uniformly in any C k -norm in B(x 0 , ρ 0 ), by Theorem 8.10 in [GT83]. In particularφ 2 ≥ ε 0 on B(x 0 , ρ 0 ).
By Theorem 1 of [AFLR07], we may approximate the distance function d 0 tox 0 inS by a smooth function u onS such that |u − d 0 | ≤ 1 and |∇u| ≤ 2. Then the sublevels B(r) = {u ≤ r} form an exhaustion ofS by compact subsets. Clearly, Furthermore, up to passing to a subsequence, we have weak convergencẽ and strong convergenceφ n | B(r) →φ| B(r) in L 2 (B(r)). Hence For any regular value r of u such that ∂B(r) intersects ∂D n transversally we have where ν = ∇u/|∇u| is the outward unit vector field along ∂B(r). Clearly, the second term on the right satisfies ∂B(r)φ n ∇φ n , ν ≤ ∂B(r) (|φ n | 2 + |∇φ n | 2 ).
Let now ε > 0 be given. Then there is a sequence of integers k m → ∞ such that, for each m, there is a subsequence of n → ∞ with where K(r) = B(r) \ B(r − 1). If this would not be the case, there would be some ε > 0 and a positive integer m such that for all integers k ≥ m there is an integer l such that, for all integers n ≥ l, We thus find, for any fixed M , an index N such that (7.3) holds for k = m, . . . , m+M and n ≥ N. If we choose M such that Mε > λ 0 (S) + 1, we get a contradiction. By the coarea formula, we have Since |∇u| ≤ 2, we then get, for n as in (7.2), that there is a regular value r n ∈ (k m − 1, k m ) of u such that where we may also assume that ∂B(r n ) intersects ∂D n transversally. We obtain It follows that for all sufficiently large m. In conclusion,

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Sinceφ ∈ H 1 0 (S), this implies thatφ is an eigenfunction of the Laplacian with eigenvalue λ 0 (S).
Nowφ is non-zero on B(x 0 , ρ 0 ). On the other hand, by the definition of the liftsφ n ,φ vanishes on any other preimages B(x, ρ 0 ) of B(x 0 , ρ 0 ) under the covering projectionS → S. Now the fundamental group of S is not trivial, hence there are such preimagesx ∈S. Thus we arrive at a contradiction to the unique continuation property for eigenfunctions of Laplacians [Aro57]. To prove Theorem 7.3, we assume the contrary and let (A n ) be a sequence of incompressible annuli in S with smooth boundary such that λ 0 (A n ) → λ 0 (S). We may assume that for all n and some fixed constants δ, θ ∈ (0, 1), by invoking Theorem 7.2 and that λ 0 (S) < λ 0 (S \ K 0 ) by the choice of K 0 in (6.2). By deforming the A n (slightly), we may also assume that ∂A n and ∂K i intersect transversally (7.5) for all n and i. Then the intersections A n ∩ K i and A n ∩ (S \K i ) are incompressible subsurfaces of S with piecewise smooth boundary.
Lemma 7.4. By passing to a subsequence, we may assume that 1) all A n are isotopic in S and, for each i > i 1 = i 1 (θ, δ), exactly one component A n of A n ∩ K i is an annulus (topologically) isotopic to A n ; 2) there is a constant 0 > 0 such that the free homotopy classes of the boundary curves of A n in A n contain curves of length at most 0 with respect to g and h; 3) there are x 0 ∈ S and ρ, ε > 0 such that B(x 0 , ρ) ⊆ A n and such that the ground states ϕ n of A n satisfy ϕ n ≥ ε on B(x 0 , ρ).
Proof. The connected components of A n ∩ ∂K i consist of embedded segments connecting two boundary points of A n and of embedded circles in the interior of A n . By the Schoenflies theorem and the topology of S, there are the following two possibilities: a) All connected components of A n ∩ K i are discs.
b) The connected components of A n ∩ K i consist of one annulus A n (topologically) isotopic to A n and discs. Now let i > i 1 = i 1 (θ, δ). Then η i−1 ϕ n is a non-zero smooth function on A n with compact support in A n ∩ K i and Rayleigh quotient by (6.5) and (7.4). Hence, if all the connected components of A n ∩ K i were discs, we would have R(η i−1 ϕ n ) ≥ Λ D (S). Therefore only case b) can occur. Then the Rayleigh quotients of η i−1 ϕ n on the discs of A n ∩ K i (on which η i−1 ϕ n does not vanish) must be at least Λ D (S). Hence the Rayleigh quotient of η i−1 ϕ n on A n must be less than λ 0 (A n ) + δ. In particular, λ 0 (A n ) ≤ λ 0 (A n ) + δ.
Note that one of the boundary circles of A n may only be piecewise smooth. Therefore A n may only be topologically isotopic to A n . Now we let i = i 1 +1. Since A n ⊆ K i , we have the uniform area bound |A n | ≤ |K i |. This together with the above estimate on λ 0 (A n ) and Lemma 5.3 implies that the length of shortest curves in A n , which are freely homotopic to the boundary circles of A n in A n , is uniformly bounded with respect to g. Then their length is also uniformly bounded with respect to h, by (6.3). In particular, there are only finitely many isotopy types of A n and, therefore also, of A n . Therefore we may pass to a subsequence so that all of them are isotopic. By Lemma 7.1 and since K i is compact, we may pass to a further subsequence so that all A n ∩ K i , and hence also all A n , contain a geodesic ball B(x 0 , ρ) such that ϕ n ≥ ε on B(x 0 , ρ) as claimed.
Proof of Theorem 7.3. By passing to a subsequence, we may assume that the sequence of A n satisfies all the properties from (7.4), (7.5), and Lemma 7.4.
Choose a shortest (with respect to h) closed h-geodesic c in the free homotopy class in S of a generator of the fundamental group of A n . This is possible since the ends of S are hyperbolic funnels with respect to h. Note that c does not depend on n since all A n are isotopic. We letŜ be the cyclic subcover ofS to which c lifts as a closedĥ-geodesicĉ, whereĥ denotes the lift of h toŜ. Note that all annuli A n are isotopic to a small tubular neighbourhood of c. Lifting the corresponding isotopies, we get liftsÂ n ⊆Ŝ of the annuli A n . Note thatÂ n is the unique compact component of π −1 (A n ) and that π :Â n → A n is a diffeomorphism which is isometric with respect to g and h and their respective liftsĝ andĥ.
Denote byx n the lift of x 0 which is contained inÂ n . Ifx n stays at bounded distance toĉ, the arguments for the case of discs in the proof of Theorem 7.2 apply again and lead to a contradiction since the fundamental group of S is not cyclic and λ 0 (Ŝ) = λ 0 (S).
Since the h-length of shortest curves, c n , in the free homotopy class of the boundary curves of A n in A n is bounded by 0 , there is a constant r 0 > 0 such that the liftŝ c n of c n toÂ n are contained in the region {|r| ≤ r 0 } ofŜ.
Letφ n be the lift of ϕ n toÂ n . Thenφ n is the ground state ofÂ n with respect toĝ and we have Since supp η j ⊆ K j+1 , the h-area of supp η j is bounded by the h-area |K j+1 | h of K j+1 . Now choose an r 1 > r 0 such that theĥ-area of either of the regions −r 1 ≤ r ≤ −r 0 and r 0 ≤ r ≤ r 1 inŜ is larger than |K j+1 | h . Finally, choose a cut off function χ onŜ such that χ = 0 on {|r| ≤ r 1 }, χ = 1 on {|r| ≥ r 2 } for some r 2 > r 1 , and such that |∇χ| 2 < δε 2 vol B(x 0 , ρ).
Computing as in the proofs of Lemmas 6.3 and 6.4 and with η = η j • π, we get where we use (7.7) and (7.8) for the passage to the last line. Sincex n → ∞ and B(x 0 , ρ) ⊆ K j by the choice of j, χ = η = 1 on B(x n , ρ) for all sufficiently large n. Combining (7.6) and (7.9), we then get R(χηφ n ) ≤ λ 0 (A n ) + 4δ < Λ D (S). (7.10) On the other hand, supp(χηφ n ) is contained in the lift B n of A n ∩ K j+1 toÂ n , intersected with {|r| ≥ r 1 }. Now the h-area of B n is bounded by |K j+1 | h and B n contains c n . Hence B n does not contain loops freely homotopic to c n in the region {|r| ≥ r 1 } ofŜ since otherwise, by the uinqueness of A n , it would contain one of the regions −r 1 ≤ r ≤ −r 0 or r 0 ≤ r ≤ r 1 . Hence B n ∩ {|r| ≥ r 1 } is a union of discs and, therefore, the Rayleigh quotient of χηφ n has to be at least Λ D (S), a contradiction to (7.10). It follows that the sequence ofx n is bounded.
Proof of Theorem 1.2 in the case χ(S) < 0. By Theorems 7.2 and 7.3, we have for any complete and connected Riemannian surface of finite type with χ(S) < 0. This implies the assertion of Theorem 1.2 in the case where S is orientable since then S does not contain cross caps. Assume now that S is not orientable and let Or(S) be the orientation covering space of S. Let C be a cross cap in S. Then the lift of C to Or(S) is an annulus A in Or(S) with λ 0 (C) ≥ λ 0 (A) ≥ Λ A (Or(S)).

Remarks, Examples, and Questions
In this section, we collect some loose ends. We start with a comment which gives another argument for calling Λ the analytic systole.

On the definition of Λ.
For complete and connected surfaces S with infinite fundamental group, an equivalent definition of the analytic systole is Λ(S) = inf F λ 0 (F ), where the infimum is taken over incompressible annuli and cross caps F with smooth boundary in S: a) For any disc D with smooth boundary and any free homotopy class [c] of closed curves in S, there is an annulus A with smooth boundary in S containing D whose soul belongs to [c]. If [c] is non-trivial, then A is incompressible. Moreover, λ 0 (A) ≤ λ 0 (D) by the domain monotonicity of λ 0 . b) For any compressible annulus A in S with smooth boundary, there is a disc D in S whose boundary ∂D is one of the boundary circles of A such that A ∪ D is a disc in S with smooth boundary, by the Schoenflies theorem, and then a) applies. c) Cross caps only occur in the case where S is not orientable. The soul of a cross cap C in S is not homotopic to zero in S. Since the fundamental group of S is torsion free, we get that C is incompressible.
However, in view of our previous articles [BMM16,BMM17], it is more natural to include discs into the definition. Moreover, it is important in our analysis to handle the case of discs separately.
8.2 On the essential spectrum. The following result, Proposition 3.6 from [BMM17] formulated for surfaces, is probably folklore. It shows that the essential spectrum of the Laplacian only depends on the geometry of the underlying surface S at infinity and that the essential spectrum of the Laplacian is empty if S is compact.
Proposition 8.1. For a complete Riemannian surface S with compact boundary (possibly empty), λ ∈ R belongs to the essential spectrum of Δ if and only if there is a Weyl sequence for λ, that is, a sequence ϕ n of smooth functions on S with compact support such that 1) for any compact K ⊆ S, supp ϕ n ∩ K = ∅ for all sufficiently large n; 2) lim sup n→∞ ϕ n 2 > 0 and lim n→∞ Δϕ n − λϕ n 2 = 0.
In work of Arne Persson and of Richard Froese and Peter Hislop, the bottom of the essential spectrum of Laplacians and more general operators has been characterized in the sense of Proposition 8.1 or, more specifically, in the sense of ( where K runs over the compact subsets of S, ordered by inclusion. Corollary 8.3. If S is a compact Riemannian surface, then the spectrum of S is discrete; that is, λ ess (S) = ∞.

Surfaces with cyclic fundamental group.
In the (unnumbered) lemma on page 551 of [Oss77], Osserman establishes the following result in the special case of domains in the Euclidean plane.
The arguments in [Oss77] also apply to the more general situation of Lemma 8.4 and therefore we skip its proof.
and such that f is monotonically decreasing in t. Since f t = 1 on K n for t ≤ 1/n and the K n exhaust S, f can be smoothly extended to [0, 1] × S by setting f 0 = 1.
Let g t = f t g. Then g t is a smooth family of conformal metrics on S and is a continuous curve of metrics with respect to the uniform distance. For t ≤ 1/n, we have Λ(S, g t ) ≤ λ 0 (F n , g t ) = λ 0 (F n , g) < e 1/n+1 λ ess (S, g).
It remains to show that the set of metrics g on S that satisfy the strict inequality Λ(S, g) > λ 0 (S,g) is an open set in the uniform C ∞ topology. This follows from the fact that two metrics that are close to each other in the uniform C ∞ topology are quasi-isometric by a quasi-isometry with quasi-isometry constant close to 1.
Remarks 8.7. 1) Any two complete Riemannian metrics g 0 , g 1 on S of finite uniform distance are quasi-isometric. This implies, that there is a constant C > 0 depending on the distance of g 0 to g 1 such that C −1 λ ess (S, g 0 ) ≤ λ ess (S, g 1 ) ≤ Cλ ess (S, g 0 ).
2) The above construction can be extended to get metrics with λ ess (S, g t ) = e t λ ess (S, g) for all t ≥ 0. 3) If S is non-compact, any complete hyperbolic metric on S satisfies λ 0 (S,g) = λ ess (S, g) = 1/4. 4) If S is non-compact, any complete Riemannian metric on S, which is in zeroth order asymptotic to a flat cylinder R/LZ × [0, ∞), has λ 0 (S, g) = λ ess (S, g) = 0. where F runs over all subsurfaces of S isotopic to T . The analytic systole is an infimum over such constants. It is interesting to ask for estimates of Λ T (S). The infimum is probably achieved by degenerate F , where ∂F is mapped onto a graph Γ in S such that S \ Γ is diffeomorphic to the interior of T . In fact, for any F isotopic to T , there is a graph Γ in S such that F ⊆ S \ Γ and such that S \ Γ is isotopic to the interior of F . Hence, by domain monotonicity, Λ T (S) is the infimum over all λ 0 (S \ Γ), where Γ runs through such graphs.
What are the optimal graphs? This circle of problems is related to the work of Helffer, Hoffman-Ostenhof, and Terracini [HHT09]. It would be interesting to see what other implications this equality has on the metric. If we rescale the metric by a function f : S → (0, 1] which is 1 outside a compact set, then λ 0 (S) can only increase, while λ ess (S) remains unaffected. Using our main theorem we can see that the new metric also satisfy the above equality. Hence one can not have a rigidity among all smooth metrics. Also, observe that points 1) and 4) of Remarks 8.7 imply that there is no such rigidity for metrics with λ ess (S) = 0. 4) (Higher dimensions) All our definitions extend in a natural way to higher dimensional manifolds. For instance, we may define the analytic systole of an n-dimensional manifold M by Λ(M ) = inf Ω λ 0 (Ω), where Ω runs over all tubular neighborhoods about essential simple loops in M . By (A.1), we have Λ(M ) ≥ λ 0 (M ). One may ask whether the strict inequality holds true under reasonable assumptions on M . Our methods seem to be too weak to adress this question.