Nonpositively curved surfaces are Loewner

We show that every closed nonpositively curved surface satisfies Loewner's systolic inequality. The proof relies on a combination of the Gauss-Bonnet formula with an averaging argument using the invariance of the Liouville measure under the geodesic flow. This enables us to find a disk with large total curvature around its center yielding a large area.


Introduction
The systole of a closed nonsimply connected surface M endowed with a Riemannian metric, denoted by sys(M ), is defined as the length of the shortest noncontractible loop of M .It is attained by the length of a noncontractible closed geodesic.The systolic area of M is defined as σ(M ) = area(M ) sys(M ) 2 .
We will say that M is Loewner if its systolic area satisfies The first systolic inequality, due to Loewner, asserts that every metric on the torus T 2 is Loewner; see [13].By [8], all surfaces with a Riemannian metric in a hyperelliptic conformal class are Loewner.In particular, every genus two surface is Loewner since every conformal class in genus two is hyperelliptic.By [7], surfaces of genus at least 20 are Loewner.Pushing this technique further, Li and Su announce that this still holds true for surfaces of genus 18 and 19, and even for surfaces of genus ≥ 11 when one restricts to nonpositively curved metrics; see [12].For the other cases the problem is still open.
In this article, we resolve this problem for nonpositively curved surfaces of any genus (as well as for nonorientable surfaces) relying on a new approach.
Actually, we prove a stronger statement; see Proposition 3.3.Let M be a closed nonpositively curved surface.For every r ∈ [0, 1  2 sys(M )], there is a ball B(r) ⊆ M of radius r with area B(r In particular, for r = 1 2 sys(M ), we have Now, if M is not a torus, we can show (see Corollary 4.2) that its systolic area is at least which is enough to conclude.
Our strategy differs from previous works.Here is the rough idea.Suppose that M is a genus g surface which is not Loewner.Normalize the metric so that sys(M ) = 1.Since the metric is nonpositively curved, every disk of radius 1  4 has area at least π 16 , which represents at least π 16 / √ 3 2 ≈ 22.6% of the total area.Now, by an averaging argument using the invariance of the Liouville measure under the geodesic flow, one should be able to find a disk D of radius 1  4 with at least 22.6% of the total curvature.The disk D + of radius 1  2 centered at the same point has a lot of (negative) curvature around its center, namely in D. This should force the disk D + to have a lot of area contradicting the √ 3 2 -bound on the surface area.At implementation level, the existence of a curvature-rich disk relies on an integral-geometric formula relating the weighted average of the curvature K on tangent disks in T M with the Euler characteristic and a comparison result between this weighted average and another weighted average of K on metric disks in M .The area lower bound on this curvature-rich disk follows from an expression relating the area of this disk with the previous weighted average of K on the same disk.

Disks and curvature
Let M be a closed surface of nonpositive Euler characteristic χ(M ) endowed with a Riemannian metric.Let π : U M → M be the canonical projection defined on the unit tangent bundle U M of M .We will sometimes denote a unit tangent vector u ∈ U M by u x ∈ U x M when we want to emphasize its basepoint x = π(u) in M .
The Liouville measure on U M decomposes as where dx is the area measure of M and du x is the canonical length measure of U x M ; see [2, §1.M].Note that the Liouville measure is invariant under the geodesic flow ϕ t : U M → U M of M and that vol(U M ) = 2π area(M ).
Let K be the Gaussian curvature of M .The Gauss-Bonnet formula for a domain D of M with piecewise smooth boundary ∂D can be written where χ(D) is the Euler characteristic of D, κ is the geodesic curvature of ∂D and τ ∂D is the sum of the angular differences of the tangent vectors at the corner points of ∂D.When D = M , the Gauss-Bonnet formula for M takes the form 3) It will be convenient to introduce the function The results of this section can be summarized as follows.First, there is an integral-geometric formula relating a weighted average of K of certain horizontal disks of radii at most r in U M in terms of the Euler characteristic and r.Furthermore, the curvature condition yields an inequality between this weighted average and the average of K of certain disks in M .The inequality between the two averages entails the existence of a curvature-rich disk.
The following function will play a key role in our approach.
Let us compute the integral of F r .
Lemma 2.2.We have Proof.By the Liouville measure decomposition (2.1), we derive Now, by the invariance of the Liouville measure under the geodesic flow, we obtain 12 where the last equality follows from the Gauss-Bonnet formula (2.3).
Let us introduce the following function in connection with the area lower bound in Proposition 3.1.
where B x (ρ) is the ball of radius ρ centered at x.
The functions F r and G r are related through the following comparison result.
Proof.By Gauss' Lemma and since r < 1 2 sys(M ), the exponential map at x in polar coordinates induces a diffeomorphism Since M is nonpositively curved, this map is distance nondecreasing by Rauch's comparison theorem, see [4, §1.11].It follows that t dt du x ≤ dy.
After integration and since K ≤ 0, this implies We can now derive our key estimate.
Proposition 2.5.Assume M is nonpositively curved.Let r ∈ (0, 1 2 sys(M )).Then there exists x 0 ∈ M such that Proof.Taking the average integral ffl M over M in Lemma 2.4 leads to By Lemma 2.2, this yields and the result immediately follows.

Disks and area
In this section, we show that disks with large (negative) curvature have a large area, and proceed to the proof of the main theorem.
We will need the following area lower bound.
Proof.First, approximate the metric on M by a real analytic metric.For this new metric, the component C x (s) of the circle of radius s ≤ r centered at x surrounding its center is a piecewise smooth curve.Moreover, the length function s → L(C x (s)) is differentiable except for a finite number of values of s, and its derivative is given by the first variation formula.Specifically, as long as B x (s) is nonempty, we have for almost every s, where κ is the geodesic curvature of the curve C x (s) and τ s is the sum of the angular difference of the tangent vectors at the corner points of C x (s).By the Gauss-Bonnet formula (2.2) for domains with boundary and since C x (s) bounds a topological disk D x (s), we derive Integrating this relation twice and using the coarea formula lead to area D x (r) ≥ πr 2 − ˆr 0 ˆρ 0 ˆDx(s) K(y) dy ds dρ (3.2) To conclude, simply observe that, when the real analytic metric approaches the initial metric on M , the domains D x (ρ) Hausdorff converge to the balls B x (ρ) (uniformly in ρ).In particular, the area of D x (r) converges to the one of B x (r), and ´Dx(ρ) K(y) dy uniformly converges to ´Bx(ρ) K(y) dy.Hence, area B x (r) ≥ πr 2 − ˆr 0 (r − ρ) ˆBx(ρ) K(y) dy dρ.
Remark 3.2.When the surface M is nonpositively curved, there is an equality in (3.1).Indeed, in this case, we can approximate the metric by a real analytic metric of negative curvature.In this case, the curve C x (s) represents the circle of radius s around x and the domain D x (s) it surrounds coincides with the metric ball of radius s around x.Because of the curvature condition, this ball is a topological disk.In this case, the coarea formula leads to an equality in (3.2) and in the following inequalities.
Putting everything together, we obtain the following area lower bound.
where M is a nonpositively curved surface.Then there exists a ball B(r) of radius r with Proof.By Proposition 2.5, there exists By Proposition 3.1, it follows that area We can conclude the proof of Theorem 1.1 as follows.
Proof of Theorem 1.1.We apply Proposition 3.3 with r = 1 2 sys(M ).Since area(M ) ≥ area B( 12 sys(M )), (3.3) we obtain Expressed in terms of the systolic area, this inequality takes the following form , It follows that for χ(M ) ≤ −1.Hence σ(M ) ≥ π 8 1 + 5 3 ≈ 0.899 and the surface M is Loewner in this case.For the Klein bottle (where χ = 0), the minimal value of the systolic area over nonpositively curved metrics is attained by a square flat metric and is equal to 1. (Alternatively, the minimal value of the systolic area of a Riemannian Klein bottle is equal to 2 √ 2 π ≈ 0.9; see [1].)Hence, every closed nonpositively curved surface is Loewner.

Corollaries
In [8] and [9], we computed the least value of the systolic area over all nonpositively curved metrics on the genus 2 surface Σ 2 and Dyck's surface 3RP 2 .For the genus 3 surface, Calabi [3] presented a CAT(0) piecewise flat metric with the lowest systolic area we know of.See Remark 4.7 below.Combined with the systolic inequality (3.4), this leads to the following result.
Corollary 4.1.We obtain the following data, where the third column represents the minimal value of the systolic area over nonpositively curved metrics.  2 sys(M ) 2 .A closed hyperbolic surface of sufficiently small systole does not contain any Loewner disk.All disks of radius half the systole have area close to that of a Euclidean disk of the same radius.However, the existence of Loewner disks is guaranteed in the following cases.
(2) If Σ g is systolically extremal then it contains a Loewner disk.
Proof.The first part of the corollary is immediate from Proposition 3.3 with r = 1  2 sys(Σ g ).For the second part, consider a rectangle R = [0, 2g − 1] × [0, 1] in the plane.Subdivide each edge of the rectangle into unit length segments.The rectangle R can be seen as a 4g-gon with unit sides labeled circularly a 1 , • • • , a 2g , ā1 , • • • , ā2g .Identifying every side a i of this 4g-gon with its corresponding side āi by a plane translation, we obtain a translation surface which is a piecewise flat genus g surface Σ g with a single conical singularity (which can be approximated by a nonpositively curved metric); see [5, p. 21].It has area 2g − 1 and systole 1.Therefore, every nonpositively curved systolically extremal surface of genus g ≥ 3 has systolic area at most 2g − 1 and contains a Loewner disk by the first part of the corollary.For g = 2, we use the minimal value of the systolic area given in Corollary 4.1.
Remark 4.6.Similarly, one can show that every nonpositively curved systolically extremal surface of negative Euler characteristic contains a Loewner disk.
Remark 4.7.Observe that nonpositively curved surfaces of genus g ≥ 4 have a systolic area greater than the minimal systolic area of a nonpositively curved surface of genus 2, which is equal to 3( √ 2 − 1) ≈ 1.242; see [8].It would be interesting to know if this still holds true in genus 3, showing a monotonicity of the minimal systolic area in terms of the genus for low genera.Observe that the best value of the systolic area we know of in genus 3 is given by a CAT(0) piecewise flat surface in the conformal class of the Klein quartic described by Calabi [3] and is equal to This metric is a critical with respect to some metric variations (see [14]) and might be extremal among all metrics without any curvature assumption.It is not surprising that such a metric is piecewise flat since optimal CAT(0) metrics are flat with finitely many singularities in every genus by [10].Note that Calabi's surface has only a slightly better systolic area than the one given by the triangle hyperbolic surface (2,3,12) of the same genus described by Schmutz [15] (and conjectured extremal among hyperbolic metrics), which is equal to 2π arcsinh(2 + √ 3) 2 ≈ 1.528.
It would be interesting to find the optimal CAT(0) metric in the conformal class of the surface described by Schmutz (or at least find a good approximation by piecewise flat metrics) to see if it has a lower systolic area than the one of Calabi's surface.Similarly, nonpositively curved surfaces nRP 2 with n ≥ 7 have a systolic area greater than the minimal systolic area of a nonpositively curved Dyck's surface 3RP (see [9]) and the question is open for n = 4, 5, 6.