Local maxima of the systole function

Abstract

We construct a sequence of closed hyperbolic surfaces that are local maxima for the systole function in their respective moduli spaces. Their systole is arbitrarily large and the number of examples grows rapidly with the genus. More precisely, for every n 3 there is some positive number L n (growing roughly linearly in n) such that the number of local maxima of the systole function in genus g with systole equal to L n grows super-exponentially in g along an arithmetic sequence of step size n. Many of these surfaces have no orientation-preserving isometries other than the identity and are the first examples of local maxima with this property.

An upperbound for the number of critical points of the systole function on the surface moduli space

“…Fortier-Bouque and Rafi [FBR20] obtained a series of local maxima of the systole function. Letting n = 3 in [FBR20, Theorem B], we have Theorem 3.7 ([FBR20], Theorem B).…”

“…Recently, Fortier-Bourque and Rafi obtained new examples of critical points (including locally maximal points) of systole function [FB19] [FBR20]. Part of their examples have relatively large systole and part of theirs have small indices.…”

“…In an unpublished manuscript [Thu86], Thurston claimed that let X g ⊂ M g be the space consisting of surfaces with filling shortest geodesics, then X g is a spine (deformation retract) of M g (See also [Ji14] [FB19], [FBR20]). Here by filling we mean that the union of the shortest geodesics cut the surfaces into polygonal disks.…”

Non-convexity of extremal length

“…This would imply that the extremal length is not convexoidal for the metric. In fact, it is conjectured in [2] that the extremal length systole attains a local maximum at the regular octahedron punctured at its vertices, which would imply that Voronoï's criterion fails for the extremal length systole, and moreover that extremal length is not convexoidal for any metric in C (see [1,Definition 1.4 and Proposition 1.5] for definitions and justification).…”