The Local Maxima of Maximal Injectivity Radius among Hyperbolic Surfaces

The function on the Teichmüller space of complete, orientable, finite-area hyperbolic surfaces of a fixed topological type that assigns to a hyperbolic surface its maximal injectivity radius has no local maxima that are not global maxima. Let Tg,n be the Teichmüller space of complete, orientable, finite-area hyperbolic surfaces of genus g with n cusps. In this paper we begin to analyze the function max : Tg,n → R that assigns to S ∈ Tg,n its maximal injectivity radius. The injectivity radius of S at x, injradx(S), is half the length of the shortest non-constant geodesic arc in S with both endpoints at x. It is not hard to see that injradx(S) varies continuously with x and approaches 0 in the cusps of S, so it attains a maximum on any fixed finite-area hyperbolic surface S. Our main theorem characterizes local maxima of max on Tg,n: Theorem 0.1. For S ∈ Tg,n, the function max attains a local maximum at S if and only if for each x ∈ S such that injradx(S) = max (S), each edge of the Delaunay tessellation of (S, x) has length 2injradx(S) and each face is a triangle or monogon. Here for a hyperbolic surface S with locally isometric universal cover π : H → S, and x ∈ S, the Delaunay tessellation of (S, x) is the projection to S of the Delaunay tessellation of π−1(x) ⊂ H, as defined by an empty circumcircles condition (see Section 2 below). In particular, a monogon is the projection to S of the convex hull of a P -orbit in π−1(x), for a maximal parabolic subgroup P of π1S acting on H by covering transformations. Theorem 5.11 of the author’s previous paper [4] characterized the global maxima of max by a condition equivalent to that of Theorem 0.1, extending work of Bavard [1]. We thus have: Corollary 0.2. All local maxima of max on Tg,n are global maxima. This contrasts the behavior of syst , the function on Tg,n that records the systole, ie. shortest geodesic, length of hyperbolic surfaces: P. Schmutz Schaller proved in [10] that for many g and n, syst has local maxima on Tg,n that are not global maxima. Comparing with syst , which is well-studied, is one motivation for studying max . (Note that for a closed hyperbolic surface S, syst(S) is twice the minimal injectivity radius of S.) As the referee has pointed out, there is a short direct argument to show that max attains a global maximum on Tg,n. (I will give details in [2]; this is also sketched in the preprint [7].) Together with this observation, Theorem 0.1 gives an alternative proof of Theorem 5.11 of [4], which is not completely independent of the results of [4] but uses only some early results from Sections 1 and Section 2.1 there. We prove Theorem 0.1 by describing explicit, injectivity radius-increasing deformations of pointed surfaces (S, x) that do not satisfy its criterion. The deformations are produced