The local maxima of maximal injectivity radius among hyperbolic surfaces

Abstract

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 T g,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 : T g,n → R + that assigns to S ∈ T g,n its maximal injectivity radius. The injectivity radius of S at x, injrad x (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 injrad x (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 T g,n :Theorem 0.1. For S ∈ T g,n , the function max attains a local maximum at S if and only if for each x ∈ S such that injrad x (S) = max (S), each edge of the Delaunay tessellation of (S, x) has length 2injrad x (S) and each face is a triangle or monogon.Here for a hyperbolic surface S with locally isometric universal cover π : H 2 → S, and x ∈ S, the Delaunay tessellation of (S, x) is the projection to S of the Delaunay tessellation of π −1 (x) ⊂ H 2 , 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 π 1 S acting on H 2 by covering transformations.Theorem 5.11 of the author's previous paper [3] 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 T g,n are global maxima.This contrasts the behavior of syst, the function on T g,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 T g,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.)The referee has sketched a direct argument to show that max attains a global maximum on T g,n . (This is also sketched in the preprint [7], and I prove a somewhat more general fact as Proposition 4.3 of [5].) Together with this observation, Theorem 0.1 gives an alternative proof of Theorem 5.11 of [3], which is not completely independent of the results of [3] 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 1 arXiv:1506.08080v2 [math.GT]

The Delaunay tessellation in hyperbolic space

“…Our results are complementary to [13] and generalise the main case of [7]. The main new case here, which is an important tool in our subsequent works [11] and [10], covers finite subsets of finite-volume non-compact hyperbolic manifolds. Compared to previous work this case exhibits substantial differences in the nature of the cells produced and of the decomposition's "geometric duality" relationship with the Voronoi tessellation.…”

Bounds for several-disk packings of hyperbolic surfaces

“…In particular, can this occur if k divides both 6χ and n, where χ is the Euler characteristic of Σ and n its number of cusps? Gendulphe answered this "no" for k = 1 [17], extending my result on the orientable case [11].…”

“…The main ingredient in the proof of Proposition 4.3 is Lemma 4.7 below. It was suggested by a referee for [11], who sketched the proof I give here. Lemma 4.7.…”

“…Key to the proof of Proposition 4.3 is Lemma 4.7, which asserts that the thick part of a surface with a short geodesic can be inflated while increasing the length of the geodesic. Proposition 4.3 and its proof were suggested by a referee for [11], and sketched by Gendulphe [17], in the case k = 1.…”