Horizontal curves of hyperbolic metrics

Melanie Rupflin,Peter M Topping

doi:10.1007/s00526-018-1386-y

Abstract

We analyse the fine convergence properties of one parameter families of hyperbolic metrics that move always in a horizontal direction, i.e. orthogonal to the action of diffeomorphisms. Such families arise naturally in the study of general curves of metrics on surfaces, and in one of the gradients flows for the harmonic map energy.

Highlights

It has been extensively studied how a general sequence of hyperbolic metrics on a fixed closed oriented surface M can degenerate
A differential geometric form of the Deligne–Mumford compactness Theorem A.4 tells us that modulo diffeomorphisms the closed hyperbolic surfaces converge to a complete, possibly noncompact, hyperbolic surface with cusp ends, after passing to a subsequence
We are concerned with the convergence of a smooth one-parameter family of hyperbolic metrics g(t), t ∈ [0, T ), as t ↑ T

Summary

Introduction

It has been extensively studied how a general sequence of hyperbolic metrics on a fixed closed oriented surface M can degenerate. By the incompleteness of the Weil–Petersson metric on Teichmüller space on surfaces of genus at least 2, the curve having finite length does not rule out degeneration of the metric, in contrast to the analogous situation on a torus. Even for such horizontal curves of finite length, close t < T has got to T the surface will still have infinitely much stretching to do, in general. Theorem 1.2 Let M be a closed oriented surface of genus γ ≥ 2, and suppose g(t) is a smooth horizontal curve in M−1, for t ∈ [0, T ), with finite length L(0) < ∞. L2 , and our results relating the geometry at different times (Lemma 3.2)

Injectivity radius along horizontal curves

Convergence of horizontal curves to noncompact hyperbolic metrics

A Appendix