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Path Connectivity of Anosov Metrics on Surfaces

Guilherme Brandão Guglielmo, R. Ruggiero

TL;DR

The article develops a non-flow approach to connect Anosov metrics on genus $>1$ surfaces to strictly negative curvature while preserving the Anosov property. By constructing Gulliver-type surfaces with no focal points and a finite set of positive-curvature bubbles, it uses Riccati-equation comparisons to ensure global Anosov dynamics and to define a conformal path $g_{\rho}=e^{2\rho w}g$ that monotonically reduces curvature in positive regions. The authors show that a suitable inverse-Laplacian conformal factor $w$ yields a path of metrics with $K_{\rho}<0$ at $\rho=1$ and preserves the separation of bubbles, ultimately proving that the entire path consists of Anosov metrics, with the endpoint being strictly negative curvature. This establishes path-connectedness within the space of Anosov metrics (without relying on Ricci flow) and provides a concrete mechanism to deform toward negative curvature while maintaining hyperbolicity.

Abstract

We construct a class of Riemannian metrics in closed surfaces of genus greater than one, having Anosov geodesic flows, and some regions of positive curvature, such that for each such surface, there exists a smooth curve of conformal deformations that preserves the Anosov property and connects the surface with a Riemannian metric of negative curvature. The conformal deformation does not arise from geometric flows like the Ricci flow, since it is known that such flows might generate conjugate points in the presence of points of positive curvature in the surface.

Path Connectivity of Anosov Metrics on Surfaces

TL;DR

The article develops a non-flow approach to connect Anosov metrics on genus surfaces to strictly negative curvature while preserving the Anosov property. By constructing Gulliver-type surfaces with no focal points and a finite set of positive-curvature bubbles, it uses Riccati-equation comparisons to ensure global Anosov dynamics and to define a conformal path that monotonically reduces curvature in positive regions. The authors show that a suitable inverse-Laplacian conformal factor yields a path of metrics with at and preserves the separation of bubbles, ultimately proving that the entire path consists of Anosov metrics, with the endpoint being strictly negative curvature. This establishes path-connectedness within the space of Anosov metrics (without relying on Ricci flow) and provides a concrete mechanism to deform toward negative curvature while maintaining hyperbolicity.

Abstract

We construct a class of Riemannian metrics in closed surfaces of genus greater than one, having Anosov geodesic flows, and some regions of positive curvature, such that for each such surface, there exists a smooth curve of conformal deformations that preserves the Anosov property and connects the surface with a Riemannian metric of negative curvature. The conformal deformation does not arise from geometric flows like the Ricci flow, since it is known that such flows might generate conjugate points in the presence of points of positive curvature in the surface.
Paper Structure (6 sections, 24 theorems, 106 equations, 7 figures)

This paper contains 6 sections, 24 theorems, 106 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Preliminaries: Surfaces free of focal points and Riccati equation
  3. Preliminaries: Compact Surfaces, Anosov Metrics and Riccati Equation
  4. Surfaces free of focal points and with Anosov metric
  5. One parameter conformal deformations of metrics
  6. Deformation without focal points and Anosov

Key Result

Theorem 1.1

Let $0<\epsilon<1$ and $\delta>0$. There exists $\Lambda=\Lambda(\epsilon,\delta)>0$ such that, if $(M,g)\in\mathcal{M}(\delta,k,\epsilon,\Lambda)$ satisfies then:

Figures (7)

  • Figure 1: Solutions of the Riccati equation with constant curvature $-\epsilon$.
  • Figure 2: Control of the Riccati solution outside the generalized bubble.
  • Figure 3: Control of Riccati solution inside and outside the generalized bubble.
  • Figure 4: Behavior of $h$.
  • Figure 5: Possible intersection between $B_{\delta e^{\mu}}^{\rho}(q_1)$ and $B_{\delta}(q_2)$ under the metric deformation.
  • ...and 2 more figures

Theorems & Definitions (53)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • proof
  • ...and 43 more