Path Connectivity of Anosov Metrics on Surfaces

TL;DR

This paper constructs a family of conformal deformations on closed surfaces with Anosov geodesic flows, connecting metrics with positive and negative curvature without relying on geometric flows like Ricci flow.

Contribution

It introduces a new class of conformal deformations that preserve Anosov properties and connect different curvature regimes on surfaces of genus greater than one.

Findings

Existence of smooth conformal curves preserving Anosov property

Deformation connects positive and negative curvature metrics

Deformation not derived from Ricci flow or similar geometric flows

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.