Flexibility of geometrical and dynamical data in fixed conformal classes

TL;DR

This paper investigates how geometric and dynamical features of hyperbolic surfaces vary within fixed conformal classes, revealing bounds on diameter and spectrum, and demonstrating the flexibility of metric entropy.

Contribution

It establishes bounds on diameter and Laplace spectrum for conformally equivalent metrics, and shows the metric entropy can vary freely, highlighting new insights into geometric-dynamical relationships.

Findings

Diameter is bounded above by a constant depending on the base metric.

Laplace spectrum is bounded below away from zero by a constant depending on the base metric.

Metric entropy of geodesic flow is flexible and can vary independently.

Abstract

Consider a smooth closed surface $M$ of fixed genus $\geqslant 2$ with a hyperbolic metric $\sigma$ of total area $A$. In this article, we study the behavior of geometric and dynamical characteristics (e.g., diameter, Laplace spectrum, Gaussian curvature and entropies) of nonpositively curved smooth metrics with total area $A$ conformally equivalent to $\sigma$. For such metrics, we show that the diameter is bounded above and the Laplace spectrum is bounded below away from zero by constants which depend on $\sigma$. On the other hand, we prove that the metric entropy of the geodesic flow with respect to the Liouville measure is flexible. Consequently, we also provide the first known example showing that the bottom of the $L^2$-spectrum of the Laplacian cannot be bounded from above by a function of the metric entropy. We also provide examples showing that our conditions are essential for the established bounds.