# On the stability of Ricci flow on hyperbolic 3-manifolds of finite volume

## TL;DR

This paper proves that the normalized Ricci-DeTurck flow on finite volume hyperbolic 3-manifolds remains stable and converges exponentially to the hyperbolic metric if starting close enough, using interpolation theory.

## Contribution

It establishes the exponential stability of Ricci flow on hyperbolic 3-manifolds of finite volume with a novel application of interpolation theory.

## Key findings

- Flow exists for all time under small initial perturbations
- Flow converges exponentially fast to hyperbolic metric
- Provides a new tool for studying minimal surface entropy

## Abstract

On a hyperbolic 3-manifold of finite volume, we prove that if the initial metric is sufficiently close to the hyperbolic metric $h_0$, then the normalized Ricci-DeTurck flow exists for all time and converges exponentially fast to $h_0$ in a weighted H\"older norm. A key ingredient of our approach is the application of interpolation theory. Furthermore, this result is a valuable tool for investigating minimal surface entropy, which quantifies the growth rate of the number of closed minimal surfaces in terms of genus. We explore this in [17].