Stability of hyperbolic manifolds with cusps under Ricci flow

TL;DR

This paper proves that finite volume hyperbolic manifolds with cusps remain stable under Ricci flow, with small perturbations returning to the hyperbolic metric without decay assumptions, using a new analytical method.

Contribution

It establishes stability of hyperbolic manifolds with cusps under Ricci flow without decay assumptions, overcoming cusp deformation challenges with a novel analytical approach.

Findings

Hyperbolic manifolds with cusps are stable under Ricci flow.

Small perturbations of the hyperbolic metric return to the original metric.

A new analytical method addresses cusp deformation stability issues.

Abstract

We show that every finite volume hyperbolic manifold of dimension greater or equal to 3 is stable under rescaled Ricci flow, i.e. that every small perturbation of the hyperbolic metric flows back to the hyperbolic metric again. Note that we do not need to make any decay assumptions on this perturbation. It will turn out that the main difficulty in the proof comes from a weak stability of the cusps which has to do with infinitesimal cusp deformations. We will overcome this weak stability by using a new analytical method developed by Koch and Lamm.