# Reaction-diffusion problems on time-dependent Riemannian manifolds: stability of periodic solutions

## TL;DR

This paper studies the stability of periodic solutions in reaction-diffusion equations on evolving Riemannian manifolds, highlighting the influence of curvature and growth on pattern formation.

## Contribution

It introduces a framework for analyzing stability of periodic solutions on time-dependent manifolds using principal eigenvalues, emphasizing Ricci curvature's role.

## Key findings

- Principal eigenvalue characterizes stability of solutions.
- Ricci curvature significantly affects pattern formation.
- Results applicable to biological models with evolving geometries.

## Abstract

We investigate the stability of time-periodic solutions of semilinear parabolic problems with Neumann boundary conditions. Such problems are posed on compact submanifolds evolving periodically in time. The discussion is based on the principal eigenvalue of periodic parabolic operators. The study is motivated by biological models on the effect of growth and curvature on patterns formation. The Ricci curvature plays an important role.