Thom's gradient conjecture for nonlinear evolution equations

TL;DR

This paper proves Thom's gradient conjecture for nonlinear evolution equations on Riemannian manifolds, confirming unique limiting directions and characterizing convergence rates in both finite and infinite dimensional contexts.

Contribution

It extends Thom's gradient conjecture to infinite dimensional problems involving geometric PDEs, providing a comprehensive analysis of limiting directions and convergence rates.

Findings

Confirmed uniqueness of limiting directions for nonlinear evolutions.

Characterized convergence rates in classical and infinite dimensional settings.

Applied results to geometric PDEs like minimal surface and harmonic map flows.

Abstract

R. Thom's gradient conjecture states that if a gradient flow of an analytic function converges to a limit, it does so along a unique limiting direction. In this paper, we extend and settle this conjecture in the context of infinite dimensional problems. Building on the foundational works of {\L}ojasiewicz, L. Simon, and the resolution of the conjecture for finite dimensional cases by Kurdyka-Mostowski-Parusinski, we focus on nonlinear evolutions on Riemannian manifolds as studied by L. Simon. This framework includes geometric PDEs such as minimal surface, harmonic map, mean curvature flow, and normalized Yamabe flow. Our main result not only confirms the uniqueness of the limiting direction but also characterizes the rate of convergence and possible limiting directions for both classical and infinite dimensional settings.