Thom's gradient conjecture for parabolic systems and the Yang-Mills and Ricci flows

TL;DR

This paper extends Thom's gradient conjecture to infinite-dimensional parabolic flows, including Yang-Mills and Ricci flows, demonstrating convergence properties near critical points under certain conditions.

Contribution

It proves the gradient conjecture for parabolic flows on Hilbert spaces, encompassing gauge-invariant flows like Yang-Mills and Ricci flows, with improvements over previous results.

Findings

Thom's gradient conjecture holds for certain infinite-dimensional flows.

Convergence to critical points is established under specific conditions.

Results include flows with gauge symmetry such as Yang-Mills and Ricci flows.

Abstract

In [8], the gradient conjecture of R. Thom was proven for gradient flows of analytic functions on Rn. This result means that the secant at a limit point converges, so that the flow cannot spiral forever. Once the trajectory becomes sufficiently close to a critical point, the flow becomes a simple scaling. Their paper is also significant in the number of auxiliary results they prove about the convergence behaviour of gradient flows, on the way to proving their main result. Many gradient flows of interest occur on infinite dimensional function spaces. And of considerable research interest today are geometric flows with a gauge or diffeomorphism symmetry. We show that the corresponding gradient conjecture holds also for parabolic flows on Hilbert spaces, including flows with a gauge symmetry such as the extensively studied Yang-Mills Flow. The same result also holds for the Ricci flow near any critical point where the assumptions are satisfied, in particular a Fredholm Hessian and a Lojasiewicz inequality with respect to an appropriately chosen functional. This version contains some small improvements on the original 2021 paper.