Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Lorenz Schabrun

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.

Thom's gradient conjecture for parabolic systems and the 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.

Paper Structure

This paper contains 8 sections, 15 theorems, 90 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Gauge invariance and the Yang-Mills flow
  4. Characteristic exponents
  5. Variants of the Lojasiewicz inequality
  6. Asymptotic critical values
  7. Estimates on a Trajectory
  8. Proof of Theorem 1

Key Result

Theorem 1

Let $V$ be a Hilbert space and let $U \subseteq V$ be an open neighborhood of $0$. Let $\mathcal{E}$ be an analytic function (in particular $\mathcal{E} \in C^2(U)$), and assume the origin is a critical point, i.e. $\mathcal{E}'(0) = 0$. Suppose that $\mathcal{E}"(0)$ is a Fredholm operator. Then fo onto the unit sphere has finite length, in particular, it converges.

Theorems & Definitions (26)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Corollary 1
  • Proposition 1
  • Proposition 2
  • proof : Proof of Theorem \ref{['Thm2']}
  • proof : Proof of Theorem \ref{['Thm3']}
  • Proposition 3
  • ...and 16 more