The long time behavior of fourth-order curvature flows

TL;DR

This paper investigates the long-term behavior of fourth-order curvature flows, providing new smoothing estimates and characterizing solutions with finite-time singularities and bounded curvature.

Contribution

It introduces improved smoothing estimates for these flows without needing Sobolev constant bounds and analyzes the asymptotic behavior of solutions.

Findings

Finite-time singularities imply unbounded curvature.

Solutions with bounded curvature have well-defined behavior at infinity.

Precompactness results for solutions under curvature bounds.

Abstract

We show precompactness results for solutions to parabolic fourth order geometric evolution equations. As part of the proof we obtain smoothing estimates for these flows in the presence of a curvature bound, an improvement on prior results which also require a Sobolev constant bound. As consequences of these results we show that for any solution with a finite time singularity, the $L^{\infty}$ norm of the curvature must go to infinity. Furthermore, we characterize the behavior at infinity of solutions with bounded curvature.