The long time behavior of fourth-order curvature flows

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.

Final version, to appear in Calc. Var. PDE

References in corpus (3)

• Fourth order curvature flows and geometric applications
• The Calabi flow on Kähler surface with bounded Sobolev constant
• The gradient flow of the $L^2$ curvature energy near the round sphere

Cited by in corpus (2)

• A concentration-collapse decomposition for $L^2$ flow singularities
• Collapsing in the $L^2$ curvature flow