Conformal Bach flow

TL;DR

This paper introduces the conformal Bach flow, proves its well-posedness and backward uniqueness on closed manifolds, and derives curvature estimates to analyze its long-term behavior.

Contribution

It establishes the well-posedness, backward uniqueness, and curvature estimates for the newly introduced conformal Bach flow.

Findings

Well-posedness of conformal Bach flow on closed manifolds

Backward uniqueness of the flow

Curvature derivative estimates under bounded curvature and pressure

Abstract

In this article we introduce conformal Bach flow and establish its well-posedness on closed manifolds. We also obtain its backward uniqueness. To give an attempt to study the long-time behavior of conformal Bach flow, assuming that the curvature and the pressure function are bounded, global and local Shi's type $L^2$-estimate of derivatives of curvatures are derived. Furthermore using the $L^2$-estimate and based on an idea from \cite{St13} we show Shi's pointwise-estimate of derivatives of curvatures without assuming Sobolev constant bound.