Short-time existence for some higher-order geometric flows

TL;DR

This paper proves short-time existence and regularity for certain higher-order geometric flows, including flows by powers of the Laplacian of the Ricci tensor and the ambient obstruction tensor, with applications to Bach flow.

Contribution

It establishes a general theorem on short-time existence for higher-order geometric flows generated by polynomial natural tensors with strongly parabolic linearizations.

Findings

Proves short-time existence for flows by powers of the Laplacian of Ricci tensor

Establishes short-time existence for flows generated by ambient obstruction tensor

Applies results to prove short-time existence for Bach flow

Abstract

We establish short-time existence and regularity for higher-order flows generated by a class of polynomial natural tensors that, after an adjustment by the Lie derivative of the metric with respect to a suitable vector field, have strongly parabolic linearizations. We apply this theorem to flows by powers of the Laplacian of the Ricci tensor, and to flows generated by the ambient obstruction tensor. As a special case, we prove short-time existence for a type of Bach flow.