Linear and nonlinear stability for the Bach flow, I

TL;DR

This paper establishes linear and nonlinear stability results for a modified Bach flow on manifolds of constant curvature, using spectral bounds and higher order identities, with implications for hyperbolic and Poincaré-Einstein spaces.

Contribution

It introduces a higher order generalization of the Koiso identity and proves stability of the Bach flow on specific classes of manifolds, extending previous understanding.

Findings

Linear stability of the gauge-modified Bach flow on complete manifolds of constant curvature.

Nonlinear stability of the Bach flow on hyperbolic and Poincaré-Einstein spaces close to the background metric.

Development of spectral bounds and higher order identities for stability analysis.

Abstract

In this paper we prove the linear stability of a gauge-modified version of the Bach flow on any complete manifold (M, h) of constant curvature. This involves some intricate calculations to obtain spectral bounds, and in particular introduces a higher order generalization of the well-known Koiso identity. We also prove nonlinear stability for the Bach flow if (M, h) is hyperbolic space, and more generally any Poincar\'e-Einstein space sufficiently close to h. In the forthcoming Part II of this project, we study the nonlinear stability question if M is either compact or else noncompact and flat, since those cases require different considerations involving a center manifold.