Stability of the Ricci Yang-Mills flow at Einstein Yang-Mills metrics

TL;DR

This paper investigates the stability of the volume-normalized Ricci Yang-Mills flow at Einstein Yang-Mills metrics on principal U(1)-bundles over closed 2D manifolds, revealing conditions for stability and the existence of a center manifold.

Contribution

It applies maximal regularity theory and abstract quasilinear PDE analysis to establish stability results for the Ricci Yang-Mills flow at Einstein Yang-Mills metrics in two dimensions.

Findings

Existence of a center manifold of fixed points in certain cases.

Presence of an asymptotically stable fixed point under specific conditions.

Analysis of the flow's asymptotic behavior using advanced PDE techniques.

Abstract

Let $P$ be a principal U(1)-bundle over a closed manifold $M$. On $P$, one can define a modified version of the Ricci flow called the Ricci Yang-Mills flow, due to these equations being a coupling of Ricci flow and the Yang-Mills heat flow. We use maximal regularity theory and ideas of Simonett concerning the asymptotic behavior of abstract quasilinear parabolic partial differential equations to study the stability of the volume-normalized Ricci Yang-Mills flow at Einstein Yang-Mills metrics in dimension two. In certain cases, we show the presence of a center manifold of fixed points, while in others, we show the existence of an asymptotically stable fixed point.