Normalized Yamabe flow on manifolds with bounded geometry

TL;DR

This paper investigates a curvature-normalized Yamabe flow on non-compact manifolds with bounded geometry, proving long-time existence and convergence under negative scalar curvature, extending previous results to infinite volume settings.

Contribution

It introduces a curvature normalization approach for Yamabe flow on non-compact manifolds and proves its long-term behavior, extending prior work to a broader class of manifolds.

Findings

Proves long-time existence of the normalized Yamabe flow under negative scalar curvature.

Establishes convergence of the flow on manifolds with bounded geometry.

Provides stronger short-time estimates for manifolds with fibered boundary metrics.

Abstract

The goal of this paper is to study Yamabe flow on a complete Riemannian manifold of bounded geometry with possibly infinite volume. In the case of infinite volume, standard volume normalization of the Yamabe flow fails and the flow may not converge. Instead, we consider a curvature normalized Yamabe flow, and assuming negative scalar curvature, prove its long-time existence and convergence. This extends the results of Su\'arez-Serrato and Tapie to a non-compact setting. In the appendix, we specify our analysis of a particular example of manifolds with bounded geometry, namely manifolds with fibered boundary metric. In this case, we obtain stronger estimates for the short-time solution using microlocal methods.