Normalized Yamabe flow on manifolds with bounded geometry

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 Suárez-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.

37 pages, 1 figure; v2: typos and reference corrected, v3: generalization to bounded geometry