Yamabe flow on non-compact manifolds with unbounded initial curvature

TL;DR

This paper proves the global existence of Yamabe flows on certain non-compact manifolds with unbounded initial curvature, extending previous results by removing curvature bounds assumptions.

Contribution

It establishes the existence of Yamabe flows on non-compact manifolds with unbounded initial curvature, assuming only boundedness of the conformal factor and specific geometric conditions.

Findings

Global existence of Yamabe flow proven under new conditions.

Initial curvature can be unbounded without growth restrictions.

Flow exists on manifolds with non-positive scalar curvature.

Abstract

We prove global existence of Yamabe flows on non-compact manifolds $M$ of dimension $m\geq3$ under the assumption that the initial metric $g_0=u_0g_M$ is conformally equivalent to a complete background metric $g_M$ of bounded, non-positive scalar curvature and positive Yamabe invariant with conformal factor $u_0$ bounded from above and below. We do not require initial curvature bounds. In particular, the scalar curvature of $(M,g_0)$ can be unbounded from above and below without growth condition.