Convergence of the Weighted Yamabe Flow

Abstract

We introduce the weighted Yamabe flow $\frac{\partial g}{\partial t}=(r_{\varphi }^m-R_{\varphi }^m)g$, $\frac{\partial \varphi }{\partial t}=\frac{m}{2}(R_{\varphi }^m-r_{\varphi }^m)$ on a smooth metric measure space $(M^n,g,e^{-\varphi }{\mathrm{dvol}}_g,m)$, where $R_{\varphi }^m$ denotes the associated weighted scalar curvature, and $r_{\varphi }^m$ denotes the mean value of the weighted scalar curvature. We prove long-time existence and convergence of the weighted Yamabe flow if the dimension $n$ satisfies $n⩾3$.

arXiv commentAdd several estimates needed in the proof of Proposition 3.3; see Proposition 4.7, 4.8 and Lemma 4.10 for precise statement