Convergence of the Weighted Yamabe Flow

TL;DR

This paper introduces the weighted Yamabe flow on smooth metric measure spaces, proving its long-term existence and convergence in dimensions three and higher, extending classical Yamabe flow results.

Contribution

The paper develops the theory of the weighted Yamabe flow on smooth metric measure spaces and establishes its long-time existence and convergence for dimensions n ≥ 3.

Findings

Proves long-time existence of the weighted Yamabe flow.

Shows convergence of the flow in dimensions n ≥ 3.

Extends classical Yamabe flow results to weighted settings.

Abstract

We introduce the weighted Yamabe flow $\frac{\partial g}{\partial t}=(r^m_{\phi}-R^m_{\phi})g$, $\frac{\partial \phi}{\partial t}=\frac{m}{2}(R^m_{\phi}-r^m_{\phi})$ on a smooth metric measure space $(M^n, g, e^{-\phi}{\rm dvol}_g, m)$, where $R^m_{\phi}$ denotes the associated weighted scalar curvature, and $r^m_{\phi}$ 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 $n\geqslant 3$.