Evolution of relative Yamabe constant under Ricci Flow

Abstract

Let $W$ be a manifold with boundary $M$ given together with a conformal class $\bar{C}$ which restricts to a conformal class $C$ on $M$. Then the relative Yamabe constant $Y_{\bar{C}}(W,M;C)$ is well-defined. We study the short-time behavior of the relative Yamabe constant $Y_{[{\bar{g}}_t]}(W,M;C)$ under the Ricci flow ${\bar{g}}_t$ on $W$ with boundary conditions that mean curvature $H_{{\bar{g}}_t}\equiv 0$ and ${\bar{g}}_t{|}_M\in C=[{\bar{g}}_0]$. In particular, we show that if the initial metric ${\bar{g}}_0$ is a Yamabe metric, then, under some natural assumptions, ${\frac{d}{dt}|}_{t=0}{Y}_{[{\bar{g}}_t]}(W,M;C)\ge 0$ and is equal to zero if and only the metric ${\bar{g}}_0$ is Einstein.

9 pages