# Evolution of relative Yamabe constant under Ricci Flow

## TL;DR

This paper investigates how the relative Yamabe constant evolves under Ricci flow on manifolds with boundary, showing it non-decreases initially and remains constant only for Einstein metrics.

## Contribution

It analyzes the short-time evolution of the relative Yamabe constant under Ricci flow with boundary conditions, establishing conditions for its initial non-decrease and characterization of stationary points.

## Key findings

- The relative Yamabe constant's derivative at initial time is non-negative.
- The derivative is zero if and only if the initial metric is Einstein.
- Under certain conditions, the Yamabe constant remains stationary during 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, $\left.\frac{d}{dt}\right|_{t=0}Y_{[\bar g_t]}(W,M;C)\geq 0$ and is equal to zero if and only the metric $\bar{g}_0$ is Einstein.