Einstein and conformally flat critical metrics of the volume functional

Details Einstein and conformally flat critical metrics of the volume functional Einstein and conformally flat critical metrics of the volume functional

2009-01-05

arxiv.org/abs/0901.0422v1

Mathematics

Differential Geometry

38 pages

Scientific paper

Let $R$ be a constant. Let $\mathcal{M}^R_\gamma$ be the space of smooth metrics $g$ on a given compact manifold $\Omega^n$ ($n\ge 3$) with smooth boundary $\Sigma $ such that $g$ has constant scalar curvature $R$ and $g|_{\Sigma}$ is a fixed metric $\gamma$ on $\Sigma$. Let $V(g)$ be the volume of $g\in\mathcal{M}^R_\gamma$. In this work, we classify all Einstein or conformally flat metrics which are critical points of $V(\cdot)$ in $\mathcal{M}^R_\gamma$.

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