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Pengzi Miao : On critical metrics on compact manifolds with boundary

It is known that, on closed manifolds, Einstein metrics of negative scalar curvature are critical points of the usual volume functional constrained to the space of metrics of constant scalar curvature. In this talk, I will discuss how this variational characterization of Einstein metrics can be localized to compact manifolds with boundary. I will present the critical point equation and focus on geometric properties of its general solutions. In particular, when a solution has zero scalar curvature and its boundary can be isometrically embedded into the Euclidean space, I will show that the volume of this critical metric is always greater than or equal to the Euclidean volume enclosed by the image of the isometric embedding and two volumes are the same if and only if the critical metric is isometric to the Euclidean metric on a standard round ball. This is a joint work with Prof. Luen-Fai Tam.

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