On the volume functional of compact manifolds with boundary with constant scalar curvature

By Pengzi Miao, Luen-Fai Tam. 2008

108 citations recorded by the source corpus.

We study the volume functional on the space of constant scalar curvature metrics with a prescribed boundary metric. We derive a sufficient and necessary condition for a metric to be a critical point, and show that the only domains in space forms, on which the standard metrics are critical points, are geodesic balls. In the zero scalar curvature case, assuming the boundary can be isometrically embedded in the Euclidean space as a compact strictly convex hypersurface, we show that the volume of a critical point is always no less than the Euclidean volume bounded by the isometric embedding of the boundary, and the two volumes are equal if and only if the critical point is isometric to a standard Euclidean ball. We also derive a second variation formula and apply it to show that, on Euclidean balls and “small” hyperbolic and spherical balls in dimensions 3 ≤ n ≤ 5, the standard space form metrics are indeed saddle points for the volume functional.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/21435e6affb5a2a6ce20d1779265e8b8fbeb4351

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In Applied Mathematics

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In Mathematical Physics

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Citations in the corpus

• Bach-Flat Critical Metrics of the Volume Functional on 4-Dimensional Manifolds with Boundary (2014, methodology): “For more details, we refer the reader to [18] and [19].”