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

Pengzi Miao, Luen Fai Tam

Research output: Contribution to journalArticle

34 Scopus citations

Abstract

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.

Original languageEnglish (US)
Pages (from-to)141-171
Number of pages31
JournalCalculus of Variations and Partial Differential Equations
Volume36
Issue number2
DOIs
StatePublished - Sep 2009
Externally publishedYes

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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