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.
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References
Agol I., Storm P.A., Thurston W.P.: Lower bounds on volumes of hyperbolic Haken 3-manifolds. J. Am. Math. Soc. 20(4), 1053–1077 (2007)
Arnowitt R., Deser S., Misner C.W.: Coordinate invariance and energy expressions in general relativity. Phys. Rev. 122(2), 997–1006 (1961)
Bartnik R.: Phase space for the Einstein equations. Comm. Anal. Geom. 13(5), 845–885 (2005)
Beig, R.: TT-tensors and conformally flat structures on 3-manifolds. Mathematics of gravitation, Part I (Warsaw, 1996), vol. 41, pp. 109–118. Banach Center Publ., Part I. Polish Acad. Sci. (1997)
Besse A.L.: Einstein Manifolds. Springer, Heidelberg (1987)
Bonnesen, T., Fenchel, W.: Theory of convex bodies [translated from the German, Boron, L., Christenson, C., Smith, B. (eds.) with the collaboration of W. Fenchel]. BCS Associates, Moscow (1987)
Corvino J.: On the existence and stability of the Penrose compactification. Ann. Henri Poincaré 8(3), 597–620 (2007)
Fischer A.E., Marsden J.E.: Deformations of the scalar curvature. Duke Math. J. 42(3), 519–547 (1975)
Fan, X.-Q., Shi, Y.-G., Tam, L.-F.: Large-sphere and small-sphere limits of the Brown-York mass. Comm. Anal. Geom. (2007, to appear). arXiv:math.DG/0711.2552
Fischer-Colbrie D., Schoen R.: The structure of complete stable minimal surfaces in 3-manifolds of nonnegative scalar curvature. Comm. Pure Appl. Math. 33(2), 199–211 (1980)
Gilbarg D., Trudinger N.S.: Elliptic Partial Differential Equations Of Second Order, 2nd edn. Springer, Heidelberg (1983)
Miao P.: Positive mass theorem on manifolds admitting corners along a hypersurface. Adv. Theor. Math. Phys. 6(6), 1163–1182 (2003)
Miao P.: On existence of static metric extensions in general relativity. Comm. Math. Phys. 241(1), 27–46 (2003)
Perelman, G.: The entropy formula for the Ricci flow and its geometric applications (2002, preprint). arXiv:math.DG/0211159
Perelman, G.: Ricci flow with surgery on three-manifolds (2003, preprint) arXiv:math.DG/0303109
Sattinger D.: Topics in stability and bifurcation theory. Lecture Notes in Math. vol. 309. Springer, Berlin (1973)
Schoen R., Yau S.-T.: On the proof of the positive mass conjecture in general relativity. Comm. Math. Phys. 65, 45–76 (1979)
Schoen, R.: Variational theory for the total scalar curvature functional for Riemannian metrics and related topics. In: Topics in calculus of variations (Montecatini Terme, 1987). Lecture Notes in Math., vol. 1365, pp. 120–154. Springer, Berlin (1989)
Shi Y.-G., Tam L.-F.: Positive mass theorem and the boundary behaviors of compact manifolds with nonnegative scalar curvature. J. Diff. Geom. 62, 79–125 (2002)
Shi Y.-G., Tam L.-F.: Rigidity of compact manifolds and positivity of quasi-local mass. Class. Quantum Gravity 24(9), 2357–2366 (2007)
Sperner E.: Zur Symmetrisierung von Funktionen auf Sphären. Math Z. 134, 317–327 (1973)
Witten E.: A new proof of the positive energy theorem. Comm. Math. Phys. 80, 381–402 (1981)
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The research of Luen-Fai Tam was partially supported by Hong Kong RGC General Research Fund # GRF 2160357.
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Miao, P., Tam, LF. On the volume functional of compact manifolds with boundary with constant scalar curvature. Calc. Var. 36, 141–171 (2009). https://doi.org/10.1007/s00526-008-0221-2
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DOI: https://doi.org/10.1007/s00526-008-0221-2