## On the volume functional of compact manifolds with boundary with constant scalar curvature.(English)Zbl 1175.49043

Summary: 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 $$3\leq n\leq 5$$, the standard space form metrics are indeed saddle points for the volume functional.

### MSC:

 49Q20 Variational problems in a geometric measure-theoretic setting 53C20 Global Riemannian geometry, including pinching
Full Text:

### References:

 [1] 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) · Zbl 1155.58016 [2] Arnowitt R., Deser S., Misner C.W.: Coordinate invariance and energy expressions in general relativity. Phys. Rev. 122(2), 997–1006 (1961) · Zbl 0094.23003 [3] Bartnik R.: Phase space for the Einstein equations. Comm. Anal. Geom. 13(5), 845–885 (2005) · Zbl 1123.83006 [4] 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) · Zbl 0894.53044 [5] Besse A.L.: Einstein Manifolds. Springer, Heidelberg (1987) · Zbl 0613.53001 [6] 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) · Zbl 0628.52001 [7] Corvino J.: On the existence and stability of the Penrose compactification. Ann. Henri Poincaré 8(3), 597–620 (2007) · Zbl 1120.83008 [8] Fischer A.E., Marsden J.E.: Deformations of the scalar curvature. Duke Math. J. 42(3), 519–547 (1975) · Zbl 0336.53032 [9] 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 [10] 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) · Zbl 0439.53060 [11] Gilbarg D., Trudinger N.S.: Elliptic Partial Differential Equations Of Second Order, 2nd edn. Springer, Heidelberg (1983) · Zbl 0562.35001 [12] Miao P.: Positive mass theorem on manifolds admitting corners along a hypersurface. Adv. Theor. Math. Phys. 6(6), 1163–1182 (2003) [13] Miao P.: On existence of static metric extensions in general relativity. Comm. Math. Phys. 241(1), 27–46 (2003) · Zbl 1149.83311 [14] Perelman, G.: The entropy formula for the Ricci flow and its geometric applications (2002, preprint). arXiv:math.DG/0211159 [15] Perelman, G.: Ricci flow with surgery on three-manifolds (2003, preprint) arXiv:math.DG/0303109 · Zbl 1130.53002 [16] Sattinger D.: Topics in stability and bifurcation theory. Lecture Notes in Math. vol. 309. Springer, Berlin (1973) · Zbl 0248.35003 [17] Schoen R., Yau S.-T.: On the proof of the positive mass conjecture in general relativity. Comm. Math. Phys. 65, 45–76 (1979) · Zbl 0405.53045 [18] 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) [19] 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) · Zbl 1071.53018 [20] Shi Y.-G., Tam L.-F.: Rigidity of compact manifolds and positivity of quasi-local mass. Class. Quantum Gravity 24(9), 2357–2366 (2007) · Zbl 1115.83006 [21] Sperner E.: Zur Symmetrisierung von Funktionen auf Sphären. Math Z. 134, 317–327 (1973) · Zbl 0283.26015 [22] Witten E.: A new proof of the positive energy theorem. Comm. Math. Phys. 80, 381–402 (1981) · Zbl 1051.83532
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.