Deformations of the hemisphere that increase scalar curvature.(English)Zbl 1227.53048

The authors disprove the Min-Oo conjecture [M. Min-Oo, “Scalar curvature rigidity of certain symmetric spaces”, CRM Proc. Lect. Notes 15, 127–136 (1998; Zbl 0911.53032)], which claims that a Riemannian metric $$g$$ on the hemisphere $$S_+^n$$, $$n \geq 3$$, which coincides with the standard metric on the boundary, and which satisfies the following two properties:
1. the boundary is totally geodesic with respect to $$g$$,
2. the scalar curvature of $$g$$ is at least $$n(n-1)$$,
is the standard spherical metric.
This conjecture was supported by a result by Miao (2002) about an analogous characterization of the Euclidean metric and by the generalization of a theorem of Miao by Y. Shi and L. F. Tam (2002).

MSC:

 53C20 Global Riemannian geometry, including pinching

Zbl 0911.53032
Full Text:

References:

 [1] Andersson, L., Dahl, M.: Scalar curvature rigidity for asymptotically locally hyperbolic manifolds. Ann. Glob. Anal. Geom. 16, 1–27 (1998) · Zbl 0946.53021 [2] Bartnik, R.: The mass of an asymptotically flat manifold. Commun. Pure Appl. Math. 39, 661–693 (1986) · Zbl 0598.53045 [3] Bartnik, R.: Energy in general relativity. In: Tsing Hua Lectures on Geometry and Analysis, Hsinchu, 1990–1991, pp. 5–27. Intl. Press, Cambridge (1997) [4] Besse, A.: Einstein Manifolds. Classics in Mathematics. Springer, Berlin (2008) · Zbl 1147.53001 [5] Boualem, H., Herzlich, M.: Rigidity at infinity for even-dimensional asymptotically complex hyperbolic spaces. Ann. Sc. Norm. Super. Pisa, Ser. V 1, 461–469 (2002) · Zbl 1170.53308 [6] Bray, H.: The Penrose inequality in general relativity and volume comparison theorems involving scalar curvature. PhD Thesis, Stanford University (1997) [7] Bray, H.: Proof of the Riemannian Penrose inequality using the positive mass theorem. J. Differ. Geom. 59, 177–267 (2001) · Zbl 1039.53034 [8] Bray, H., Brendle, S., Eichmair, M., Neves, A.: Area-minimizing projective planes in three-manifolds. Commun. Pure Appl. Math. 63, 1237–1247 (2010) · Zbl 1200.53053 [9] Bray, H., Brendle, S., Neves, A.: Rigidity of area-minimizing two-spheres in three-manifolds. Commun. Anal. Geom. (to appear) · Zbl 1226.53054 [10] Brendle, S.: Blow-up phenomena for the Yamabe equation. J. Am. Math. Soc. 21, 951–979 (2008) · Zbl 1206.53041 [11] Brendle, S.: Ricci Flow and the Sphere Theorem. Graduate Studies in Mathematics, vol. 111. Am. Math. Soc., Providence (2010) · Zbl 1196.53001 [12] Brendle, S.: Rigidity phenomena involving scalar curvature. Surv. Differ. Geom. (to appear) · Zbl 1382.53012 [13] Brendle, S., Marques, F.C.: Blow-up phenomena for the Yamabe equation II. J. Differ. Geom. 81, 225–250 (2009) · Zbl 1166.53025 [14] Brendle, S., Marques, F.C.: Scalar curvature rigidity of geodesic balls in S n . arxiv:1005.2782 · Zbl 1237.53037 [15] Chruściel, P.T., Herzlich, M.: The mass of asymptotically hyperbolic Riemannian manifolds. Pac. J. Math. 212, 231–264 (2003) · Zbl 1056.53025 [16] Chruściel, P.T., Nagy, G.: The mass of spacelike hypersurfaces in asymptotically anti-de-Sitter space-times. Adv. Theor. Math. Phys. 5, 697–754 (2001) · Zbl 1033.53061 [17] Corvino, J.: Scalar curvature deformation and a gluing construction for the Einstein constraint equations. Commun. Math. Phys. 214, 137–189 (2000) · Zbl 1031.53064 [18] Eichmair, M.: The size of isoperimetric surfaces in 3-manifolds and a rigidity result for the upper hemisphere. Proc. Am. Math. Soc. 137, 2733–2740 (2009) · Zbl 1187.53038 [19] Fischer, A.E., Marsden, J.E.: Deformations of the scalar curvature. Duke Math. J. 42, 519–547 (1975) · Zbl 0336.53032 [20] Gromov, M.: Positive curvature, macroscopic dimension, spectral gaps and higher signatures. In: Functional Analysis on the Eve of the 21st Century, Vol. II, New Brunswick, 1993. Progr. Math., vol. 132, pp. 1–213. Birkhäuser, Boston (1996) [21] Gromov, M., Lawson, H.B.: Spin and scalar curvature in the presence of a fundamental group. Ann. Math. 111, 209–230 (1980) · Zbl 0445.53025 [22] Gromov, M., Lawson, H.B.: Positive scalar curvature and the Dirac operator on complete Riemannian manifolds. Publ. Math. IHÉS 58, 83–196 (1983) · Zbl 0538.53047 [23] Hang, F., Wang, X.: Rigidity and non-rigidity results on the sphere. Commun. Anal. Geom. 14, 91–106 (2006) · Zbl 1119.53029 [24] Hang, F., Wang, X.: Rigidity theorems for compact manifolds with boundary and positive Ricci curvature. J. Geom. Anal. 19, 628–642 (2009) · Zbl 1175.53056 [25] Herzlich, M.: Scalar curvature and rigidity of odd-dimensional complex hyperbolic spaces. Math. Ann. 312, 641–657 (1998) · Zbl 0946.53022 [26] Huang, L., Wu, D.: Rigidity theorems on hemispheres in non-positive space forms. Commun. Anal. Geom. 18, 339–363 (2010) · Zbl 1217.53041 [27] Huisken, G., Polden, A.: Geometric evolution equations for hypersurfaces. In: Calculus of Variations and Geometric Evolution Problems, Cetraro, 1996. Lecture Notes in Mathematics, vol. 1713, pp. 45–84. Springer, Berlin (1999) · Zbl 0942.35047 [28] Li, P.: Lecture Notes on Geometric Analysis. Lecture Notes Series, vol. 6. Seoul National University, Seoul (1993) · Zbl 0822.58001 [29] Listing, M.: Scalar curvature on compact symmetric spaces. arxiv:1007.1832 · JFM 07.0699.03 [30] Llarull, M.: Sharp estimates and the Dirac operator. Math. Ann. 310, 55–71 (1998) · Zbl 0895.53037 [31] Lohkamp, J.: Metrics of negative Ricci curvature. Ann. Math. 140, 655–683 (1994) · Zbl 0824.53033 [32] Lohkamp, J.: Scalar curvature and hammocks. Math. Ann. 313, 385–407 (1999) · Zbl 0943.53024 [33] Miao, P.: Positive mass theorem on manifolds admitting corners along a hypersurface. Adv. Theor. Math. Phys. 6, 1163–1182 (2002) [34] Min-Oo, M.: Scalar curvature rigidity of asymptotically hyperbolic spin manifolds. Math. Ann. 285, 527–539 (1989) · Zbl 0686.53038 [35] Min-Oo, M.: Scalar curvature rigidity of certain symmetric spaces. In: Geometry, Topology, and Dynamics, Montreal, 1995. CRM Proc. Lecture Notes, vol. 15, pp. 127–137. Am. Math. Soc., Providence (1998) [36] Parker, T., Taubes, C.H.: On Witten’s proof of the positive energy theorem. Commun. Math. Phys. 84, 223–238 (1982) · Zbl 0528.58040 [37] Schoen, R., Yau, S.T.: On the proof of the positive mass conjecture in general relativity. Commun. Math. Phys. 65, 45–76 (1979) · Zbl 0405.53045 [38] Schoen, R., Yau, S.T.: Existence of incompressible minimal surfaces and the topology of three dimensional manifolds of non-negative scalar curvature. Ann. Math. 110, 127–142 (1979) · Zbl 0431.53051 [39] Schoen, R., Yau, S.T.: On the structure of manifolds with positive scalar curvature. Manuscr. Math. 28, 159–183 (1979) · Zbl 0423.53032 [40] Shi, Y., Tam, L.F.: Positive mass theorem and the boundary behaviors of compact manifolds with nonnegative scalar curvature. J. Differ. Geom. 62, 79–125 (2002) · Zbl 1071.53018 [41] Toponogov, V.: Evaluation of the length of a closed geodesic on a convex surface. Dokl. Akad. Nauk SSSR 124, 282–284 (1959) · Zbl 0092.14603 [42] Wang, X.: The mass of asymptotically hyperbolic manifolds. J. Differ. Geom. 57, 273–299 (2001) · Zbl 1037.53017 [43] Witten, E.: A new proof of the positive energy theorem. Commun. Math. Phys. 80, 381–402 (1981) · Zbl 1051.83532
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