Abstract
In this paper we extend the local scalar curvature rigidity result in Brendle and Marques (J Differ Geom 88:379–394, 2011) to a small domain on general vacuum static spaces, which confirms the interesting dichotomy of local surjectivity and local rigidity about the scalar curvature in general in the light of the paper (Corvino, Commun Math Phys 214:137–189, 2000). We obtain the local scalar curvature rigidity of bounded domains in hyperbolic spaces. We also obtain the global scalar curvature rigidity for conformal deformations of metrics in the domains, where the lapse functions are positive, on vacuum static spaces with positive scalar curvature, and show such domains are maximal, which generalizes the work in Hang and Wang (Commun Anal Geom 14:91–106, 2006).
Similar content being viewed by others
References
Andersson, L., Cai, M., Galloway, G.: Rigidity and positivity of mass for asymptotically hyperbolic manifolds. Ann. Henri Poincaré 9(1), 1–33 (2008)
Andersson, L., Dahl, M.: Scalar curvature rigidity for asymptotically locally hyperbolic manifolds. Ann. Global Anal. Geom. 16, 1–27 (1998)
Barbosa, E., Mirandola, H., Vitorio, F.: Rigidity theorems of conformal class on compact manifolds with boundary. (2014). arXiv:1411.6851
Bonini, V., Qing, J.: A positive mass theorem on asymptotically hyperbolic manifolds with corners along a hypersurface. Ann. Henri Poincaré 9(2), 347–372 (2008)
Brendle, S.: Rigidity phenomena involving scalar curvature. (2010). arXiv:1008.3097
Brendle, S., Marques, F.C., Neves, A.: Deformations of the hemisphere that increase scalar curvature. Invent. Math. 185, 175–197 (2011)
Brendle, S., Marques, F.C.: Scalar curvature rigidity of geodesic balls in \(S^n\). J. Differ. Geom. 88, 379–394 (2011)
Cao, H.-D., Catino, G., Chen, Q., Mantegazza, C., Mazzieri, L.: Bach-flat gradient steady Ricci solitons. (2011). arXiv:1107.4591
Cao, H.-D., Chen, Q.: On locally conformally flat gradient steady Ricci solitons. Trans. Am. Math. Soc. 364, 2377–2391 (2012)
Chruściel, P., Herzlic, M.: The mass of asymptotically hyperbolic Riemannian manifolds. Pacific J. Math. 212(2), 231–264 (2003)
Corvino, J.: Scalar curvature deformation and a gluing construction for the Einstein constraint equations. Commun. Math. Phys. 214, 137–189 (2000)
Cox, G., Miao, P., Tam, L.F.: Remarks on a scalar curvature rigidity theorem of Brendle and Marques. Asian J. Math. 17, 457–470 (2013)
Chen, W., Li, C.: Methods on nonlinear elliptic equations. AIMS Book Series on Differential Equations & Dynamics, Systems, 4. AIMS (2010)
Cruz, C., Lima, L., Montenegro, J.: Deforming the scalar curvature of the de Sitter-Schwarzschild space. (2014). arXiv:1411.1600
Ebin, D.: The manifold of Riemannian metrics. In: Proc. Sympos. Pure Math., vol. XV (Berkeley, Calif.), pp. 11-40. Amer. Math. Soc., Providence (1968)
Fischer, A., Marsden, J.: Deformations of the scalar curvature. Duke Math. J. 42(3), 519–547 (1975)
Hang, F., Wang, X.: Rigidity and non-rigidity results on the sphere. Commun. Anal. Geom. 14, 91–106 (2006)
Hang, F., Wang, X.: Rigidity theorems for compact manifolds with boundary and positive Ricci curvature. J. Geom. Anal. 19, 628–642 (2009)
Hawkins, S., Ellis, G.: The large scale structure of space-time. Cambridge University Press (1975)
Karp, L., Pinsky, M.: The first eigenvalue of a small geodesic ball in a Riemannian manifold. Bulletin des Sciences Mathématiques 111, 222–239 (1987)
Kobayashi, O.: A differential equation arising from scalar curvature function. J. Math. Soc. Jpn. 34(4), 665–675 (1982)
Kobayashi, O., Obata, M.: Conformally-flatness and static space-time. Manifolds and Lie groups, progress in mathematics, vol. 14, pp. 197-206. Birkhäuser (1981)
Lafontaine, J.: Sur la géométrie d’une généralisation de l’équation différentielle d’Obata. J. Math. Pures Appliquées 62, 63–72 (1983)
Miao, P.: Posotive mass theorem on manifolds admitting corners along a hypersurface. Adv. Theor. Math. Phys. 6, 1163–1182 (2002)
Miao, P., Tam, L.F.: Scalar curvature rigidity with a volume constraint. Commun. Anal. Geom. 20, 1–30 (2012)
Min-Oo, M.: Scalar curvature rigidity of asymptotically hyperbolic spin manifolds. Math. Ann. 285, 527–539 (1989)
Min-Oo, M.: Scalar curvature rigidity of certain symmetric spaces. In: Geometry, topology, and dynamics (Montreal, 1995), vol. 15, pp. 127–137. CRM Proc. Lecture Notes. American Mathematical Society, Providence RI (1998)
Qing, J.: On the uniqueness of AdS spacetime in higher dimensions. Ann. Henri Poincaré 5, 245–260 (2004)
Qing, J., Yuan, W.: A note on static spaces and related problems. J. Geom. Phys 74, 18–27 (2013)
Shi, Y., Tam, L.F.: Positive mass theorem and boundary behaviors of compact manifolds with nonnegative scalar curvature. J. Diff. Geom. 62, 79–125 (2002)
Schoen, R., Yau, S.T.: On the proof of positive mass conjecture in general relativity. Commun. Math. Phys. 65, 45–76 (1979)
Schoen, R., Yau, S.T.: Proof of positive mass theorem II. Commun. Math. Phys. 79, 231–260 (1981)
Wang, X.: The mass of asymptotically hyperbolic manifolds. J. Differ. Geom. 57(2), 273–299 (2001)
Witten, E.: A new proof of the positive mass theorem. Commun. Math. Phys. 80, 381–402 (1981)
Acknowledgments
We would like to thank Professors Justin Corvino, Pengzi Miao and Carla Cerderbaum for their interests and stimulating discussions.
Author information
Authors and Affiliations
Corresponding author
Additional information
Research of the authors supported by NSF Grant DMS-1005295 and DMS-1303543.
Rights and permissions
About this article
Cite this article
Qing, J., Yuan, W. On scalar curvature rigidity of vacuum static spaces. Math. Ann. 365, 1257–1277 (2016). https://doi.org/10.1007/s00208-015-1302-0
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00208-015-1302-0