Some characterizations of space-forms

Pigola, Stefano; Rigoli, Marco; Setti, Alberto

doi:10.1090/S0002-9947-06-04091-8

Some characterizations of space-forms
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by Stefano Pigola, Marco Rigoli and Alberto G. Setti PDF

Trans. Amer. Math. Soc. 359 (2007), 1817-1828 Request permission

Erratum: Trans. Amer. Math. Soc. 360 (2008), 3943-3944.

Abstract:

Integral conditions on the traceless Ricci tensor are used to characterize Euclidean and hyperbolic spaces among complete, locally conformally flat manifolds of constant scalar curvature. The main tools are vanishing-type results for $L^{p}$-solutions of a large class of differential inequalities. Further applications of the technique are also given.

References

M. Anderson, The compactification of a minimal submanifold in Euclidean space by the Gauss map. Preprint.
Pierre Bérard, Remarques sur l’équation de J. Simons, Differential geometry, Pitman Monogr. Surveys Pure Appl. Math., vol. 52, Longman Sci. Tech., Harlow, 1991, pp. 47–57 (French, with English summary). MR 1173032
Jean-Pierre Bourguignon, The “magic” of Weitzenböck formulas, Variational methods (Paris, 1988) Progr. Nonlinear Differential Equations Appl., vol. 4, Birkhäuser Boston, Boston, MA, 1990, pp. 251–271. MR 1205158
Gilles Carron, Une suite exacte en $L^2$-cohomologie, Duke Math. J. 95 (1998), no. 2, 343–372 (French). MR 1652017, DOI 10.1215/S0012-7094-98-09510-2
Gilles Carron, $L^2$-cohomologie et inégalités de Sobolev, Math. Ann. 314 (1999), no. 4, 613–639 (French). MR 1709104, DOI 10.1007/s002080050310
Luis A. Caffarelli, Basilis Gidas, and Joel Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math. 42 (1989), no. 3, 271–297. MR 982351, DOI 10.1002/cpa.3160420304
Huai-Dong Cao, Ying Shen, and Shunhui Zhu, The structure of stable minimal hypersurfaces in $\textbf {R}^{n+1}$, Math. Res. Lett. 4 (1997), no. 5, 637–644. MR 1484695, DOI 10.4310/MRL.1997.v4.n5.a2
Bao Qiang Wu and Qing Ming Cheng, The generalized maximum principle and conformally flat spaces, Northeast. Math. J. 8 (1992), no. 1, 54–56. MR 1164748
Avner Friedman, Partial differential equations, Holt, Rinehart and Winston, Inc., New York-Montreal, Que.-London, 1969. MR 0445088
S. I. Goldberg, An application of Yau’s maximum principle to conformally flat spaces, Proc. Amer. Math. Soc. 79 (1980), no. 2, 268–270. MR 565352, DOI 10.1090/S0002-9939-1980-0565352-8
Th. Hasanis, Conformally flat spaces and a pinching problem on the Ricci tensor, Proc. Amer. Math. Soc. 86 (1982), no. 2, 312–315. MR 667296, DOI 10.1090/S0002-9939-1982-0667296-1
David Hoffman and Joel Spruck, Sobolev and isoperimetric inequalities for Riemannian submanifolds, Comm. Pure Appl. Math. 27 (1974), 715–727. MR 365424, DOI 10.1002/cpa.3160270601
N. H. Kuiper, On conformally-flat spaces in the large, Ann. of Math. (2) 50 (1949), 916–924. MR 31310, DOI 10.2307/1969587
Peter Li and Jiaping Wang, Complete manifolds with positive spectrum, J. Differential Geom. 58 (2001), no. 3, 501–534. MR 1906784
Lei Ni, Gap theorems for minimal submanifolds in $\textbf {R}^{n+1}$, Comm. Anal. Geom. 9 (2001), no. 3, 641–656. MR 1895136, DOI 10.4310/CAG.2001.v9.n3.a2
Masafumi Okumura, Hypersurfaces and a pinching problem on the second fundamental tensor, Amer. J. Math. 96 (1974), 207–213. MR 353216, DOI 10.2307/2373587
Masafumi Okumura, Submanifolds and a pinching problem on the second fundamental tensors, Trans. Amer. Math. Soc. 178 (1973), 285–291. MR 317246, DOI 10.1090/S0002-9947-1973-0317246-1
Stefano Pigola, Marco Rigoli, and Alberto G. Setti, Vanishing theorems on Riemannian manifolds, and geometric applications, J. Funct. Anal. 229 (2005), no. 2, 424–461. MR 2182595, DOI 10.1016/j.jfa.2005.05.007
R. Schoen and S.-T. Yau, Lectures on differential geometry, Conference Proceedings and Lecture Notes in Geometry and Topology, I, International Press, Cambridge, MA, 1994. Lecture notes prepared by Wei Yue Ding, Kung Ching Chang [Gong Qing Zhang], Jia Qing Zhong and Yi Chao Xu; Translated from the Chinese by Ding and S. Y. Cheng; With a preface translated from the Chinese by Kaising Tso. MR 1333601
Mariko Tani, On a conformally flat Riemannian space with positive Ricci curvature, Tohoku Math. J. (2) 19 (1967), 227–231. MR 220213, DOI 10.2748/tmj/1178243319
Shun-Hui Zhu, The classification of complete locally conformally flat manifolds of nonnegative Ricci curvature, Pacific J. Math. 163 (1994), no. 1, 189–199. MR 1256184, DOI 10.2140/pjm.1994.163.189