Abstract
Themotivation of this paper is to study a second order elliptic operatorwhich appears naturally in Riemannian geometry, for instance in the study of hypersurfaces with constant r-mean curvature. We prove a generalizedBochner-type formula for such a kind of operators and as applicationswe obtain some sharp estimates for the first nonzero eigenvalues in two special cases. These results can be considered as generalizations of the Lichnerowicz-Obata Theorem.
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Hilário Alencar, Manfredo do Carmo and Maria Fernanda Elbert. Stability of hypersurfaces with vanishing r-mean curvatures in Euclidean spaces. J. Reine Angew. Math., 554 (2003), 201–216, DOI 10. 1515/crll. 2003. 006. MR1952173 (2003k:53061).
Hilário Alencar, Manfredo do Carmo and Walcy Santos. A gap theorem for hypersurfaces of the sphere with constant scalar curvature one. Comment. Math. Helv., 77(3) (2002), 549–562, DOI 10. 1007/s00014-002-8351-1. MR1933789 (2003m:53098).
Hilário Alencar, Harold Rosenberg and Walcy Santos. On the Gauss map of hypersurfaces with constant scalar curvature in spheres. Proc. Amer. Math. Soc., 132(12) (2004), 3731–3739 (electronic), DOI 10. 1090/S0002-9939-04-07493-3. MR2084098 (2005h:53099).
Hilário Alencar, Walcy Santos and Detang Zhou. Stable hypersurfaces with constant scalar curvature. Proc. Amer. Math. Soc., 138(9) (2010), 3301–3312, DOI 10. 1090/S0002-9939-10-10388-8. MR2653960 (2011m:53095)
Alexander Alexandrov. Die innere Geometrie der konvexen Flachen, in Russian. German translation: Akademie, Berlin (1955).
Luis J. Alías and J. Miguel Malacarne. On the first eigenvalue of the linearized operator of the higher order mean curvature for closed hypersurfaces in space forms. Illinois J. Math., 48(1) (2004), 219–240. MR2048223 (2005b:53093)
João Lucas Marques Barbosa and Antônio Gervásio Colares. Stability of hypersurfaces with constant r-mean curvature. Ann. Global Anal. Geom., 15(3) (1997), 277–297, DOI 10.1023/A:1006514303828. MR1456513 (98h:53091).
Arthur L. Besse. Einstein manifolds, Classics in Mathematics. Springer-Verlag, Berlin, 2008. Reprint of the 1987 edition. MR2371700 (2008k:53084).
Qing-Ming Cheng. Compact locally conformally flat Riemannianmanifolds. Bull. London Math. Soc., 33(4) (2001), 459–465, DOI 10.1017/S0024609301008074. MR1832558 (2002g:53045).
Xu Cheng. An almost-Schur type lemma for symmetric (2, 0) tensors and applications. Pacific J. Math., 267(2) (2014), 325–340, DOI 10. 2140/pjm.2014.267.325, MR3207587.
Shiu Yuen Cheng and Shing Tung Yau. Hypersurfaces with constant scalar curvature. Math. Ann., 225(3) (1977), 195–204. MR0431043 (55 #4045).
M. P. do Carmo and F. W. Warner. Rigidity and convexity of hypersurfaces in spheres. J. Differential Geometry, 4 (1970), 133–144. MR0266105 (42 #1014).
Maria Fernanda Elbert. Constant positive 2-mean curvature hypersurfaces. Illinois J. Math., 46(1) (2002), 247–267. MR1936088 (2003g:53103).
David Gilbarg and Neil S. Trudinger. Elliptic partial differential equations of second order. Classics in Mathematics, Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition. MR1814364 (2001k:35004).
Jaques Hadamard. Sur certaines proprits des trajectories en dynamique. J. Math. Pures Appl., 3 (1897), 331–387.
André Lichnerowicz. Géométrie des groupes de transformations. Travaux et Recherches Mathématiques, III. Dunod, Paris, 1958 (French). MR0124009 (23 #A1329).
Sebastián Montiel and Antonio Ros. Compact hypersurfaces: the Alexandrov theorem for higher order mean curvatures. Differential geometry, Pitman Monogr. Surveys Pure Appl. Math., vol. 52, Longman Sci. Tech., Harlow, 1991, pp. 279–296. MR1173047 (93h:53062).
Morio Obata. Certain conditions for a Riemannian manifold to be isometric with a sphere. J. Math. Soc. Japan, 14 (1962), 333–340. MR0142086 (25 #5479).
Peter Petersen. Riemannian geometry. 2nd ed., Graduate Texts in Mathematics, vol. 171, Springer, New York, 2006. MR2243772 (2007a:53001).
Robert C. Reilly. Variational properties of functions of the mean curvatures for hypersurfaces in space forms. J. Differential Geometry, 8 (1973), 465–477. MR0341351 (49 #6102).
Harold Rosenberg. Hypersurfaces of constant curvature in space forms. Bull. Sci. Math., 117(2) (1993), 211–239. MR1216008 (94b:53097).
R. Schoen, L. Simon and S. T. Yau. Curvature estimates forminimal hypersurfaces. Acta Math., 134(3-4) (1975), 275–288. MR0423263 (54 #11243).
Jeff A. Viaclovsky. Conformal geometry, contact geometry, and the calculus of variations. DukeMath. J., 101(2) (2000), 283–316, DOI 10. 1215/S0012-7094-00-10127-5. MR1738176 (2001b:53038).
K. Voss. Einige differentialgeometrischeKongruenzsätze für geschlossene Flächen und Hyperflächen. Math. Ann., 131 (1956), 180–218 (German). MR0080327 (18,229f).
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Corresponding author. Partially supported by CNPq and Fapeal of Brazil.
Partially supported by CNPq and Faperj of Brazil.
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Alencar, H., Neto, G.S. & Zhou, D. Eigenvalue estimates for a class of elliptic differential operators on compact manifolds. Bull Braz Math Soc, New Series 46, 491–514 (2015). https://doi.org/10.1007/s00574-015-0102-1
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DOI: https://doi.org/10.1007/s00574-015-0102-1