Abstract
In this paper, we first investigate several rigidity problems for hypersurfaces in the warped product manifolds with constant linear combinations of higher order mean curvatures as well as “weighted” mean curvatures, which extend the work (Brendle in Publ Math Inst Hautes Études Sci 117:247–269, 2013; Brendle and Eichmair in J Differ Geom 94(94):387–407, 2013; Montiel in Indiana Univ Math J 48:711–748, 1999) considering constant mean curvature functions. Secondly, we obtain the rigidity results for hypersurfaces in the space forms with constant linear combinations of intrinsic Gauss–Bonnet curvatures \(L_k\). To achieve this, we develop some new kind of Newton–Maclaurin type inequalities on \(L_k\) which may have independent interest.
Similar content being viewed by others
References
Alás, L.J., Impera, D., Rigoli, M.: Hypersurfaces of constant higher order mean curvature in warped products. Trans. Am. Math. Soc. 365, 591–621 (2013)
Alexandrov, A.D.: Uniqueness theorems for surfaces in the large I-V. Vestn. Leningr. Univ. 11, 5–17 (1956); 12, 15–44 (1957); 13, 14–26 (1958); 13, 27–34 (1958); 13, 5–8 (1958). English transl. in Am. Math. Soc. Transl. 21, 341–354, 354–388, 389–403, 403–411, 412–416 (1962)
Aquino, C.P., de Lima, H.F.: On the unicity of complete hypersurfaces immersed in a semi- Riemannian warped product. J. Geom. Anal. (2012). doi:10.1007/s12220-012-9366-5
Alías, L.J., de Lira, J.H.S., Malacarne, J.M.: Constant higher-order mean curvature hypersurfaces in Riemannian spaces. J. Inst. Math. Jussieu 5(4), 527–562 (2006)
Brendle, S.: Constant mean curvature surfaces in warped product manifolds. Publ. Math. Inst. Hautes Études Sci. 117, 247–269 (2013)
Brendle, S., Eichmair, M.: Isoperimetric and Weingarten surfaces in the Schwarzschild manifold. J. Differ. Geom. 94(94), 387–407 (2013)
Dahl, M., Gicquaud, R., Sakovich, A.: Penrose type inequalities for asymptotically hyperbolic graphs. Ann. Henri Poincaré 14(5), 1135–1168 (2013)
Gårding, L.: An inequality for hyperbolic polynomials. J. Math. Mech. 8, 957–965 (1959)
Ge, Y., Wang, G., Wu, J.: A new mass for asymptotically flat manifolds. arXiv:1211.3645
Ge, Y., Wang, G., Wu, J.: The Gauss–Bonnet–Chern mass of conformally flat manifolds. IMRN. arXiv:1212.3213 (to appear)
Ge, Y., Wang, G., Wu, J.: Hyperbolic Alexandrov–Fenchel quermassintegral inequalities II. arXiv:1304.1417
Ge, Y., Wang, G., Wu, J.: The GBC mass for asymptotically hyperbolic manifolds. arXiv:1306.4233
Guan, P.: Topics in Geometric Fully Nonlinear Equations. Lecture Notes. http://www.math.mcgill.ca/guan/notes.html
He, Y., Li, H., Ma, H., Ge, J.: Compact embedded hypersurfaces with constant higher order anisotropic mean curvatures. Indiana Univ. Math. J. 58(2), 853–868 (2009)
Hijazi, O., Montiel, S., Zhang, X.: Dirac operator on embedded hypersurfaces. Math. Res. Lett. 8, 195–208 (2001)
Hsiang, W.-Y., Teng, Z.-H., Yu, W.-C.: New examples of constant mean curvature immersions of (2k-1)-spheres into Euclidean 2k-space. Ann. Math. (2) 117(3), 609–625 (1983)
Hsiung, C.C.: Some integral formulas for closed hypersurfaces. Math. Scand. 2, 286–294 (1954)
Koh, S.E.: Sphere theorem by means of the ratio of mean curvature functions. Glasg. Math. J. 42, 91–95 (2000)
Korevaar, N.J.: Sphere theorems via Alexsandrov for constant Weingarten curvature hypersurfaces—appendix to a note of A. Ros. J. Differ. Geom. 27, 221–223 (1988)
Lanczos, C.: A remarkable property of the Riemann–Christoffel tensor in four dimensions. Ann. Math. (2) 39(4), 842–850 (1938)
Liebmann, H.: Eine neue Eigenschaft der Kugel. Nachr. Akad. Wiss. Göttingen. 1899, 44–55 (1899)
Montiel, S.: Unicity of constant mean curvature hypersurfaces in some Riemannian manifolds. Indiana Univ. Math. J. 48, 711–748 (1999)
Montiel, S.: Uniqueness of spacelike hypersurfaces of constant mean curvature in foliated spacetimes. Math. Ann. 314(3), 529–553 (1999)
Montiel, S., Ros, A.: Compact hypersurfaces: The Alexandrov theorem for higher order mean curvatures. In: Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 52 (in honor of M.P. do Carmo; edited by B. Lawson and K. Tenenblat), pp. 279–296 (1991)
Reilly, R.: Applications of the Hessian operator in a Riemannian manifold. Indiana Univ. Math. J. 26, 459–472 (1977)
Ros, A.: Compact hypersurfaces with constant scalar curvature and a congruence theorem. J. Differ. Geom. 27, 215–220 (1988)
Ros, A.: Compact hypersurfaces with constant higher order mean curvatures. Revista Mathmática Iberoamericana 3, 447–453 (1987)
Süss, W.: Über Kennzeichnungen der Kugeln und Affinsphären durch Herrn K.-P. Grotemeyer. Arch. Math. (Basel) 3, 311–313 (1952)
Wente, H.C.: Counterexample to a conjecture of H. Hopf. Pac. J. Math. 121(1), 193–243 (1986)
Wu, J.: A new characterization of geodesic spheres in the hyperbolic space. Proc. Am. Math. Soc. arXiv:1305.2805 (to appear)
Acknowledgments
Both authors would like to thank Professors Guofang Wang, Yuxin Ge and Dr. Wei Wang for helpful discussions.
Author information
Authors and Affiliations
Corresponding author
Additional information
J. Wu is partly supported by SFB/TR71 “Geometric partial differential equations” of DFG. C. Xia is supported by funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC Grant Agreement No. 267087.
Rights and permissions
About this article
Cite this article
Wu, J., Xia, C. On rigidity of hypersurfaces with constant curvature functions in warped product manifolds. Ann Glob Anal Geom 46, 1–22 (2014). https://doi.org/10.1007/s10455-013-9405-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10455-013-9405-x