×

Compact graphs over a sphere of constant second order mean curvature. (English) Zbl 1189.53058

If \(x: M^n\rightarrow \mathbb{R}^{n+1}(c)\) is an oriented hypersurface in the Euclidean space and \(k_1,\dots, k_n\) the principal curvatures of \((M^n,x)\), one may consider \(S_r\), the \(r^{th}\) elementary symmetric functions of the \(k_i\), as a generalization of the mean curvature: \(S_r = k_{i1} \dots k_{ir},\, r = 0, 1,\dots, n\). Many well known theorems on constant mean curvature hypersurfaces can be extended by means of the \(S_r\).
For instance, for a given \(r\), A. Ros has proved that a round sphere is the unique compact embedded hypersurface in Euclidean space for which \(S_r\) is constant [J. Differ. Geom. 27, No. 2, 215–220 (1988; Zbl 0638.53051); Rev. Mat. Iberoam. 3, No. 3–4, 447–453 (1987; Zbl 0673.53003)]. Later on, S. Montiel and A. Ros extended this result to any compact embedded hypersurface contained in an open hemisphere of the sphere \(\mathbb{S}^{n+1}\) [in: Differential geometry. A symposium in honour of Manfredo do Carmo, Proc. Int. Conf., Rio de Janeiro/Bras. 1988, Pitman Monogr. Surv. Pure Appl. Math. 52, 279–296 (1991; Zbl 0723.53032)].
In the paper under review, the authors obtain a similar result for compact star-shaped hypersurfaces of the sphere, whose \(S_2\) function is a positive constant, and without assuming that it is contained in an open hemisphere.
Let \(x: M^n\rightarrow M^{n+1}(c)\) be an oriented hypersurface with unit normal vector field \(N\) in a real space form \(M^{n+1}(c)\). For a conformal vector field \(V\) on \(M^{n+1}(c)\), the authors define a support function on \(M^{n}\) by \(g(p) = <V,N>(x(p))\). They consider also the \(L_r\) operator (an extension of the Laplacian) given by: \(L_r : D(M) \rightarrow D(M)\), \(L_r(f) = div (P_r\nabla f)\), where \(P_r\) is the \(r^{th}\) Newton tensor. Then, they obtain a formula for \(L_r(g)\) acting on the support function \(g\) in terms of the \(S_r\) (whose Laplacian version has been previously obtained by several authors), and which is the main ingredient to prove: Let \(x: M^n\rightarrow \mathbb{S}^{n+1}\) be a compact star-shaped hypersurface with positive constant \(S_2\). Then \((M^n,x)\) is totally umbilic. As an easy consequence, they also prove that a compact star-shaped minimal hypersurface \(x: M^n\rightarrow \mathbb{S}^{n+1}\) has to be totally geodesic.

MSC:

53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53C45 Global surface theory (convex surfaces à la A. D. Aleksandrov)
53C65 Integral geometry
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Luis J. Alías and A. Gervasio Colares, Uniqueness of spacelike hypersurfaces with constant higher order mean curvature in generalized Robertson-Walker spacetimes, Math. Proc. Cambridge Philos. Soc. 143 (2007), no. 3, 703 – 729. · Zbl 1131.53035
[2] Luis J. Alías, Marcos Dajczer, and Jaime Ripoll, A Bernstein-type theorem for Riemannian manifolds with a Killing field, Ann. Global Anal. Geom. 31 (2007), no. 4, 363 – 373. · Zbl 1125.53005
[3] Luis J. Alías, Jorge H. S. de Lira, and J. Miguel Malacarne, Constant higher-order mean curvature hypersurfaces in Riemannian spaces, J. Inst. Math. Jussieu 5 (2006), no. 4, 527 – 562. · Zbl 1118.53038
[4] Abdênago Barros, Aldir Brasil, and Antonio Caminha, Stability of spacelike hypersurfaces in foliated spacetimes, Differential Geom. Appl. 26 (2008), no. 4, 357 – 365. · Zbl 1145.53046
[5] Barros, A. and Sousa, P., An extension of Jellett’s theorem, Bull. Sci. Math., 133 (2009), 190-197. · Zbl 1162.53044
[6] A. Caminha, On spacelike hypersurfaces of constant sectional curvature Lorentz manifolds, J. Geom. Phys. 56 (2006), no. 7, 1144 – 1174. · Zbl 1102.53044
[7] Susana Fornari and Jaime Ripoll, Killing fields, mean curvature, translation maps, Illinois J. Math. 48 (2004), no. 4, 1385 – 1403. · Zbl 1072.53020
[8] R. E. Greene and H. Wu, Function theory on manifolds which possess a pole, Lecture Notes in Mathematics, vol. 699, Springer, Berlin, 1979. · Zbl 0414.53043
[9] Ernst Heintze, Extrinsic upper bounds for \?\(_{1}\), Math. Ann. 280 (1988), no. 3, 389 – 402. · Zbl 0628.53044
[10] Chuan-Chih Hsiung, Some integral formulas for closed hypersurfaces, Math. Scand. 2 (1954), 286 – 294. · Zbl 0057.14603
[11] Jellett, J., La surface dont la courbure moyenne est constant, Jour. de Math. Pures et Appliquées, XVIII (1853), 163-167.
[12] 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. · Zbl 0723.53032
[13] Robert C. Reilly, Variational properties of functions of the mean curvatures for hypersurfaces in space forms, J. Differential Geometry 8 (1973), 465 – 477. · Zbl 0277.53030
[14] Antonio Ros, Compact hypersurfaces with constant scalar curvature and a congruence theorem, J. Differential Geom. 27 (1988), no. 2, 215 – 223. With an appendix by Nicholas J. Korevaar. · Zbl 0638.53051
[15] Antonio Ros, Compact hypersurfaces with constant higher order mean curvatures, Rev. Mat. Iberoamericana 3 (1987), no. 3-4, 447 – 453. · Zbl 0673.53003
[16] Harold Rosenberg, Hypersurfaces of constant curvature in space forms, Bull. Sci. Math. 117 (1993), no. 2, 211 – 239. · Zbl 0787.53046
[17] Sousa, P., O Laplaciano de uma função tipo suporte e aplicações, Master thesis, Univ. Federal do Ceará, 2003.
[18] W. Süss, Über Kennzeichnungen der Kugeln und Affinsphären durch Herrn K.-P. Grotemeyer, Arch. Math. (Basel) 3 (1952), 311 – 313 (German). · Zbl 0048.15502
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.