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Hypersurfaces with null higher order mean curvature. (English) Zbl 1210.53057
The constancy condition of higher elementary symmetric functions of the principal curvatures (= higher mean curvatures H r ) is an interesting problem in the global theory of hypersurfaces. For the compact case and a non-vanishing constant see [R. Walter, Math. Ann. 270, 125–145 (1985; Zbl 0536.53054)]. In the present paper, the authors deal with the case of vanishing higher mean curvature of complete and orientable hypersurfaces of dimension n in space forms. They make the assumption that an open part does not consist of totally geodesic hypersurfaces. Another assumption is that H r+1 vanishes identically and H r never vanishes. One of the results states that, under these assumptions, the manifold is foliated by complete totally geodesic submanifolds of dimension n-r.
53C42Immersions (differential geometry)
53A10Minimal surfaces, surfaces with prescribed mean curvature
[1]H. Alencar. Hipersuperfícies Mínimas de 2m Invariantes por SO(m) × SO(m). Doctor Thesis, IMPA – Brazil (1988).
[2]H. Alencar and A.G. Colares. Integral Formulas for the r-Mean Curvature Linearized Operator of a Hypersurface. Annals of Global Analysis and Geometry, 16 (1998), 203–220. · Zbl 0920.53030 · doi:10.1023/A:1006555603714
[3]H. Alencar, M. do Carmo and M.F. Elbert. Stability of Hypersurface with Vanishing r-Mean Curvatures in Euclidean spaces. J. Reine Angew. Math., 554 (2003), 201–216. · Zbl 1093.53063 · doi:10.1515/crll.2003.006
[4]H. Alencar and K. Frensel. Hypersurface Whose Tangent Geodesic Omit a Nonempty Set. Differential Geometry – A Symposium In Honour of Manfredo do Carmo, ed. New York: Longman Scientific & Technical (1991), 1–13.
[5]A. Caminha. On Hypersurface into Riemannian Space of Constant Sectional Curvature. Kodai Math J., 29 (2006), 185–210. · Zbl 1107.53037 · doi:10.2996/kmj/1151936435
[6]A. Caminha. Complete Spacelike Hypersurfaces in Conformally Stationary Lorentz manifolds. Gen. Relativ Gravit, 41 (2009), 173–189. · Zbl 1162.83304 · doi:10.1007/s10714-008-0663-z
[7]M. Dajczer et al. Submanifolds and Isometric Immersions. Publish or Perish, Houston (1990).
[8]M. Dacjzer and D. Gromoll. Gauss Parametrizations and Rigidity Aspects of Submanifolds. J. Differential Geometry, 22 (1985), 1–12.
[9]M. Dacjzer and D. Gromoll. On Spherical Submanifolds with Nullity. Proc. Am. Math. Soc., 93 (1985), 99–100. · doi:10.1090/S0002-9939-1985-0766536-0
[10]M.F. Elbert. Constant Positive 2-Mean Curvature Hypersurfaces. Illinois J. Math., 46(1) (2002), 247–267.
[11]T. Hasanis and D. Koutroufiots. A Property of Complete Minimal Surfaces. Trans. Amer. Math. Soc., 281 (1984), 833–843. · doi:10.1090/S0002-9947-1984-0722778-5
[12]H. Rosenberg. Hypersurfaces of Constant Curvature in Space Forms. Bull. Sc. Math., 117 (1993), 217–239.
[13]J. Sato. Stability of O(p + 1) × O(p + 1)-Invariant Hypersurfaces with Zero Scalar Curvature in Euclidean Space. Annals of Global Analysis and Geometry, 22 (2002), 135–153. · Zbl 1036.53039 · doi:10.1023/A:1019536730847