Abstract
A classical result of Alexandrov [10] states that a compact hypersurface with constant mean curvature embedded in Euclidean space must be a round sphere. The original proof is based on a clever use of the maximum principle for elliptic partial differential equations. This method, now called the Alexandrov’s reflexion method, also works for hypersurfaces in ambient spaces having a sufficiently large number of isometric reflexions, for instance in the hyperbolic space. It is worth to observe that, in an analytical context, this is the root of what has been called the “moving plane” technique, initiated by the pioneering work of Serrin, [254], and over and over successfully applied to prove special symmetries of solutions to certain PDE’s.
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References
A.D. Alexandrov, A characteristic property of spheres. Ann. Mat. Pura Appl. (4) 58, 303–315 (1962)
L.J. Alías, M. Dajczer, Uniqueness of constant mean curvature surfaces properly immersed in a slab. Comment. Math. Helv. 81(3), 653–663 (2006)
L.J. Alías, M. Dajczer, Constant mean curvature hypersurfaces in warped product spaces. Proc. Edinb. Math. Soc. (2) 50(3), 511–526 (2007)
L.J. Alías, M. Dajczer, Normal geodesic graphs of constant mean curvature. J. Differ. Geom. 75(3), 387–401 (2007)
L.J. Alías, J.H.S. de Lira, M. Rigoli, Geometric elliptic functionals and mean curvature. Ann. Sc. Norm. Super. Pisa Cl. Sci. To appear in Vol. XV (2016). Submitted: 2013-07-30 Accepted: 2014-07-09 pp. 44 DOI Number: 10.2422/2036-2145.201308_003
L.J. Alías, D. Impera, M. Rigoli, Hypersurfaces of constant higher order mean curvature in warped products. Trans. Am. Math. Soc. 365(2), 591–621 (2013)
X. Cheng, H. Rosenberg, Embedded positive constant r-mean curvature hypersurfaces in M m ×R. An. Acad. Brasil. Ciênc. 77(2), 183–199 (2005)
M. Dajczer, P.A. Hinojosa, J.H.S. de Lira, Killing graphs with prescribed mean curvature. Calc. Var. Partial Differ. Equ. 33(2), 231–248 (2008)
S.C. García-Martínez, D. Impera, M. Rigoli, A sharp height estimate for compact hypersurfaces with constant k-mean curvature in warped product spaces. Proc. Edinb. Math. Soc. (2) 58(2), 403–419 (2015)
D. Hoffman, J.H.S. de Lira, H. Rosenberg, Constant mean curvature surfaces in M 2 ×R. Trans. Am. Math. Soc. 358(2), 491–507 (2006)
S. Montiel, Unicity of constant mean curvature hypersurfaces in some Riemannian manifolds. Indiana Univ. Math. J. 48(2), 711–748 (1999)
S. Montiel, A. Ros, Compact hypersurfaces: the Alexandrov theorem for higher order mean curvatures, in Differential Geometry. Pitman Monographs & Surveys in Pure & Applied Mathematics, vol. 52 (Longman Scientific & Technical, Harlow, 1991), pp. 279–296
S. Pigola, M. Rigoli, A.G. Setti, A remark on the maximum principle and stochastic completeness. Proc. Am. Math. Soc. 131(4), 1283–1288 (2003) (electronic)
S. Pigola, M. Rigoli, A.G. Setti, A Liouville-type result for quasi-linear elliptic equations on complete Riemannian manifolds. J. Funct. Anal. 219(2), 400–432 (2005)
J. Serrin, A symmetry problem in potential theory. Arch. Ration. Mech. Anal. 43, 304–318 (1971)
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Alías, L.J., Mastrolia, P., Rigoli, M. (2016). Hypersurfaces in Warped Products. In: Maximum Principles and Geometric Applications. Springer Monographs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-24337-5_7
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