Some geometric properties of hypersurfaces with constant -mean curvature in Euclidean space

Authors:
Debora Impera, Luciano Mari and Marco Rigoli

Journal:
Proc. Amer. Math. Soc. **139** (2011), 2207-2215

MSC (2010):
Primary 53C21, 53C42; Secondary 58J50, 53A10

DOI:
https://doi.org/10.1090/S0002-9939-2010-10649-4

Published electronically:
November 29, 2010

Erratum:
Proc. Amer. Math. Soc. 141 (2013), 2221-2223

MathSciNet review:
2775398

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let be an isometrically immersed hypersurface. In this paper, we exploit recent results due to the authors to analyze the stability of the differential operator associated with the th Newton tensor of . This appears in the Jacobi operator for the variational problem of minimizing the -mean curvature . Two natural applications are found. The first one ensures that under a mild condition on the integral of over geodesic spheres, the Gauss map meets each equator of infinitely many times. The second one deals with hypersurfaces with zero -mean curvature. Under similar growth assumptions, we prove that the affine tangent spaces , , fill the whole .

**1.**H. Alencar and A. G. Colares,*Integral formulas for the -mean curvature linearized operator of a hypersurface.*, Ann. Global Anal. Geom.**16**(1998), 203-220. MR**1626663 (99k:53075)****2.**N. Aronszajn,*A unique continuation theorem for solutions of elliptic partial differential equations or inequalities of second order*, J. Math. Pures Appl.**36**(1957), 235-249. MR**0092067 (19:1056c)****3.**J. L. M. Barbosa and A. G. Colares,*Stability of hypersurfaces with constant -mean curvature*, Ann. Glob. Anal. Geom.**15**(1997), 277-297. MR**1456513 (98h:53091)****4.**B. Bianchini, L. Mari, and M. Rigoli,*Spectral radius, index estimates for Schrödinger operators and geometric applications*, Journ. Funct. Anal.**256**(2009), 1769-1820. MR**2498559 (2010a:58038)****5.**S. Y. Cheng and S. T. Yau,*Hypersurfaces with constant scalar curvature*, Math. Ann.**225**(1977), 195-204. MR**0431043 (55:4045)****6.**M. F. Elbert,*Constant positive -mean curvature hypersurfaces*, Ill. J. Math.**46**(2002), 247-267. MR**1936088 (2003g:53103)****7.**D. Fisher-Colbrie,*On complete minimal surfaces with finite Morse index in three manifolds*, Invent. Math.**82**(1985), 121-132. MR**808112 (87b:53090)****8.**D. Fisher-Colbrie and R. Schoen,*The structure of complete stable minimal surfaces in -manifolds of nonnegative scalar curvature*, Comm. Pure Appl. Math.**XXXIII**(1980), 199-211. MR**0562550 (81i:53044)****9.**G. H. Hardy, J. E. Littlewood, and G. Polya,*Inequalities*(2nd ed.), Cambridge University Press, 1952. MR**0046395 (13:727e)****10.**J. Hounie and M. L. Leite,*The maximum principle for hypersurfaces with vanishing curvature functions*, J. Differential Geom.**41**(1995), 247-258. MR**1331967 (96b:53080)****11.**W. F. Moss and J. Piepenbrink,*Positive solutions of elliptic equations*, Pac. J. Math.**75**(1978), 219-226. MR**500041 (80b:35008)****12.**S. Pigola, M. Rigoli, and A. G. Setti,*Vanishing and finiteness results in geometric analysis. A generalization of the Bochner technique*, Progress in Mathematics, vol. 266, Birkhäuser Verlag, Basel, 2008. MR**2401291 (2009m:58001)****13.**H. Rosenberg,*Hypersurfaces of constant curvature in space forms*, Bull. Sc. Math., Série**117**(1993), 211-239. MR**1216008 (94b:53097)**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2010):
53C21,
53C42,
58J50,
53A10

Retrieve articles in all journals with MSC (2010): 53C21, 53C42, 58J50, 53A10

Additional Information

**Debora Impera**

Affiliation:
Dipartimento di Matematica, Università degli Studi di Milano, Via Saldini 50, I-20133 Milano, Italy

Email:
debora.impera@unimi.it

**Luciano Mari**

Affiliation:
Dipartimento di Matematica, Università degli Studi di Milano, Via Saldini 50, I-20133 Milano, Italy

Email:
luciano.mari@unimi.it

**Marco Rigoli**

Affiliation:
Dipartimento di Matematica, Università degli Studi di Milano, Via Saldini 50, I-20133 Milano, Italy

Email:
marco.rigoli@unimi.it

DOI:
https://doi.org/10.1090/S0002-9939-2010-10649-4

Received by editor(s):
March 31, 2010

Received by editor(s) in revised form:
June 14, 2010

Published electronically:
November 29, 2010

Communicated by:
Jianguo Cao

Article copyright:
© Copyright 2010
American Mathematical Society