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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Some geometric properties of hypersurfaces with constant $ r$-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
Published electronically: November 29, 2010
Erratum: Proc. Amer. Math. Soc. 141 (2013), 2221-2223
MathSciNet review: 2775398
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Abstract: Let $ f:M\rightarrow\mathbb{R}^{m+1}$ be an isometrically immersed hypersurface. In this paper, we exploit recent results due to the authors to analyze the stability of the differential operator $ L_r$ associated with the $ r$th Newton tensor of $ f$. This appears in the Jacobi operator for the variational problem of minimizing the $ r$-mean curvature $ H_r$. Two natural applications are found. The first one ensures that under a mild condition on the integral of $ H_r$ over geodesic spheres, the Gauss map meets each equator of $ \mathbb{S}^m$ infinitely many times. The second one deals with hypersurfaces with zero $ (r+1)$-mean curvature. Under similar growth assumptions, we prove that the affine tangent spaces $ f_*T_pM$, $ p\in M$, fill the whole $ \mathbb{R}^{m+1}$.


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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: http://dx.doi.org/10.1090/S0002-9939-2010-10649-4
PII: S 0002-9939(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