Some geometric properties of hypersurfaces with constant $r$-mean curvature in Euclidean space
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- by Debora Impera, Luciano Mari and Marco Rigoli PDF
- Proc. Amer. Math. Soc. 139 (2011), 2207-2215 Request permission
Erratum: Proc. Amer. Math. Soc. 141 (2013), 2221-2223.
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 $\mathbb {R}^{m+1}$.References
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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
- MR Author ID: 148315
- Email: marco.rigoli@unimi.it
- 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
- © Copyright 2010 American Mathematical Society
- 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
- MathSciNet review: 2775398