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• W1970784998 abstract "Let $f:Mrightarrow mathbb {R}^{m+1}$ 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 $f_*T_pM$, $pin M$, fill the whole $mathbb {R}^{m+1}$." @default.
• W1970784998 created "2016-06-24" @default.
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• W1970784998 date "2010-11-29" @default.
• W1970784998 modified "2023-10-18" @default.
• W1970784998 title "Some geometric properties of hypersurfaces with constant $r$-mean curvature in Euclidean space" @default.
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• W1970784998 doi "https://doi.org/10.1090/s0002-9939-2010-10649-4" @default.
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