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

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

2010-03-31

arxiv.org/abs/1003.6035v3

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

Mathematics

Differential Geometry

10 pages, corrected typos

Scientific paper

Let $f:M\ra \erre^{m+1}$ be an isometrically immersed hypersurface. In this paper, we exploit recent results due to the authors in \cite{bimari} 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 the mild condition that the integral of $H_r$ over geodesic spheres grows sufficiently fast, the Gauss map meets each equator of $\esse^m$ 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 $\erre^{m+1}$.

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