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

TL;DR

This paper investigates geometric properties of hypersurfaces with constant r-mean curvature in Euclidean space, focusing on stability analysis of associated differential operators and implications for the Gauss map and tangent spaces.

Contribution

It applies recent stability results to derive new geometric consequences for hypersurfaces with constant r-mean curvature, including Gauss map behavior and tangent space coverage.

Findings

Gauss map intersects each equator infinitely often under certain conditions.

Hypersurfaces with zero (r+1)-mean curvature have tangent spaces filling the entire Euclidean space.

Stability analysis links curvature growth to geometric properties.

Abstract

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 $\erre^{m+1}$.