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

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 $\mathbb {R}^{m+1}$.