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

Let M be an isometrically immersed hypersurface in the Euclidean (m+1)-dimensional space. In this paper, we exploit recent results due to the authors to analyze the stability of the differential operator associated with the rth Newton tensor of the immersion. This appears in the Jacobi operator for the variational problem of minimizing the r-mean curvature. Two natural applications are found. The first one ensures that under a mild condition on the integral of the r-mean curvature over geodesic spheres, the Gauss map meets each equator of the m-dimensional unit sphere 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 fill the whole euclidean space.