  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 $\erre^{m+1}$.
