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Let $f:M\ra \erre^{m+1}$ be an isometrically immersed hypersurface. In thispaper, we exploit recent results due to the authors in \cite{bimari} to analyzethe stability of the differential operator $L_r$ associated with the $r$-thNewton tensor of $f$. This appears in the Jacobi operator for the variationalproblem of minimizing the $r$-mean curvature $H_r$. Two natural applicationsare found. The first one ensures that, under the mild condition that theintegral of $H_r$ over geodesic spheres grows sufficiently fast, the Gauss mapmeets each equator of $\esse^m$ infinitely many times. The second one dealswith hypersurfaces with zero $(r+1)$-mean curvature. Under similar growthassumptions, we prove that the affine tangent spaces $f_*T_pM$, $p\in M$, fillthe whole $\erre^{m+1}$.

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