Stable constant mean curvature hypersurfaces in \(\mathbb R^{n+1}\) and \(\mathbb H^{n+1}(-1)\)
A hypersurface \(x\colon M^n\to \overline N^{n+1}\) in a Riemannian manifold with constant mean curvature \(H\) is called strongly stable if, for all \(f\in C_0^\infty(M)\), \[ \int_M\{| \nabla f| ^2-(\overline{\text{Ric}}(\nu,\nu)+| A| ^2)f^2\}\, dv \geq 0, \] where \(\nabla f\) is the gradient of \(f\). It is called stable if the above inequality holds only for those \(f\) that satisfy \(\int_M f\,dv=0\). In this paper, it is proved that complete stable hypersurfaces of \(\mathbb R^{n+1}\) (or \(\mathbb H^{n+1}(-1)\)), \(n=3,4,5\), with constant mean curvature \(H>0\) (or \(H>1\), in the case of \(\mathbb H^{n+1}(-1)\)) and finite \(L^2\)-norm of the traceless second fundamental form \(\phi\) are geodesic spheres. The proof depends on a clever use of various analytic inequalities which imply the following. Let \(M^n\) \((n\leq 7)\) be a complete, noncompact, stable hypersurface in a simply connected Riemannian manifold \(Q^{n+1}(k)\) with constant curvature \(k\). Assume that the hypersurface has constant mean curvature \(H\geq-k\) and that \(\int_M| \phi| ^2<\infty\). Then \(\int_M| \phi| ^5<\infty\) (this shows why one needs \(3\leq n\leq 5\)). Putting this together with a result of do Carmo-Cheung-Santos which state that such hypersurfaces are compact and the fact that compact constant mean curvature hypersurfaces of \(Q^{n+1}(k)\) are geodesic spheres, one obtains the results of this paper.
scientific article; zbMATH DE number 2176478