Complete hypersurfaces with constant mean curvature and finite index in hyperbolic spaces

Caccioppoli’s inequalities on constant mean curvature hypersurfaces in Riemannian manifolds

“…Rearranging terms in (40), one obtains (7). Now, we state the result of Theorem 3.1 in the particular case of N being a product of manifolds with constant curvature.…”

“…We restrict ourselves to the complete noncompact case since in R n+1 , S n+1 and H n+1 , the only weakly stable compact hypersurfaces of constant mean curvature are geodesic spheres [7]. We recall that the classification of stable constant mean curvature surfaces in R 3 It is proved in [27], [16] and [40] that, for n = 3, 4, in R n+1 (respectively in H n+1 ) there is no finite index, complete, noncompact hypersurface with constant mean curvature H = 0 (respectively H large enough). The analogous problem in higher dimension is still open.…”

“…Furthermore, G. de Oliveira and the third author [20] found many examples of stable minimal surfaces in H 3 . It is proved in [27], [16] and [40] that, for n = 3, 4, in R n+1 (respectively in H n+1 ) there is no finite index, complete, noncompact hypersurface with constant mean curvature H = 0 (respectively H large enough). The analogous problem in higher dimension is still open.…”