General curvature estimates for stable \(H\)-surfaces immersed into a space form
Let \(M\) be a stable constant mean curvature \(H\) surface immersed in the simply connected 3-manifold \(\overline{M}(c)\) with constant curvature \(c.\) Using ideas as in \textit{R. Schoen} [Ann. Math. Stud. 103, 111-126 (1983; Zbl 0532.53042)], the authors derive estimates for the norm of the traceless second fundamental form and the Gaussian curvature at the center of a relatively compact geodesic ball with the stability operator bounded from below. These estimates imply that the two notions of weak and strong Morse indices coincide for complete noncompact constant mean curvature surfaces and that a complete stable constant mean curvature \(1\) surface in the usual hyperbolic space is a horosphere, which was proved by \textit{A. da Silveira} [Math. Ann. 277, 629-638 (1987; Zbl 0627.53045)].
scientific article; zbMATH DE number 1344448