Complete hypersurfaces with constant mean curvature and finite total curvature (Q1389889)

Let \(x:M^n \to \overline {M}^{n+1}(c) \) be an isometric immersion of a complete hypersurface into a simply connected complete Riemannian manifold \(\overline {M}^{n+1}(c) \) with constant sectional curvature \(c\) \((c\leq 0)\). We denote by \(A\) and \(H\) the Weingarten endomorphism and the mean curvature of \(M^n .\) The hypersurface \(M^n\) is called \(H\)-hypersurface if it has constant mean curvature \((H=\text{const.})\) and we say that \(M^n\) has finite total curvature if \((*)\) \(\int _M| {A^0}| ^n dv_M<\infty\), where \(A^0 =A-H I\) is the traceless fundamental tensor of \(M^n\). It has been observed by \textit{H. Alencar} and \textit{M. P. do Carmo} [An. Acad. Bras. Cienc. 66, 265-274 (1994; Zbl 0810.53051)] that many theorems on minimal surfaces involving \(A\) extend to nonzero constant mean curvature if \(A\) is replaced by \(A^0\). It is thus a natural question to ask what the consequences are of the condition \((*)\) for complete \(H\)-hypersurfaces. In this paper, the authors firstly prove that a complete \(H\)-hypersurface with finite total curvature satisfies nice Sobolev inequalities and an equation of Simons' type. Then, by using these, they prove the following main result. Let \(x:M^n \to \overline {M}^{n+1}(c)\), \(c\leq 0\), be a complete \(H\)-hypersurface with finite total curvature. Then, given any \(\varepsilon >0\) and any \(p_0\in M\), there exists a geodesic ball \(B_{R}(p_0)\) with center \(p_0\) and radius \(R\) such that \(| A^0|(p)<\varepsilon \) for all \(p\in M\setminus B_R(p_0)\). As applications of the main result, the authors obtain several interesting corollaries. We mention the following which answers a question posed by \textit{H. Alencar} and \textit{M. P. do Carmo} [loc.cit.]. Let \(x:M^2 \to \overline {M}^3(c)\), with \(c= 0\) or \(-1\), be a complete \(H\)-surface with \(H^2>| c| \) and with finite total curvature. Then \(M^2\) is compact.