Curvature Estimates On Stable Hypersurfaces With Constant Mean Curvature In Euclidean Spaces

In this paper, using the technique in Schoen-Simon-Yau’s paper, we obtain an Lpestimate for the length of the traceless second fundamental form|φ|on strongly stablehypersurfaces with constant mean curvature in space forms. With this Lpestimate andthe mean value inequality for subharmonic functions, we derive a Colding-Minicozzi typecurvature estimate for strongly stable hypersurfaces with constant mean curvature inRn+1of arbitrary dimension, which shows that the locally controlled volume growth of extrin-sic geodesic balls yields the globally controlled volume growth of intrinsic geodesic ballsif Rn+1M=. Furthermore, we can prove an Schoen-Simon-Yau type curvature estimatefor strongly stable hypersurfaces with constant mean curvature inRn+1of arbitrary di-mension under such boundary condition. Moreover, we deduce a Bernstein-type theoremfor complete hypersurfaces with constant mean curvature of arbitrary dimension, given afinite Lp-norm total curvature condition.