Geometry of Total Curvature on Negatively Curved Manifolds

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In 1997 we studied how the total curvature changes through the mean curvature flow by using the method of Hamilton and Huisken. As a by-product, we got a Bernstein type thorem for minimal sub-manifolds in the Euclidean space.Theorem 1 Suppose that u=(u^1, ..., u^p) : R^n*R^p satisfies the system of minimal surface equation and its graph graph(u) has flat normal connection. If<<numerical formula>>then all u^i are linear functions.In 1998, we extended the halfspace theorem for minimal hypersurfaces by Hoffman and Meeks (1990) to hypersurfaces with 0 higher order mean curvature. Let M^n * R^<n+1> be a hypersurface. We define k-th mean curvatue H_k byH_k= SIGMA__<i_1<...<i_k> lambda_1 ... lambda_<ik>,where lambda_1, ..., lambda_n are principal curvatures of M.Suppose k is odd. We call M elliptic type if the following condition holds everywhere : */(mbda) H_k > 0 for *i. Note that this condition does not depend on the choice of the unit normal vector since k is odd.Theorem 2 Let k be an odd integer, n an ineger satisfying 1 <less than or equal> k < n <less than or equal> 2k. If M^n * R^<n+1> is a properly immersed elliptic type complete hypersurf ace with H_k = 0, then M cannot be contained in any Euclidean halfspace.