Geometry of Total Curvature on Negatively Curved Manifolds

1997年，我们用哈密顿和惠斯肯的方法研究了总曲率如何通过平均曲率流变化。作为一个副产品，我们得到了欧氏空间中极小子流形的一个Bernstein型定理.定理1假设u=(u^1，…，u^p)：r^n*R^p满足极小曲面方程组且其图(U)具有平坦的法连通.1998年，我们将Hoffman和Meek(1990)关于极小超曲面的半空间定理推广到高阶平均曲率为0的超曲面。设M^n*R^&lt；n+1；是一个超曲面。我们定义k次平均曲率H_k为：H_k=Sigma__&lt；I_1&lt；...&lt；I_k&gt；lambda_1……其中lambda_1，…，lambda_n是M的主曲率。定理2设k是奇数，n是满足1&lt；小于或等于&gt；2k的整数。如果M^n*R^&lt；n+1&gt；是适当浸入的椭圆型完备超曲面，且H_k=0，则M不能包含在任何欧几里德半空间中。

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.