Geometry of Total Curvature on Negatively Curved Manifolds

负曲流形上总曲率的几何

• 批准号: 09640099
• 负责人: OKAYASU Takashi
• 金额: $ 1.98万
• 依托单位: Toyama University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1997
• 资助国家: 日本
• 起止时间: 1997 至 1998
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09640099/
• 关键词: Bernstein theorem; higher codimensional graph; higher order mean curvature; halfspace theorem; normal connection; 最大値原理; Bennstein Problem; 極小部分多様体

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.

1997年我们用汉密尔顿和Huisken的方法研究了总曲率如何通过平均曲率流而变化。作为副产品，我们得到了欧氏空间中极小子流形的一个伯恩斯坦型定理：定理1设u=（u^1，.，u^p）：R^n*R^p满足极小曲面方程组且其图（u）具有平坦法联络。1998<numerical formula>年，我们将霍夫曼和米克斯（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&gt;是超曲面.我们定义第k阶平均曲率H_k为H_k= SIGMA_i &lt;i_1&lt;. <i_k>联系我们lambda_<ik>，其中lambda_1，.，λ_n是M的主曲率。我们称M为椭圆型，如果以下条件处处成立：*/（mbda）H_k &gt; 0，对于 *i.定理2设k为奇整数，n为满足1 <less than or equal>k &lt;n2 <less than or equal>k的整数。若M^n * R^&lt;n+1&gt;是H_k = 0的真浸入椭圆型完备超曲面，则M不能包含在任何欧氏半空间中.

项目成果

Takashi Okayasu: "An extension of Chern-Lashof theorem to other space forms" Proceedings of the Pacific Rim Geometry Conference, International Press. (to appear).

Takashi Okasu：“陈-拉斯霍夫定理到其他空间形式的延伸”环太平洋几何会议记录，国际出版社。