偶数维黎曼流形直径估计及曲面法向演化问题

【作者】 董莲莲；

【导师】 王培合；

【作者基本信息】 曲阜师范大学 ， 基础数学， 2009， 硕士

【摘要】 本文研究偶数维Riemannian流形的直径及曲面法向演化问题,共分四节.第一二节为本文的引言与预备知识.第三节首先介绍了Hausdorff距离及Gromov-Hausdorff收敛的概念.设M²ⁿ是具有度量g的2n维紧致无边单连通的Riemannian流形,S²ⁿ是欧氏空间R²ⁿ⁺¹中的单位球面.若流形M²ⁿ满足截面曲率K_M∈(0,1],体积0<V（M）≤2（1+η）V(B_?),本文得到M²ⁿ直径的一个上界估计.这里,η是某个仅依赖于n的正数,（?）为S²ⁿ上半径是;（?）π的测地球.然后给出了这类流形上的一个gap现象及流形上Laplacian算子的第一特征值的一个下界估计,即λ₁>（?）.最后利用Hausdor（?）收敛得到了这类流形直径更精确的一个上界估计及更宽的一个gap现象.第四节介绍了超曲面的概念.设Mⁿ是Rⁿ⁺¹中紧致无边连通的超曲面.若Mⁿ满足初始状态是严格凸的,得到它沿外法向演化的渐近性态在某种意义下是Rⁿ⁺¹中n维超球面.更多还原

【Abstract】 This dissertation consists of four sections. We shall investigate the diameter on even dimensionalRiemannian manifolds and asymptotic analysis to the flow of supersurfaces along their normal direction.The first and second sections are introduction and preliminaries respectively.In the third section, we first introduce the concept of Hausdorff distance and Gromov-Hausdorffconvergence. Let M²ⁿ be a 2n-dimensional compact, simply connected Riemannian manifold without boundary and S²ⁿ be the unit sphere in Euclidean space R²ⁿ⁺¹. We derive an upper bound of the diameter in this note whenever the manifold concerned satisfies that the sectional curvature K_M varies in (0,1] and the volume V（M） is not larger than 2（l +η）V(B_?π) for some positive numberηdepending only on n, where B_?πis the geodesic ball on S²ⁿ with radius （?）π. A gap phenomenon of the manifold concerned will be given out. Then we give a lower bound of the first eigenvalue of Laplacian operator on manifold M, which is,λ₁ is biggerthan （?）. Finally, by using the method of Hausdorff convergence, we get a more concise estimation of the diameter and prove the existence of a broader gap on the manifold mentioned.In the forth section, we introduce the concept of hypersurface. Let Mⁿ be a n（≥3）-dimensionalcompact, connected hypersurface without boundary in Rⁿ⁺¹. We derive asymptotic analysis of this hypersurface along its outer normal direction which is a super-sphere in some sense whenever the initial hypersurface concerned satisfies that it is strictly convex hypersurface.更多还原