Even Wei Liman Manifold Diameter Estimates And Surface Methodology To The Question Of The Evolution

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.