Studies On Eigenvalues And Curvature Flows For Submanifolds

The topics related to eigenvalues of geometric operators and geometric curvature flows are very important in geometric analysis.In this paper,some new results are proved on the basis of previous studies.On the one hand,we first study the P(?)lya conjecture on the eigenvalues of the Laplacian and get some improved lower bounds for the Dirichlet eigenvalues problem.Besides,we also get an improved lower bound for the clamed plate eigenvalue problem.Under an intergarl cuvature pinching condition,we prove the gradient estimate for the positive solution of the Allen-Cahn equation.Moreover,we obtain new lower bounds for the first eigenvalues of the f-Laplacain and the p-Laplacian.On the other hand,when we study the Dirichlet eigenvalue problems,we get an improved estimate for the rato of k-th eigenvale to the first eigenvalue.In addition,we obtain some comparison theorems for several famous eigenvalues on an complete weighted Riemannian manifold.These comparison theorems not only improve the previous results in numerical value,but also remove some curvature constraints.For the curvature flows,we obtain a cylindrical estimation for the mean curvature flow in hyperbolic space,which generalizes the previous results of cylindrical estimations in the Euclidean space and the spherical space.And we give a rigdity theorem for the ancient solution of the mean curvature flow in the four-dimensional Euclidean space.For a class of the inverse curvature flows,we prove that if a hypersurface in an n + 1-dim Reissner-Nordstr¨om-ADs manifold is star-shaped and k-convex,then the flow exist for t ? [0,?)and the second fundamental forms convergece to a constant.