Studies On Geometric Curvature Flow And Rigidity Problem For Manifolds

In this thesis,we mainly study the geometric curvature flows of manifolds and rigidity problem for submanifolds.To construct the Yamabe metric,Hamilton introduced the(normalized)Yam-abe flow and proved the long time existence and the uniqueness of flow.Chow proved the smooth convergence theorem for the Yamabe flow on compact and locally confor-mally flat Riemannian manifold with initial metric having positive Ricci curvature.Ye further studied the smooth convergence of the Yamabe flow on compact Rieman-nian manifold.We prove that under an integral curvature pinching condition,the Yamabe flow on a locally conformally flat Riemannian manifold converges smoothly to a metric with constant sectional curvature.The ancient solution is a class of mean curvature flow solution.The translating solution is a special ancient solution,the time slice of which is called a translator.Bao-Shi obtained a Bernstein type theorem for translators in Rⁿ⁺¹.Kunikawa and Xin investigated the Bernstein type theorem for translators with arbitrary codimen-sion.Kunikawa also proved a Bernstein type theorem for ancient solutions of the hypersurface mean curvature flow in Rⁿ⁺¹.We prove a Bernstein type theorem for ancient solutions of the mean curvature flow in the Euclidean space with arbitrary codimension under the pinching condition of the slope function.Motivated by the smooth convergence of the mean curvature flow,Huisken-Sinestrari,Lynch-Nguyen and Risa-Sinestrari etc.proved rigidity theorems for the ancient solutions of the mean curvature flow in spheres.Lei-Xu-Zhao obtained a rigidity theorem for ancient solutions of the mean curvature flow in the sphere with arbitrary codimension.Based on the work of Pu-Su-Xu on the mean curvature flow,we prove a rigidity theorem for ancient solutions of the surface mean curvature flow in 4-dimensional sphere under the pinching condition of the normal curvature.Gauchman proved a rigidity theorem for closed minimal submanifolds satisfying?(u)?1/3 in the sphere.Gauchman,Coulton etc.obtained a similar theorem for compact and totally real minimal submanifold in the complex projective space and the quaternion projective space.Xu-Huang-Zhao proved a pinching theorem for the geometric invariant?for submanifolds with parallel mean curvature in the sphere.We prove a pinching theorem of?for complete submanifolds with parallel mean curvature in the complex projective space and the quaternion projective space.