Mean Curvature Type Flow And The Application Of Curvature Flow In Eigenvalue Isoperemetric Problem

Our main goal is to study the application of mean curvature flows. Especially, we give a further discussion for the mean curvature type flow, which was established firstly by Guan and Li in their research for the quermassintegral inequality. Meanwhile. We also study the first nonzero eigenvalues for the Laplace and P-Laplace operators on closed hypersurfaces in Euclidean space by inverse mean curvature, on the basis of proving inequalities which preserving along the inverse mean curvature flow, we give isoperimetric estimates for both eigenvalues.In the first chapter, we mainly introduce the background for curvature flows and their applications. The Ricc flow, Yamabe flow and their applications were referred firstly, then we give a simple classification for all mean curvature flows and presented their major results and recent progress. At last, applications of mean curvature flows were introduced in detail, which included some questions we concerned.In the second chapter, we give some fact about star-shaped hypersurfaces in Eu-clidean space and rotationally symmetric space, also the convex cone in Euclidean s-pace.In the third chapter, we study the mean curvature type flow with perpendicular Neumann boundary condition. Specifically, for a star-shaped hypersurface inside a con-vex cone in Euclidean space, let them contact perpendicularly, then evolves the surface by the mean curvature type flow. The question is equivalent to a divergent parabolic equation with perpendicular Neumann boundary condition. By the parabolic maximal principle with boundary condition, we obtain uniform estimates for both norm and gra-dient of the equation, and then the long time existence. At last, we proved that the evolving hypersurfaces converge exponentially to some part of a sphere inside the cone and centered at the vertex of it, also, the volume of the domain enclosed by the surface and the cone is invariant in all this process.In chapter four, we study the mean curvature flow in rotationally symmetric s-paces. After a comprehensive investigation for the rotationally symmetric space and the closed hypersurface in it, we defined the mean curvature type flow in this space, and proved that the volume enclosed by the surface is invariant and the area for it is decline monotonously for all time the flow exist. On the condition that the initial hpersurface is star-shaped, this question equivalent to an initial-value problem of a divergent parabolic equation defined on the unit sphere. By a complex calculation, we obtain the uniform estimates of both norm and gradient for it, and then the long time existence. At last, the flow converge to some sphere exponentially. As a result, we extended Guan’s corresponding conclusions in space forms.In chapter five, we study the first nonzero eigenvalues of Laplace and P-Laplace operators on closed hypersurfaces in Euclidean space by inverse mean curvature flow. Firstly, we obtained the evolution equations for both eigenvalues along the general form of mean curvature flow in space forms, which unified the previous results. Then, we proved new inequalities preserving along the inverse mean curvature flow, on the basis of obtaining the monotonous qualities of the eigenvalues, the isoperimetric estimates obtained by the help of the inequalities we proved.