Some Geometric Problems About Hypersurfaces With Free Boundary In Geodesic Balls In Space Forms

In recent years,hypersurfaces with free boundary have attracted more and more attention from geometric scholars,and the study of geometric inequalities and geometric properties on them has gradually become a hot issue in the direction of modern geometric analysis.This thesis mainly studies the prescribed mean curvature on free boundary hypersurfaces and a class of volume preserving mean curvature flows.In the first part of this thesis,we recall the classical mean curvature expression on closed hypersurfaces in Euclidean space,and consider that under a special class of conformal differential homeomorphic transformations,the mean curvature expressions in hyperbolic space and spherical space are calculated by using the formula of mean curvature under conformal transformations.In the second part of the thesis,we discuss the prescribed mean curvature of hypersurfaces with free boundary on the sphere Bn+1,that is,given the smooth function ψ on the region sandwiched by two spherical caps in the sphere,is there a hypersurface ∑ with free boundary in the sphere,so that the mean curvature of the hypersurface is the given functionψ?By constructing a conformal differential homeomorphism between the upper half space and the sphere,we transform the problem into an equivalent existence problem of solutions of quasilinear elliptic partial differential equations with Neumann boundary value in the upper half space with conformal flat metric.When the prescribed function ψ satisfies certain conditions,the uniform C0 estimate and gradient estimate of the solution of the equation can be obtained.Finally,we use the method of topological degree to prove that there is a star shaped hypersurface ∑ with free boundary in Bn+1,so that the mean curvature on E is the given function ψ.In the third part of the thesis,we study a class of volume preserving mean curvature flow problems on hypersurfaces with free boundary in the hyperbolic space BrH and spherical space Sn+1.By using Minkowski formula and simply calculating the evolution equations of volume and area,it can be obtained that this kind of mean curvature flow has the properties of preserving volume and area monotonicity,that is,the volume of the closed region enclosed by the hypersurface remains unchanged during the flow process,and the area of the hypersurface does not increase during the flow process.We prove that when the initial hypersurface is a star-shaped hypersurface with free boundary,this kind of flow equation has the property of smooth convergence,and prove that when t→∞,the limit hypersurface converged to is a spherical cap.At the same time,we give a flow method to prove the relative isoperimetric problem of star shaped hypersurfaces with free boundary in BrH and Sn+1.