Nonlinear Flows And Curvature Estimates

In this thesis,we research exterior non-collapsing estimates and interior noncollapsing estimates for a fully nonlinear inverse curvature flow for inverse-concave speed functions in space forms,and longtime existence of a class of piece-wise logarithmic Gauss curvature flow.This thesis is divided into five chapters to illustrate.Chapter 1 is introduction,which introduces the background and the developments of curvature flows,inverse curvature flows and Minkowski problem,and the main results of this thesis.In Chapter 2,we introduce the preliminaries of this thesis.It contains the properties of inverse-concave curvature function,viscosity solution and geometry of ball curvature,convex geometry in the sphere,the piece-wise logarithmic Gauss curvature flow,the fundamental formulas and the projection coordinate.In Chapter 3,we study the special properties of ball curvature,which is a twopoints function.We consider the sphere and the hyperbolic space as the embedded submanifold of Euclidean space and Minkowski space,respectively.By a suitable choice of the coordinates system,we obtain a differential inequality by maximum principle of a two-point function.Then,we consider exterior ball curvature estimates for inverse curvature flow,for which the speed is a monotone increasing,symmetric,homogeneous of degree one and inverse-concave function.Selecting a suitable auxiliary function and using the differential inequality,we derive exterior non-collapsing estimates for a fully nonlinear inverse curvature flow.Finally,we investigate terms of the differential inequality in previous article,and obtain optimal inscribed estimates by choosing a different auxiliary function.In Chapter 4,we mainly research longtime existence of a class of piece-wise logarithmic Gauss curvature flow.By the projection of hemisphere onto the hyperplane of Euclidean space,we transform convex hypersurface in the sphere to convex hypersurface in Euclidean hyperplane,and get the a priori estimates of these piece-wise logarithmic Gauss curvature flows.Then,we obtain longtime existence of a class of piece-wise logarithmic Gauss curvature flow in giving index condition.Finally,we give an application of these piece-wise logarithmic Gauss curvature flows.Make use of different value of the monotone functional J,we give that prescribed curvature problem has at least two solutions when β > 0.In Chapter 5,we summarize the main contents of this thesis.Moreover,we put forward some questions that will be further studied in the future.