A Class Of Nonhomogeneous Inverse Curvature Flows In Euclidean Space

As we all know,the study of curvature flow originates from the study of geomet-ric inequalities,in which the expansibility of curvature flow plays an important role in proving the inequalities of hypersurfaces,which attracts many scholars,the most fa-mous of which are Huisken and Ilmanen who use the inverse mean curvature flow to prove the Riemannian Penrose inequality Convergence and convergence problems not only play an important role in proving geometric inequalities,but also have importan-t applications in Minkowski problems of convex geometry,etc.therefore,they also attract the interest of many experts and scholars.In this paper,we study the evolution of a star and-admissible initial hypersur-face in Euclidean space Rⁿ⁺¹under a class of non-homogeneous inverse curvature flow,where the evolution rate is a non-homogeneous function of the principal curvature.By using the theory of completely nonlinear parabolic equation,we prove that the solu-tion of this kind of flow exists for a long time and the solution shrinks to the sphere after stretching transformation.This result can be regarded as a reference for Gerhardt and Urbas generalizations of the results for non contractive invariant inverse curvature flows.The content of the paper is arranged as follows.In the first chapter,the research background and progress of inverse curvature flow are introduced,and the problems and main results of this paper are also discussed.In the second chapter,we mainly introduce the basic knowledge of inverse curva-ture flow,including the extremum principle of linear parabolic equation,the geometry of Submanifolds and so on,and we derive some basic geometric expressions of Hy-persurfaces in Euclidean space.At the same time,according to the basic knowledge,we transform this kind of inverse curvature flow equation into a completely nonlinear parabolic equation.In the third chapter,we obtain the regularity results of the completely nonlinear parabolic equation in the second chapter,including C⁰ estimator,C¹ estimator andC² estimator.In Chapter four,we prove the main theorem of this paper,that is,we obtain the long-time existence of flows and characterize the asymptotic behavior.In the fifth chapter,we summarize the related work and look forward to the future work.