A Class Of Inverse Mean Curvature Flows In Rⁿ⁺¹

The problem of inverse curvature flow not only comes from the proof of the Pen-rose inequality in physics,but also has important research significance in mathematics.In particular,the long-term existence of inverse curvature flow(such as inverse mean curvature flow,inverse Gaussian curvature flow,etc.)and the characterization of its asymptotic behavior;short-term existence and singularity classification within a limit-ed time are the current research hot spot.At the same time,the study of such problems has also promoted the development of interdisciplinary subjects such as submanifold geometry,partial differential equations,and functional analysis.In this paper,we discuss a kind of inverse mean curvature flow with isotropy in Euclidean space.This flow is a natural generalization of inverse mean curvature flow.We transform the flow equation into a second-order fully nonlinear parabolic equation by re parameterizing the flow.Then we use the theory of parabolic equation to obtain the long-term existence of this kind of inverse mean curvature flow,and gradually near behavior is characterized.The content of this paper is as follows:In the first chapter,we mainly introduce the research background and current situ-ation of the inverse curvature flow.At the same time,we give the main theorem of this paper,and emphasize that the conclusion of this paper is a natural generalization of the previous results.In the second chapter,we give some basic knowledge.We not only introduce the maximum principle of parabolic equation,but also express some geometric quantities of hypersurface,which is a special sub-manifold.At the same time,we give the basic geometric quantities of developing hypersurface.In the third chapter,we introduce in detail how to transform a vector equation system in the form of flow into a single second-order fully nonlinear parabolic equation by means of reparameterization.At the same time,we calculate some basic geometric evolution equations.In the fourth chapter,we prove the main theorem of this paper,that is,we obtain the long-time existence of flow,and give the characterization of asymptotic behavior.