Hyperbolic Geometric Flow

This article is a review article on hyperbolic geometric flow.It mainly introduces three typical hyperbolic geometric flows on the Riemann surface proposed by De-Xing Kong and his team:standard hyperbolic geometric flow,Einstein's hyperbolic geometric flow and dissipative hyperbolic geometric flows,and studies their properties and the lifespan of classical solutions.The main content consists of the following chapters.In the first chapter,the research background and some basic properties of the hyperbolic geometric flow on the Riemann surface are first introduced.Then the hyperbolic geometric flow is extended to the Finsler manifold,and we consider its short-term existence and wave character of curvature in the fast-decreasing Berwald space.Finally,we introduce the source of the harmonic hyperbolic geometric flow and its important geometric properties.Through the contents of the three sub-sections of this chapter,the preliminary introduction to the hyperbolic geometric flow is completed,which reflects its importance in geometry and theoretical physics.The second chapter introduces the standard hyperbolic geometric flow in detail,mainly including the following three parts.The first part introduces the form and morphological analysis of the exact solution of the standard hyperbolic geometric flow.The second part introduces the existence of the classical solutions of one-dimensional and two-dimensional standard hyperbolic geometric flow on the Riemann surface,and gives an accurate lifespan.In the third part,we introduce the existence of classical solutions to the Cauchy problem for multidimensional hyperbolic geometric flow and introduce the lower bound estimation of the lifespan for classical solutions under small initial values.In the third chapter,the basic form and morphological analysis of an exact solution of Einstein's hyperbolic geometric flow are given firstly.Then the equivalence between Einstein metrics and the all-umbilical condition is introduced,and the existence of classical solutions for Einstein's hyperbolic geometric flow solution and the phenomenon of cracking is studied.Finally,we discuss the difference between the hyperbolic geometric flow and the logistic flow,which deepens our understanding of the Einstein's equation and the basic characteristics of the hyperbolic geometric flow.Chapter 4 details the hyperbolic geometric flow with general dissipative terms.It mainly includes the following five parts.The first part introduces the forms and morphological analysis of some typical exact solutions.The second part gives the concept of hyperbolic Ricci soliton.The third part introduces the geometric properties of short-term existence and wave character of curvature.In the fourth part,we give the global existence of the classical solutions and the asymptotic behavior to the Cauchy problem under the condition of small initial value.Finally,Finally,the definition and some good properties of the dissipative hyperbolic geometric flow on the modified Riemann extensions are introduced.