Geometric Equations On Graphs And Discrete Curvature Flows

Graph theory is a branch of discrete mathematics,which is widely used in many fields.In recent decades,using geometric and analytical methods to study the related problems on graphs has attracted wide attentions.In particular,discretizing some important problems from differential manifolds to graphs is very representative.In this thesis,we mainly study the existence of strictly positive solutions of Yamabe type equations with discrete p-Laplacian on locally finite graphs,and discuss the existence of non-negative solutions of a type of nonlinear equations derived from Yamabetype equations on graphs.In addition,by generalizing the discrete Laplacian from p = 2to p > 1,we also study the problem of finding constant curvature metric on surfaces with combinatorial p-Calabi flows under the background of Euclidean geometry and hyperbolic geometry.This thesis is organized as follows:The first chapter is the introduction.In Section 1.1.1,we mainly introduce the historical background and research status of the Yamabe-type problems on weighted graphs,and we mainly introduce that of the combinatorial Calabi flow in Section 1.1.2.Finally,in Section 1.2,we give our main five results of this thesis.In Chapter 2,we introduce some basic definitions of weighted graphs and some related concepts of discrete Laplacican operators in Section 2.1.Section 2.2 mainly introduces circle packing metric,combinatorial Gauss curvature and combinatorial pCalabi flow.In Chapter 3,let G =?V,E?be a connected infinite and locally finite weighted graph,?pbe the p-th discrete graph Laplacian.In this chapter,we consider the p-th Yamabe type equation???on G,where h and g are known,2 < ? ? p.The prototype of this equation comes from the smooth Yamabe equation on an open manifold.We prove that the above equation has at least one positive solution on G.In Chapter 4.Let G =?V,E?be a locally finite connected weighted graph,?pbe the p-th graph Laplacian.We consider the p-th nonlinear equation???on G,where p > 2,h,f satisfy certain assumptions.Grigor'yan-Lin-Yang[1]proved the existence of the solution to the above nonlinear equation in a bounded domain ? ? V.In this chapter,we show that there exists a strictly positive global solution to the above nonlinear equation in the infinite set V.To the m-order differential operator ?_m,p,we also prove the existence of the nontrivial solution to the analogous nonlinear equation.In Chapter 5,for triangulated surfaces and any p > 1,we introduce the combinatorial p-th Calabi flows which precisely equal the combinatorial Calabi flows first introduced in H.Ge's thesis[3]?or see H.Ge[2]?when p = 2.The difficulties for the generalizations come from the nonlinearity of the p-th flow equation when p 2.Adopting different approaches,we show that the solution to the combinatorial p-th Calabi flow exists for all time and converges if and only if there exists a circle packing metric of constant?zero resp.?curvature in Euclidean?hyperbolic resp.?background geometry.Our results generalize the work of H.Ge[2],Ge-Xu[4]and Ge-Hua[5]on the combinatorial Calabi flow from p = 2 to any p > 1.In Chapter 6,for any p > 1 and triangulated surfaces,we introduce the combinatorial p-th Ricci flow which exactly equals the combinatorial Ricci flows first introduced by Chow-Luo[6]when p = 2.Then we show the long time existence and partial convergence of the solution to the combinatorial p-th Ricci flow.Additionally,different from the p = 2 case,we show that the solution to the combinatorial p-th Ricci flow has no exponential convergence for general p > 1,p 2.Our results partially generalize Chow-Luo's work on the combinatorial Ricci flow from p = 2 to any p > 1.