Some Generalizations Of Three Circles Theorems In Riemannian And Discrete Geometry

Hadamard three circles theorem on the maximum of modules of holomorphic functions in classical complex analysis has many forms of generalization,which is used to characterize the behavior of some functions or solutions of partial differential equations at infinity.Recently,Liu Gang proved the three circles theorem on K¨ahler manifold,and used it to study holomorphic functions with polynomial growth.in the Riemann case,Shing-Tung Yau studied the harmonic functions with polynomial growth and introduced the frequency function of harmonic functions.The monotone nondecreasing property of frequency function is equivalent to the three circles theorem in integral form.At the same time,discrete geometry,especially discrete geometry analysis,is also flourishing.As an interdisciplinary field of graph theory,geometry,discrete group theory and probability theory,discrete geometric analysis is becoming more and more important both in theory and in application.However,no object similar with the complex structure on complex manifold had been found on graphs,so it is not realistic to directly extend the three circle theorem in complex geometry to graphs.Therefore,we first obtain some form of three circle theorem for Riemannian manifolds.Secondly,we try to extend the three circles theorem to discrete graphs.We hope to use the three circles theorem to study the further geometric properties of graphs.