Global Pinching Theorem And The Iteration Procedure Of Moser On Manifolds

In this paper, we mainly study the application of global pinching theorem in harmonic functions and the iteration procedure of Moser .It consists of four sections.In part one, we briefly talk about the history and background of global pinching theorem and the iteration procedure of Moser. In part two, we give some inequalities, formulas and definitions such as Sobolev constant, cut-off function, space form and so on. In part three,firstly we introduce the global pinching lemma from [2],then under the same pinching condition, we obtain the gradients of the harmonic functions on the non-compact complete manifolds are constants. Moreover, when the volumes of the manifolds are infinite, we have that these functions themselves are constants. In the last part, we first introduce the iteration procedure of Moser and then we estimate locally the weak solution of the Poisson equation.