Some Results On F-harmonic Maps

In this paper,we generalize some gap results known for harmonic maps to F-harmonic maps and F-harmonic maps with potential.In particular, we prove that, under a certain level of energy depending on the curvature of the domain and target manifolds,the only F-harmonic maps are the constant ones.The main tools are Bochner-Weitzenbock and Reilly-type formulas involving the F-laplace operator.This paper is made up of five parts as follows:In chapter one,we will give a general introduction to the history and the recent researches on this field.In chapter two, we mainly introduce the preliminary knowledge for the thesis.In chapter three, we mainly use a Bochner-type formula of F-harmonic maps to study the Liouville property of F-harmonic maps.In chapter four,we will make use of a Bochner-type formula and Reilly-type formula of F-harmonic maps to study some gap properties of F-harmonic maps and F-harmonic maps with potential.In chapter five,we will discuss the gap level for energy density of F-harmonic maps and gain an important inequality.