Gradient Estimates And Liouville Type Theorems For The Nonlinear Sigma Model

This paper spreads around the nonlinear sigma model,that is,the promotion of Dirac-harmonic maps.We study the gradient estimates for the nonlinear sigma model with curvature term and gravitino.Finally,we obtain the Liouville type theorem by using gradient estimates.Chapter one indicates the research background and motivation of the whole thesis.Chapter two,we simply introduces the preliminaries of this thesis.It contains Euler-Lagrange equations,Weitzenb?ck formulas,Kato-Yau inequalities and some important estimates.Chapter three is the main part of the article.We apply the maximum principle to prove the gradient estimates for the nonlinear sigma model with gravitino.Furthermore,we can obtain the gradient estimates for Dirac-harmonic maps with curvature term when gravitino vanishes.Finally,in chapter four,we get the Liouville type theorem for the nonlinear sigma model with curvature term and gravitino respectively by using gradient estimates.