Upper Bound Estimation Of K(?)hler Angle On Symplectic Critical Surface

In this paper,we mainly study the upper bound estimation of the K(?)hler angle α on a symplectic critical surface.For a closed two-dimensional β-symplectic critical surface E in the K(?)hler surface M.α is defined as the K(?)hler angle on E.If Lq(∑):=∫∑1/cosqαdV∑is bounded for some q>3,then there exists a uniform upper bound estimate for the K(?)hler angle by Morse iterative.For q>4,a uniform upper bound estimate for α is known and we carry out the range of q.The paper contains three parts as following:In Chapter 1,we firstly introduce the research background and present situation of Holomorphic curves in K(?)hler surfaces,then the relationship between the upper bound estimation of K(?)hler angle α and the existence of Holomorphic curves in the K(?)hler surfaces are expounded.Finally,we give the main results of this paper.In Chapter 2,we introduce some basic definitions,then derive a Sobolev inequality on manifolds and two important inequalities on β-symplectic critical surfaces,which are mainly used for Morse iteration.In Chapter 3,using the method of Morse iteration,an upper bound estimate of K(?)hler angle α is obtained.