On The Lagrangian Angle And The K(?)hler Angle Of Immersed Surface In The Complex Plane

This dissertation consists of two chapters where, for a given immersed surface x M2? C2 in the complex plane C2, the Lagrangian angle ?x, the K(?)hler angle and the related rigidity problems are studied (see the preprint [1]).In chapter 1, we mainly discuss the Lagrangian angle of a given immersed surface x:M2?C2 with constant K(?)hler angle ?. Firstly, we provide extensions, respectively, of Lagrangian angle, Maslov form and Maslov class to more general immersed surfaces in C2 than Lagrangian surfaces. Then we naturally extend relevant theorems to surfaces of constant K(?)hler angle, and get the following result that the mean curvature vector H, the canonical complex structure J on C2 and the Lagrangian angle ?x meet the following equation (see Theorem 1.2): sin2?x*(??x)=-(JH)T, where T and ? denote, respectively, the projection onto the tangent space TM2 of x and the gradient operator on the surface M2. Finally, an application of the result shows that the Maslov class of a compact self-shrinker surface with constant K(?)hler angle is non-trivial (see Theorem 1.3).In chapter 2, we consider the rigidity of the K(?)hler angle of x:M2 ? C2 and the classification problem for self-shrinker surfaces in terms of the K(?)hler angle ?. As the re-sult, two pinching theorems are proven:For a given self-shrinker surface x:M2 ? C2, let h and dVM denote the second fundamental form of x and volume form of the surface M2, respectively. If and the K(?)hler angle ? satisfies some additional conditions, then x(M2) is either a Lagrangian surface or a plane (see Theorem 2.4 and Theorem 2.5).