On the Lagrangian angle and the Kähler angle of immersed surfaces in the complex plane $\bbc^2$

In this paper, we discuss the Lagrangian angle and the Kahler angle of immersed surfaces in ℂ2. Firstly, we provide an extension of Lagrangian angle, Maslov form and Maslov class to more general surfaces in ℂ2 than Lagrangian surfaces, and then naturally extend a theorem by J.-M. Morvan to surfaces of constant Kahler angle, together with an application showing that the Maslov class of a compact self-shrinker surface with constant Kahler angle is generally non-vanishing. Secondly, we obtain two pinching results for the Kahler angle which imply rigidity theorems of self-shrinkers with Kahler angle under the condition that \({\smallint _M}{\left| h \right|^2}{{\rm{e}}^{ - {{{{\left| x \right|}^2}} \over 2}}}{\rm{d}}{V_M}\; < \;\infty \), where h and x denote, respectively, the second fundamental form and the position vector of the surface.