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

Abstract

In this paper, we discuss the Lagrangian angle and the K\"ahler angle of immersed surfaces in $\mathbb C^2$. Firstly, we provide an extension of Lagrangian angle, Maslov form and Maslov class to more general surfaces in $\mathbb C^2$ than Lagrangian surfaces, and then naturally extend a theorem by J.-M. Morvan to surfaces of constant K\"ahler angle, together with an application showing that the Maslov class of a compact self-shrinker surface with constant K\"ahler angle is generally non-vanishing. Secondly, we obtain two pinching results for the K\"ahler angle which imply rigidity theorems of self-shrinkers with K\"ahler angle under the condition that $\int_M |h|^2e^{-\frac{|x|^2}{2}}dV_M<\infty$, where $h$ and $x$ denote, respectively, the second fundamental form and the position vector of the surface.