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

Abstract

In this paper, we discuss the Lagrangian angle and the Kähler 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 Kähler angle, together with an application showing that the Maslov class of a compact self-shrinker surface with constant Kähler angle is generally non-vanishing. Secondly, we obtain two pinching results for the Kähler angle which imply rigidity theorems of self-shrinkers with Kähler angle under the condition that ${\int }_M|h{|}^2{e}^{-\frac{|x{|}^2}{2}}d{V}_M<\infty$, where $h$ and $x$ denote, respectively, the second fundamental form and the position vector of the surface.

References in corpus (1)

• Maslov class and minimality in Calabi-Yau manifolds