Abstract
In this paper, we discuss the Lagrangian angle and the Kähler 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 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 \({\smallint _M}{\left| h \right|^2}{{\rm{e}}^{ - {{{{\left| x \right|}^2}} \over 2}}}{\rm{d}}{V_M}\; < \;\infty \), where h and x denote, respectively, the second fundamental form and the position vector of the surface.
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The first author was supported by National Natural Science Foundation of China (11671121, 11871197).
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Li, X., Li, X. On the Lagrangian Angle and the Kähler Angle of Immersed Surfaces in the Complex Plane ℂ2. Acta Math Sci 39, 1695–1712 (2019). https://doi.org/10.1007/s10473-019-0617-4
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DOI: https://doi.org/10.1007/s10473-019-0617-4