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 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 ∫M|h|2e-|x|2/2 dVM<∞, where h and x denote, respectively, the second fundamental form and the position vector of the surface.
Xingxiao LI
,
Xiao LI
. ON THE LAGRANGIAN ANGLE AND THE KÄHLER ANGLE OF IMMERSED SURFACES IN THE COMPLEX PLANE C2[J]. Acta mathematica scientia, Series B, 2019
, 39(6)
: 1695
-1712
.
DOI: 10.1007/s10473-019-0617-4
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