Geodesic
Symplectic critical surfaces in Kähler surfaces
Journal of the European Mathematical Society, Tome 12 (2010) no. 2, pp. 505-527

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Let M be a Kähler surface and Σ be a closed symplectic surface which is smoothly immersed in M. Let α be the Kähler angle of Σ in M. We first deduce the Euler–Lagrange equation of the functional L=∫Σ​cosα1​dμ in the class of symplectic surfaces. It is cos3αH=(J(J∇cosα)⊤)⊥, where H is the mean curvature vector of Σ in M, and J is the complex structure compatible with the Kähler form ω in M; it is an elliptic equation. We call a surface satisfying a this equation a symplectic critical surface. We show that, if M is a Kähler–Einstein surface with a nonnegative scalar curvature, each symplectic critical surface is holomorphic. We also study the topological properties of symplectic critical surfaces. By our formula and Webster’s formula, we deduce that the Kähler angle of a compact symplectic critical surface is constant, which is not true a for noncompact symplectic critical surfaces.
DOI : 10.4171/jems/207
Classification : 53-XX, 58-XX, 00-XX
Keywords: Symplectic surface, holomorphic curve, Kähler surface 
Xiaoli Han; Jiayu Li. Symplectic critical surfaces in Kähler surfaces. Journal of the European Mathematical Society, Tome 12 (2010) no. 2, pp. 505-527. doi: 10.4171/jems/207
@article{JEMS_2010_12_2_a9,
     author = {Xiaoli Han and Jiayu Li},
     title = {Symplectic critical surfaces in {K\"ahler} surfaces},
     journal = {Journal of the European Mathematical Society},
     pages = {505--527},
     year = {2010},
     volume = {12},
     number = {2},
     doi = {10.4171/jems/207},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/jems/207/}
}
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