Symplectic critical surfaces in Kähler surfaces

  • Xiaoli Han

    Tsinghua University, Beijing, China
  • Jiayu Li

    Chinese Academy of Sciences, Beijing, China

Abstract

Let be a Kähler surface and be a closed symplectic surface which is smoothly immersed in . Let be the Kähler angle of in . We first deduce the Euler–Lagrange equation of the functional in the class of symplectic surfaces. It is , where is the mean curvature vector of in , and is the complex structure compatible with the Kähler form in ; it is an elliptic equation. We call a surface satisfying a this equation a symplectic critical surface. We show that, if 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.

Cite this article

Xiaoli Han, Jiayu Li, Symplectic critical surfaces in Kähler surfaces. J. Eur. Math. Soc. 12 (2010), no. 2, pp. 505–527

DOI 10.4171/JEMS/207