# Symplectic critical surfaces in Kähler surfaces

### Xiaoli Han

Tsinghua University, Beijing, China### Jiayu Li

Chinese Academy of Sciences, Beijing, China

## Abstract

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 $cos_{3}αH=(J(J∇cosα)_{⊤})_{⊥}$, 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.

## 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