# Symplectic critical surfaces in Kähler surfaces

• Volume: 012, Issue: 2, page 505-527
• ISSN: 1435-9855

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## Abstract

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Let $M$ be a Kähler surface and $\Sigma$ be a closed symplectic surface which is smoothly immersed in $M$. Let $\alpha$ be the Kähler angle of $\Sigma$ in $M$. We first deduce the Euler-Lagrange equation of the functional $L={\int }_{\Sigma }\frac{1}{cos\alpha }d\mu$ in the class of symplectic surfaces. It is ${cos}^{3}\alpha H={\left(J{\left(J\nabla cos\alpha \right)}^{\top }\right)}^{\perp }$, where $H$ is the mean curvature vector of $\Sigma$ in $M$, $J$ is the complex structure compatible with the Kähler form $\omega$ in $M$, which is an elliptic equation. We call such a surface a symplectic critical surface. We show that, if $M$ is a Kähler-Einstein surface with nonnegative scalar curvature, each symplectic critical surface is holomorphic. We also study the topological properties of the symplectic critical surfaces.

## How to cite

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Han, Xiaoli, and Li, Jiayu. "Symplectic critical surfaces in Kähler surfaces." Journal of the European Mathematical Society 012.2 (2010): 505-527. <http://eudml.org/doc/277290>.

@article{Han2010,
abstract = {Let $M$ be a Kähler surface and $\Sigma$ be a closed symplectic surface which is smoothly immersed in $M$. Let $\alpha$ be the Kähler angle of $\Sigma$ in $M$. We first deduce the Euler-Lagrange equation of the functional $L=\int _\{\Sigma \}\frac\{1\}\{\cos \alpha \}d\mu$ in the class of symplectic surfaces. It is $\cos ^3\alpha H=(J(J\nabla \cos \alpha )^\top )^\bot$, where $H$ is the mean curvature vector of $\Sigma$ in $M$, $J$ is the complex structure compatible with the Kähler form $\omega$ in $M$, which is an elliptic equation. We call such a surface a symplectic critical surface. We show that, if $M$ is a Kähler-Einstein surface with nonnegative scalar curvature, each symplectic critical surface is holomorphic. We also study the topological properties of the symplectic critical surfaces.},
author = {Han, Xiaoli, Li, Jiayu},
journal = {Journal of the European Mathematical Society},
keywords = {symplectic surface; holomorphic curve; Kähler surface; symplectic surface; holomorphic curve; Kähler surfaces},
language = {eng},
number = {2},
pages = {505-527},
publisher = {European Mathematical Society Publishing House},
title = {Symplectic critical surfaces in Kähler surfaces},
url = {http://eudml.org/doc/277290},
volume = {012},
year = {2010},
}

TY - JOUR
AU - Han, Xiaoli
AU - Li, Jiayu
TI - Symplectic critical surfaces in Kähler surfaces
JO - Journal of the European Mathematical Society
PY - 2010
PB - European Mathematical Society Publishing House
VL - 012
IS - 2
SP - 505
EP - 527
AB - Let $M$ be a Kähler surface and $\Sigma$ be a closed symplectic surface which is smoothly immersed in $M$. Let $\alpha$ be the Kähler angle of $\Sigma$ in $M$. We first deduce the Euler-Lagrange equation of the functional $L=\int _{\Sigma }\frac{1}{\cos \alpha }d\mu$ in the class of symplectic surfaces. It is $\cos ^3\alpha H=(J(J\nabla \cos \alpha )^\top )^\bot$, where $H$ is the mean curvature vector of $\Sigma$ in $M$, $J$ is the complex structure compatible with the Kähler form $\omega$ in $M$, which is an elliptic equation. We call such a surface a symplectic critical surface. We show that, if $M$ is a Kähler-Einstein surface with nonnegative scalar curvature, each symplectic critical surface is holomorphic. We also study the topological properties of the symplectic critical surfaces.
LA - eng
KW - symplectic surface; holomorphic curve; Kähler surface; symplectic surface; holomorphic curve; Kähler surfaces
UR - http://eudml.org/doc/277290
ER -

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