# Symplectic critical surfaces in Kähler surfaces

Journal of the European Mathematical Society (2010)

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

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topHan, 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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