Symplectic critical surfaces in Kahler surfaces

Let M be a Kahler surface and Sigma be a closed symplectic surface which is smoothly immersed in M. Let alpha be the Kahler angle of Sigma in M. We first deduce the Euler-Lagrange equation of the functional L = integral(Sigma) = 1/cos alpha d mu in the class of symplectic surfaces. It is cos(3) alpha H = (J(J del cos alpha)(inverted perpendicular))(perpendicular to), where H is the mean curvature vector of Sigma in M, and J is the complex structure compatible with the Kahler form omega in M; it is an elliptic equation. We call a surface satisfying this equation a symplectic critical surface. We show that, if M is a Kahler - Einstein surface with 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 Kahler angle of a compact symplectic critical surface is constant, which is not true for noncompact symplectic critical surfaces.