The Second Variation of the Functional $$L$$ L of Symplectic Critical Surfaces in Kähler Surfaces

Abstract

Let M be a complete Kähler surface and be a symplectic surface which is smoothly immersed in M. Let α be the Kähler angle of in M. In the previous paper Han and Li (JEMS 12:505-527, 2010) 2010, we study the symplectic critical surfaces, which are critical points of the functional L = 1 cos α dμ in the class of symplectic surfaces. In this paper, we calculate the second variation of the functional L and derive some consequences. In particular, we show that, if the scalar curvature of M is positive, is a stable symplectic critical surface with cos α ≥ δ > 0, whose normal bundle admits a holomorphic section X ∈ L 2 ( ), then is holomorphic. We construct symplectic critical surfaces in C 2 . We also prove a Liouville theorem for symplectic critical surfaces in C 2 .

The deformation of symplectic critical surfaces in a Kähler surface-II—compactness

“…We called such a surface a symplectic critical surface. We studied the properties of the symplectic critical surfaces in [7] and also examined the second variation formula of the functional L in [8].…”

“…We also saw that β-symplectic critical surfaces share many properties with minimal surfaces (c.f. [3], [7], [8], [12], [17], [18]). We ( [10]) constructed explicit β-symplectic critical surfaces in C 2 (0 ≤ β < ∞) and showed that it converges to a plane as β → ∞.…”