Gradient flow for $\beta$-symplectic critical surfaces

Abstract

In this paper we consider a gradient flow for the $L_{\beta }$ -functional introduced by the authors in Han–Li–Sun (2018). We first prove that the “symplectic” property is preserved along the gradient flow. Then we prove a monotonicity formula and an $\epsilon$-regularity theorem for the flow. As consequences, we show that the $\lambda$-tangent cone of the flow consists of finite flat planes. Another application is that we can show that the flow exists globally and converges to a holomorphic curve if the initial surface is sufficiently close to a holomorphic curve and the ambient Käher–Einstein surface has positive scalar curvature.