Gradient flow for \(\beta\)-symplectic critical surfaces (Q6586920)

Let \((M,g,J)\) be a Kähler surface. Let \(\omega\) be its Kähler form. For a compact, real surface \(\Sigma\) smoothly immersed in \(M\), the Kähler angle of \(\sigma\) in \(M\) is defined by \(\omega_{\Sigma}=\cos\alpha d\mu_{\Sigma}\), where \(d\mu_{\Sigma}\) for the area element of \(\Sigma\) of the induced metric \(g\). \(\Sigma\) is called a symplectic surface if \(\cos\alpha>0\). For \(\beta\ge0\) the authors consider the functionals \N\[\NL_{\beta}(\Sigma)=\int_{\sigma}\frac1{\cos^{\beta}\alpha}d\mu\N\]\Nand their gradient flows. The authors prove that the symplectic property is preserved along the flow. They also prove that if \((M,g,J)\) is Kähler-Eintein of positive scalar curvature and \(\Sigma\) is sufficiently close to a holomorphic curve, then the flow exists globally and converges to a holomorphic curve. The authors also prove a monotonicity formula and an \(\epsilon\)-regularity theorem for the flow.