Gradient flow for \(\beta\)-symplectic critical surfaces
type: Item
instance of: scholarly article
title: Gradient flow for \(\beta\)-symplectic critical surfaces
author: Xiao Li Han
author: Jiayu Li
author: Jun Sun
published in: Annales de l'Institut Henri Poincaré. Analyse Non Linéaire
publication date: 2024-08-13T00:00:00Z
review text: 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.
reviewed by: Włodzimierz Jelonek
Mathematics Subject Classification ID: 53E50
Mathematics Subject Classification ID: 53E30
Mathematics Subject Classification ID: 53C42
zbMATH DE Number: 7896316
zbMATH Keywords: \(\beta\)-symplectic critical surfaces
zbMATH Keywords: gradient flow
zbMATH Keywords: monotonicity formulas
zbMATH Keywords: \(\varepsilon\)-regularity
MaRDI profile type: Publication
cites work: On the first variation of a varifold
cites work: Q4515454
cites work: Mean curvature flow of surface in 4-manifolds
cites work: Singularity of mean curvature flow of Lagrangian submanifolds
cites work: Minimal Surfaces by Moving Frames
cites work: Generic mean curvature flow. I: Generic singularities
cites work: Q5694433
cites work: Symplectic critical surfaces in Kähler surfaces
cites work: The second type singularities of symplectic and Lagrangian mean curvature flows
cites work: The deformation of symplectic critical surfaces in a Kähler surface. II: Compactness
cites work: The Deformation of Symplectic Critical Surfaces in a Kähler Surface-I
cites work: Gradient Flow of the $L_β$-Functional
cites work: Translating solitons to symplectic mean curvature flows
cites work: Flow by mean curvature of convex surfaces into spheres
cites work: Asymptotic behavior for singularities of the mean curvature flow
cites work: Q3137116
cites work: Elliptic regularization and partial regularity for motion by mean curvature
cites work: Singularities of Lagrangian mean curvature flow: zero-Maslov class case
cites work: Q3336268
cites work: Mean curvature flow of surfaces in Einstein four-manifolds.
cites work: A local regularity theorem for mean curvature flow
cites work: Minimal surfaces in Kähler surfaces and Ricci curvature
DOI: 10.4171/AIHPC/100
description: scientific article; zbMATH DE number 7896316