Gradient Flow of the $L_β$-Functional

Abstract

In this paper, we start to study the gradient flow of the functional $L_β$ introduced by Han-Li-Sun in [8]. As a first step, we show that if the initial surface is symplectic in a Kähler surface, then the symplectic property is preserved along the gradient flow. Then we show that the singularity of the flow is characterized by the maximal norm of the second fundamental form. When $β$=1, we derive a monotonicity formula for the flow. As applications, we show that the $λ$-tangent cone of the flow consists of the finite flat planes.