Stability of line bundle mean curvature flow

Abstract

Let $(X,\omega)$ be a compact K\"ahler manifold of complex dimension $n$ and $(L,h)$ be a holomorphic line bundle over $X$. The line bundle mean curvature flow was introduced in \cite{JY} in order to find deformed Hermitian-Yang-Mills metrics on $L$. In this paper, we consider the stability of the line bundle mean curvature flow. Suppose there exists a deformed Hermitian Yang-Mills metric $\hat h$ on $L$. We prove that the line bundle mean curvature flow converges to $\hat h$ exponentially in $C^\infty$ sense as long as the initial metric is close to $\hat h$ in $C^2$-norm.