# An $\varepsilon$-regularity theorem for line bundle mean curvature flow

## TL;DR

This paper establishes an $ ext{epsilon}$-regularity theorem for the line bundle mean curvature flow, a geometric flow used to find special metrics on Kähler manifolds, by introducing a scale-invariant monotone quantity and analyzing self-shrinker solutions.

## Contribution

It introduces an $ ext{epsilon}$-regularity theorem for the flow, along with a new monotone quantity and a Liouville type theorem for self-shrinkers, advancing understanding of flow regularity.

## Key findings

- Proved an $ ext{epsilon}$-regularity theorem for the flow.
- Defined a scale-invariant monotone quantity.
- Established a Liouville type theorem for self-shrinkers.

## Abstract

In this paper, we study the line bundle mean curvature flow defined by Jacob and Yau. The line bundle mean curvature flow is a kind of parabolic flows to obtain deformed Hermitian Yang-Mills metrics on a given K\"ahler manifold. The goal of this paper is to give an $\varepsilon$-regularity theorem for the line bundle mean curvature flow. To establish the theorem, we provide a scale invariant monotone quantity. As a critical point of this quantity, we define self-shrinker solution of the line bundle mean curvature flow. The Liouville type theorem for self-shrinkers is also given. It plays an important role in the proof of the $\varepsilon$-regularity theorem.