Line bundle mean curvature flow

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.  In this talk I will give an $\varepsilon$-regularity theorem for the line bundle mean curvature flow. To establish the theorem, we provide a scale invariant monotone quantity and as the critical point of this quantity we define self-shrinker solution of the line bundle mean curvature flow. The Liouville type theorem for self-shrinkers also are given and it plays an important role in the proof of the $\varepsilon$-regularity theorem.