Regularity for the Harmonic-Einstein equation

We establish a regularity theorem for the Harmonic-Einstein equation. As a byproduct of the local regularity, we also have a compactness theorem on the Harmonic-Einstein equation. The method is mainly the Moser iteration technique which has been used and developed by Bando et al. (1989) [4], Tian (1990) [22], Tian and Viaclovsky (2005) [23] and others.