We study some Monege-Ampere type equations in $R^4$ and derive a new geometric monotonicity formula. As an application in analysis, we prove a Liouville type theorem for a class of fully non-linear equations. As applications in conformal geometry, we give some of mass-like inequalities asymptotically hyperbolic 4-discs. We will also discuss some on-going works in conformally compact spaces in dimension 2 and 4.

Parts of works are joint with Biao Ma, Wei Wei and Jinyang Wu.​