In this talk, I will discuss some recent joint work with Biao Ma on finding constant Q-curvature metric on conic 4-manifolds. 

In dimension 2, classical uniformization theory has been generalized by Troyanov and others to discuss conic surfaces with constant scalar curvature. In dimension 4, classical works of Branson-Chang-Yang and Chang-Gursky-Yang reveal an analogue theory for constant Q-curvature metrics, which depends on sharp analytic inequalities of Moser, Trudinger, Adams and Bechner. We generalize Adams’ inequality to a version with singular weights and apply it to prove that for subcritical conformal conic 4-folds, the variational approach produces a constant Q-curvature metric. We will also discuss the critical case, where some delicate solutions are constructed.