In the first part, we study the constant Q-curvature metrics on conic 4 manifolds. We introduce 3 different types of configurations of conic singularities: stable, semi-stable, and unstable. In the stable case, we prove the existence of a constant Q-curvature metric in the conformal class. For conic 4-sphere with at most 2 singularities, we classify the solutions. They are rotational symmetric metrics on conic 4-sphere with identical cone type at each singularity and they correspond to the semi-stable configuration. Along the study, we derive an asymptotic expansion of the constant Q-curvature metric near conic singularities which is of independent interest.

In the second part, we study the curve-shortening-flow (CSF) on Riemann surfaces with possible ambient conic singularities. We generalize a comparison function due to Huisken which was originally defined for CSF on plane to surfaces. We obtain a new proof of the classic Gage-Hamilton-Grayson theorem on surfaces by studying the first time singularities of the flow. If the ambient Riemann surface admits conic singularities, we prove that under the flow, embedded simple closed curves cannot touch conic singularities with cone angles less than or equal to π.

In the last part of the thesis, we prove a Liouville type theorem for an one parameter family of 2-Hessian type fully non-linear elliptic equations on R^4 . When the parameter ρ runs from 1 to 0, the equations interpolate between the σ2 -Yamabe equation and the 2-Hessian equation. Our theorem holds for ρ > 0. Our approach contains 2 important ingredients: one is a monotonic quasi-local mass type quantity defined on the level sets of the solution; another is a series of estimates which controls the asymptotic shape of level sets of u at infinity.

In the third part, we generalize the Andrews’ inequality to manifolds with conic singularities and with boundary. We establish the rigidity result when the equality holds.

This thesis consists of 4 parts on the analysis of singularities that arise from conformal geometry, geometric flows, and geometric inequalities.