UC Irvine

In this dissertation, we study the prescribed scalar curvature problem in a conformal class on orbifolds with isolated singularities. This problem is more subtle than the manifold case since the positive mass theorem does not hold for ALE metrics in general.

In Chapter 2, we give a positive mass theorem on AF orbifolds. Together with the negative mass ALE space example -- LeBrun's metric, it shows that mass can be discontinuous on compact orbifolds. This further implies that the study of the Yamabe equation on orbifolds can be substantially different from the manifold case.

From 3 to Chapter 5, we prove the compactness and existence theorem for the orbifold prescribed scalar curvature problem, in dimension 4, which can also be generalized to higher dimensions. The proof is by carefully analyzing the Pohozaev Identity and the blow-up behavior of a sequence of solutions.

In Chapter 6, we provide another version of the existence theorem, by looking into the Yamabe energy for all functions, and also of a certain testing function. This existence theorem differs but also overlaps with the previous one, and they both have important use cases.

By applying our theorems to the families of Calabi metrics and LeBrun's metrics, we obtain lots of existence examples. Moreover, in Chapter 7, after playing some tricks on LeBrun's metrics, we get some non-existence examples, and thus establish the full theory of Leray-Schauder degree for radially symmetric solutions. Especially, we observe an interesting "wall" in the prescribed scalar curvature space such that the Leray-Schauder degree jumps by 1 upon crossing this "wall".