Some Elliptic PDE Problems in Conformal Geometry

Abstract

In this thesis, divided into two parts, we explore two distinct types of elliptic partialdierential equation (PDE) problems in conformal geometry. In the first part of thesis, we prove that any closed Riemannian manifold with positive Yamabe invariant and total Q-curvature can be conformally deformed into a metric with positive scalar curvature and constant Q-curvature. Moreover, we derive analogous results for the Q-curvature and T-curvature of a Riemannian manifold with an umbilic boundary, subject to the condition that the first Yamabe constant and total (Q; T)-curvature are positive. The proof utilizes on the method of continuity and the Leray-Schauder degree theory. In the second part, we prove that the zero function is the sole solution to a certain degenerate PDE defined in the upper halfspace under some geometric assumptions. This result implies that the Euclidean metric is the only adapted compactification of the standard half-plane model of hyperbolic space when the scalar curvature of the compactified metric has a certain sign. iii