In this work, we delve into geometric analysis, particularly examining the interplay between lower bounds on Ricci curvature and specific functionals. Our exploration begins with an investigation into the implications of Yamabe invariants for asymptotically Poincare-Einstein manifolds and their conformal boundaries under conditions of "Ric"[greater than or equal to] - (n-1)g. We establish a relationship wherein the type II Yamabe invariant of the conformal compactification of the manifold is bounded below by the Yamabe invariant of its conformal boundary. Additionally, we focus on compact manifolds with boundary where "Ric"[greater than or equal to] 0 and "II"[greater than or equal to] 1, obtaining partial results concerning Wang's conjecture. [The dissertation citations contained here are published with the permission of ProQuest LLC. Further reproduction is prohibited without permission. Copies of dissertations may be obtained by Telephone (800) 1-800-521-0600. Web page: http://www.proquest.com/en-US/products/dissertations/individuals.shtml.]