The structure of fully nonlinear equations and its applications to prescribed problems on complete conformal metrics

Abstract: This paper investigates the structure of fully nonlinear equations and their applications to geometric problems. We solve some fully nonlinear version of the Loewner-Nirenberg and Yamabe problems. Notably, we introduce Morse theory techniques to construct admissible metrics under a weak condition on the underlying metric, which can be further relaxed in a broad setting. Furthermore, we provide some topological obstruction to demonstrate the optimality of our structural conditions.