Liouville theorems for conformally invariant fully nonlinear equations. I

A fundamental theorem of Liouville asserts that positive entire harmonic functions in Euclidean spaces must be constant. A remarkable Liouville-type theorem of Caffarelli-Gidas-Spruck states that positive entire solutions of $-\Delta u=u^{ {(n+2)}/{(n-2)} }$, $n\ge 3$, are unique modulo M\"obius transformations. Far-reaching extensions were established for general fully nonlinear conformally invariant equations through the works of Chang-Gursky-Yang, Li-Li, Li, and Viaclovsky. In this paper, we derive necessary and sufficient conditions for the validity of such Liouville-type theorems. This leads to necessary and sufficient conditions for local gradient estimates of solutions to hold, assuming a one-sided bound on the solutions, for a wide class of fully nonlinear elliptic equations involving Schouten tensors. A pivotal advancement in proving these Liouville-type theorems is our enhanced understanding of solutions to such equations near isolated singularities. In particular, we utilize earlier results of Caffarelli-Li-Nirenberg on lower- and upper-conical singularities. For general conformally invariant fully nonlinear elliptic equations, we prove that a viscosity super- (sub-)solution can be extended across an isolated singularity if and only if it is a lower- (upper-)conical singularity. We also provide necessary and sufficient conditions for lower- (upper-)conical behavior of a function near isolated singularities in terms of its conformal Hessian. As an application of our Liouville theorems and local gradient estimates, we establish new existence and compactness results for conformal metrics on a closed Riemannian manifold with prescribed symmetric functions of the Schouten (Ricci) tensor, allowing the scalar curvature of the conformal metrics to have varying signs.