Differential inclusions for the Schouten tensor and nonlinear eigenvalue problems in conformal geometry

Abstract

Let $g_0$ be a smooth Riemannian metric on a closed manifold $M^n$ of dimension $n\geq 3$. We study the existence of a smooth metric $g$ conformal to $g_0$ whose Schouten tensor $A_g$ satisfies the differential inclusion $\lambda(g^{-1}A_g)\in\Gamma$ on $M^n$, where $\Gamma\subset\mathbb{R}^n$ is a cone satisfying standard assumptions. Inclusions of this type are often assumed in the existence theory for fully nonlinear elliptic equations in conformal geometry. We assume the existence of a continuous metric $g_1$ conformal to $g_0$ satisfying $\lambda(g_1^{-1}A_{g_1})\in\bar{\Gamma'}$ in the viscosity sense on $M^n$, together with a nondegenerate ellipticity condition, where $\Gamma' = \Gamma$ or $\Gamma'$ is a cone slightly smaller than $\Gamma$. In fact, we prove not only the existence of metrics satisfying such differential inclusions, but also existence and uniqueness results for fully nonlinear eigenvalue problems for the Schouten tensor. We also give a number of geometric applications of our results. We show that the solvability of the $\sigma_2$-Yamabe problem is equivalent to positivity of a nonlinear eigenvalue for the $\sigma_2$-operator in three dimensions. We also give a generalisation of a theorem of Aubin and Ehrlick on pinching of the Ricci curvature, and an application in the study of Green's functions for fully nonlinear Yamabe problems.