  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.
