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

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 $λ(g^{-1}A_g)\inΓ$ on $M^n$, where $Γ\subset\mathbb{R}^n$ 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 $λ(g_1^{-1}A_{g_1})\in\bar{Γ'}$ in the viscosity sense on $M^n$, together with a nondegenerate ellipticity condition, where $Γ' = Γ$ or $Γ'$ is a cone slightly smaller than $Γ$. 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 $σ_2$-Yamabe problem is equivalent to positivity of a nonlinear eigenvalue for the $σ_2$-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.

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