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

TL;DR

This paper establishes existence and uniqueness of conformal metrics satisfying differential inclusions involving the Schouten tensor, with applications to nonlinear eigenvalue problems and geometric curvature conditions in conformal geometry.

Contribution

It introduces a framework for solving differential inclusions for the Schouten tensor and proves new existence and uniqueness results for nonlinear eigenvalue problems in conformal geometry.

Findings

Existence of conformal metrics satisfying Schouten tensor inclusions.

Equivalence of the $\sigma_2$-Yamabe problem solvability to eigenvalue positivity.

Generalization of pinching theorems for Ricci curvature.

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.