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

Journal article

Let g₀ be a smooth Riemannian metric on a closed manifold Mⁿ of dimension n ≥ 3. We study the existence of a smooth metric g conformal to g₀ whose Schouten tensor A_g satisfies the differential inclusion λ(g⁻¹A_g) ∈ Γ on Mⁿ, where Γ ⊂ Rⁿ 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 g₁ conformal to g₀ satisfying λ(g₁⁻¹A_g1) ∈ Γ̅' in the viscosity sense on Mⁿ, 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 sign of a nonlinear eigenvalue for the σ₂ operator is a conformal invariant in three dimensions. We also give a generalisation of a theorem of Aubin & Ehrlick on pinching of the Ricci curvature, and an application in the study of Green’s functions for fully nonlinear Yamabe problems.

Authors: Duncan, JAJ; Nguyen, L

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Elsevier
• Journal: Advances in Mathematics
• Volume: 432
• Article number: 109263
• Publication date: 2023-08-24
• Acceptance date: 2023-08-05
• DOI: 10.1016/j.aim.2023.109263
• ISSN: 0001-8708
• Language: English
• Keywords: differential inclusion; nonlinear eigenvalues; Schouten tensor; nonlinear Yamabe problem