The σₖ-Loewner-Nirenberg problem on Riemannian manifolds for k=n/2 and beyond

Let (Mⁿ,g₀) be a smooth compact Riemannian manifold of dimension n≥ 3 with smooth non-empty boundary ∂ M. Let Γⁿ be a symmetric convex cone and f a symmetric defining function for $Γ$ satisfying standard assumptions. Denoting by Agᵤ the Schouten tensor of a conformal metric gᵤ = u⁻²g₀, we show that the associated fully nonlinear Loewner-Nirenberg problem {align*} {cases} f(λ(-g_u⁻¹Ag_u)) = 1/2, λ(-g_u⁻¹Ag_u)∈Γ& on M ∂ M u = 0 & on ∂ M {cases} {align*} admits a solution if μ_Γ⁺ > 1-δ, where μ_Γ⁺ is defined by (-μ_Γ⁺,1,,1)∈∂Γ and $δ>0$ is a constant depending on certain geometric data. In particular, we solve the σₖ-Loewner-Nirenberg problem for all k≤ n/2, which extends recent work of the authors to include the important threshold case k=n/2. In the process, we establish that the fully nonlinear Loewner-Nirenberg problem and corresponding Dirichlet boundary value problem with positive boundary data admit solutions if there exists a conformal metric g∈[g₀] such that λ(-g⁻¹Ag)∈Γ on M; these latter results require no assumption on μ_Γ⁺ and are new when (1,0,,0)∈∂Γ.