A flow approach to the generalized Loewner-Nirenberg problem of the $σ_k$-Ricci equation

Abstract

We introduce a flow approach to the generalized Loewner-Nirenberg problem (1.5) − (1.7) of the σ k -Ricci equation on a compact manifold (M n , g) with boundary. We prove that for initial data u 0 ∈ C 4,α (M) which is a subsolution to the σ k -Ricci equation (1.5), the Cauchy-Dirichlet problem (3.1)−(3.3) has a unique solution u which converges in C 4 loc (M • ) to the solution u ∞ of the problem (1.5) − (1.7), as t → ∞.