Boundary Expansions of Complete Conformal Metrics with Negative Ricci Curvatures

TL;DR

This paper investigates the boundary behavior of complete conformal metrics with negative Ricci curvatures, providing asymptotic expansions and regularity estimates near the boundary.

Contribution

It introduces new boundary asymptotic expansions and regularity estimates for solutions to the $\sigma_k$-Ricci problem on manifolds with boundary.

Findings

Established boundary asymptotic expansions for the conformal metric.

Derived $C^1$ and $C^2$ estimates near the boundary.

Enhanced understanding of boundary regularity for $\sigma_k$-Ricci solutions.

Abstract

We study the boundary behaviors of a complete conformal metric which solves the $\sigma_k$-Ricci problem on the interior of a manifold with boundary. We establish asymptotic expansions and also $C^1$ and $C^2$ estimates for this metric multiplied by the square of the distance in a small neighborhood of the boundary.