On The Negativity of Ricci Curvatures of Complete Conformal Metrics

TL;DR

This paper investigates the Ricci curvature of complete conformal metrics arising from a singular Yamabe problem, proving negativity in convex domains and exploring conditions where negativity fails.

Contribution

It establishes the negativity of Ricci curvatures in convex domains and constructs examples where this negativity does not hold, depending on boundary proximity.

Findings

Complete conformal metrics in convex domains have negative Ricci curvature.

Negativity of Ricci curvature can fail in domains close to low-dimensional boundary sets.

Green's function expansion and positive mass theorem are key tools in the analysis.

Abstract

A version of the singular Yamabe problem in bounded domains yields complete conformal metrics with negative constant scalar curvatures. In this paper, we study whether these metrics have negative Ricci curvatures. Affirmatively, we prove that these metrics indeed have negative Ricci curvatures in bounded convex domains in the Euclidean space. On the other hand, we provide a general construction of domains in compact manifolds and demonstrate that the negativity of Ricci curvatures does not hold if the boundary is close to certain sets of low dimension. The expansion of the Green's function and the positive mass theorem play essential roles in certain cases.