The existence of negatively curved metrics on locally conformally flat manifolds with boundary

TL;DR

This work proves that every compact connected locally conformally flat manifold with smooth boundary admits a smooth conformal metric $g_u = e^{2u} g$ with negative sectional curvature, extending a Morse-theoretic construction to higher dimensions. It also shows that, without assuming local conformal flatness, one can construct a conformal metric with positive Einstein tensor in the same setting. The approach combines a Morse-theoretic construction of a nondegenerate function $v$ (yielding $u = e^{N v}$ for large $N$) with a curvature-tensor framework based on the Schouten tensor and its conformal deformations, enforcing eigenvalue conditions $ abla^2$-dependent via a cone $oldsymbol{ m abla}oldsymbol{ extGamma}$. The results extend 3D constructions to higher dimensions and clarify how conformal deformations control curvature signs through admissible function theory.

Abstract

We use certain Morse functions to construct conformal metrics with negative sectional curvature on locally conformally flat manifolds with boundary. Moreover, without conformally flatness assumption, we also construct conformal metric of positive Einstein tensor.

Theorems & Definitions (14)

• Theorem 1.1: yuan-PUE3-construction
• Theorem 1.2
• Theorem 1.3
• Remark 1.4
• Lemma 2.1
• proof
• Remark 2.2
• Theorem 2.3
• Remark 2.4
• proof