On locally conformally flat manifolds with positive pinched Ricci curvature

TL;DR

The paper proves that a complete, locally conformally flat $n$-manifold with a positive pinching of Ricci curvature, $\text{Rc}\ge \varepsilon R g>0$, must be compact. It achieves this by developing and employing a locally conformally flat Yamabe flow, establishing distance-change and curvature-estimate tools, and constructing local flows with uniform existence times that can be pasted into a global flow. A key technical achievement is a local $\frac{c}{t}$ curvature bound under preserved pinching, together with a robust maximum-principle framework to handle noncompact settings. The resulting blow-up analysis yields a contradiction for noncompact limits, thereby proving the compactness result and extending Hamilton’s pinching conjecture to higher dimensions under local conformal flatness.

Abstract

By using the Yamabe flow, we prove that if $(M^n,g)$, $n\geq3$, is an $n$-dimensional locally conformally flat complete Riemannian manifold $Rc\geq εRg>0$, where $ε>0$ is a uniformly constant, then $M^n$ must be compact. Our result shows that Hamilton's pinching conjecture also holds for higher dimensional case if we assume additionally the metric is locally conformally flat.

Theorems & Definitions (24)

• Theorem 1.1
• Remark 1.2
• Theorem 1.3
• Lemma 2.1: Lemma 2.2 and Lemma 2.4 in chow
• Remark 2.2
• Theorem 2.3: Corollaire IV.1.2 in Ho
• Theorem 2.4
• proof
• Lemma 3.1
• proof