Vacuum Static Spaces with Positive Isotropic Curvature

TL;DR

This work studies compact vacuum static spaces with positive isotropic curvature (PIC) and derives sharp rigidity results. By introducing a structural 3-tensor $T$ and a 2-form $\omega$ and relating them to Bach and Cotton tensors, the authors show that $\omega=0$ forces Bach-flatness and imposes strong level-set geometry; using warped-product representations and existing classifications (notably Qing–Yuan), they classify up to finite cover the manifolds as ${\mathbb S}^n$ or ${\mathbb S}^1\times{\mathbb S}^{n-1}$. When PIC is assumed, the rigidity extends to all dimensions considered, with special treatment in low dimensions via Chen–Tang–Zhu and related results. The appendix provides the technical minimum-set analysis, showing the minimum (and maximum) level sets are either points or totally geodesic stable minimal hypersurfaces, which underpins the global classification. Overall, the paper achieves a definitive rigidity result for compact static vacuum spaces with PIC, linking geometric analysis, conformal invariants, and general relativity-inspired structures.

Abstract

In this paper, we study vacuum static spaces with positive isotropic curvature. We prove that if $(M^n, g, f)$, $n \ge 4$, is a compact vacuum static space with positive isotropic curvature, then up to finite cover, $M$ is isometric to a sphere ${\Bbb S}^n$ or the product of a circle ${\Bbb S}^1$ with an $(n-1)$-dimensional sphere ${\Bbb S}^{n-1}$.

Theorems & Definitions (53)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Lemma 2.1
• Theorem 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4
• proof