On the structure of complete 3-manifolds with nonnegative scalar curvature

TL;DR

This paper classifies the local structure of complete 3-manifolds with nonnegative scalar curvature containing certain minimal surfaces, showing they are locally isometric to standard product spaces under specific topological conditions.

Contribution

It proves that such manifolds with a complete area-minimizing cylinder are locally isometric to standard product spaces, extending understanding of their geometric and topological structure.

Findings

Manifolds with a complete area-minimizing cylinder are locally isometric to  imes \u211b^2 or  imes  imes \u211b.

If the fundamental group contains a subgroup isomorphic to a surface of positive genus, the manifold is locally  imes  imes .

The results hold under bounded sectional curvature and nonnegative scalar curvature assumptions.

Abstract

In this paper we will show the following result: Let $\mathcal{N} $ be a complete (noncompact) connected orientable Riemannian three-manifold with nonnegative scalar curvature $S \geq 0$ and bounded sectional curvature $ K_{s} \leq K $. Suposse that $\Sigma \subset \mathcal{N} $ is a complete orientable connected area-minimizing cylinder so that $\pi_1 (\Sigma) \in \pi_1 (\mathcal{N})$. Then $\mathcal{N}$ is locally isometric either to $\mathbb{S} ^1 \times \mathbb{R} ^2 $ or $\mathbb{S}^1 \times \mathbb{S}^1 \times \mathbb{R}$ (with the standard product metric). As a corollary, we will obtain: Let $\mathcal{N} $ be a complete (noncompact) connected orientable Riemannian three-manifold with nonnegative scalar curvature $S \geq 0$ and bounded sectional curvature $ K_{s} \leq K $. Assume that $\pi_1 (\mathcal{N})$ contains a subgroup which is isomorphic to the fundamental group of a compact surface of positive genus. Then, $\mathcal{N}$ is locally isometric to $\mathbb{S}^1 \times \mathbb{S}^1 \times \mathbb{R}$ (with the standard product metric).