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

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 $S\ge 0$ and bounded sectional curvature $ K_{s} \leq K $\mathrm{.}Supossethat$\Sigma \subset $\mathcal{N}$ $ is a complete orientable connected area-minimizing cylinder so that $\pi_1 (\Sigma) \in \pi_1 ($\mathcal{N}$)$\mathrm{.}Then$$\mathcal{N}$$islocallyisometriceitherto$$\mathbb{S}$ ^1 \times $\mathbb{R}$ ^2 $or$$\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 $S\ge 0$ and bounded sectional curvature $K_s\le K$. 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).