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

2011/12/05 by Espinar, Jose M.
#Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1112.0878

Abstract

In this paper we will show the following result: Let N be a complete (noncompact) connected orientable Riemannian three-manifold with nonnegative scalar curvature S ≥ 0 and bounded sectional curvature K_s ≤ K . Suposse that Σ⊂ N is a complete orientable connected area-minimizing cylinder so that π₁ (Σ) ∈ π₁ (N). Then N is locally isometric either to \mathbbS ¹ × ℝ ² or \mathbbS¹ × \mathbbS¹ × ℝ (with the standard product metric). As a corollary, we will obtain: Let N be a complete (noncompact) connected orientable Riemannian three-manifold with nonnegative scalar curvature S ≥ 0 and bounded sectional curvature K_s ≤ K . Assume that π₁ (N) contains a subgroup which is isomorphic to the fundamental group of a compact surface of positive genus. Then, N is locally isometric to \mathbbS¹ × \mathbbS¹ × ℝ (with the standard product metric).

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