Complete 3-manifolds with scalar curvature lower bound, finitely many ends, and finite first Betti number satisfy a sharp bottom spectrum upper bound and are parabolic under positive scalar curvature.
Liu, A sharp inequality between scalar curvature and the bottom spectrum on complete manifolds, arXiv:2603.20864
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Bottom spectrum and parabolicity of 3-manifolds with scalar curvature lower bound
Complete 3-manifolds with scalar curvature lower bound, finitely many ends, and finite first Betti number satisfy a sharp bottom spectrum upper bound and are parabolic under positive scalar curvature.