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.
Geometric Rigidity via Harmonic Twisted Spinors
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abstract
We study Gromov's exact-lift two-form method in scalar-curvature geometry. For a closed Riemannian spin manifold carrying a homologically non-trivial closed two-form whose lift to the universal cover is exact, we prove the sharp hyperbolic scalar-curvature comparison with the bottom of the spectrum of the universal Riemannian covering. The two-form enters through Gromov's twisted \(L^2\)-index, which produces harmonic spinors for a family of small unitary twists. We analyze the equality case by interpreting the refined Kato equality defect conformally and use the harmonic spinors to construct a parallel spinor with respect to a suitable conformally related metric. This yields that the original metric is Einstein. In the positive-spectrum case, this method implies that the universal cover is real hyperbolic.
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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.