For closed manifolds with sectional curvature at most -1, reaching McKean's lower spectral bound forces the universal cover to be hyperbolic space, in both the Laplacian and p-Laplacian cases.
Bottom spectrum and parabolicity of 3-manifolds with scalar curvature lower bound
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abstract
Under a necessary topological assumption, two global results are established for complete three dimensional manifolds. The first one provides a sharp upper bound for the bottom spectrum in terms of the scalar curvature lower bound. The second one shows that such manifolds do not admit any positive Green's function if the scalar curvature is bounded from below by a positive constant.
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2026 1verdicts
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McKean Rigidity for Cocompact Negatively Curved Manifolds and the \(p\)-Laplacian
For closed manifolds with sectional curvature at most -1, reaching McKean's lower spectral bound forces the universal cover to be hyperbolic space, in both the Laplacian and p-Laplacian cases.