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.
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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.