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Sharp bottom spectrum and scalar curvature rigidity

· 2024 · math.DG · arXiv 2408.08245

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abstract

We establish a sharp upper bound for the bottom spectrum of the Beltrami Laplacian on universal covers of closed Riemannian manifolds with a scalar curvature lower bound. Moreover, we prove a scalar curvature rigidity theorem when this bound is achieved. Additionally, we prove a net characterization of scalar curvature for general complete noncompact Riemannian manifolds.

fields

math.DG 2

years

2026 2

verdicts

UNVERDICTED 2

representative citing papers

$L^\infty$-metrics on tori and Schoen's conjecture

math.DG · 2026-06-19 · unverdicted · novelty 5.0

Under a non-surjectivity assumption on the fundamental group homomorphism from the singular set, an L^∞ metric on a torus with non-negative scalar curvature outside a Minkowski dimension ≤ n-3+(n-1)^{-1} singular set extends to a smooth flat metric, proved via weighted scalar curvature and the relat

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Showing 2 of 2 citing papers.

  • $L^\infty$-metrics on tori and Schoen's conjecture math.DG · 2026-06-19 · unverdicted · none · ref 24 · internal anchor

    Under a non-surjectivity assumption on the fundamental group homomorphism from the singular set, an L^∞ metric on a torus with non-negative scalar curvature outside a Minkowski dimension ≤ n-3+(n-1)^{-1} singular set extends to a smooth flat metric, proved via weighted scalar curvature and the relat

  • Scalar curvature, sharp bottom spectrum and geometric rigidity math.DG · 2026-06-10 · unverdicted · none · ref 11 · internal anchor

    Closed manifolds with Sc_g ≥ -n(n-1) and λ₁(̃M,̃g)=(n-1)²/4 are hyperbolic under topological assumptions from previous work.