Spectral scalar curvature and uniform Urysohn width
For every n ≥ 4, a complete connected smooth Riemannian n -manifold without boundary satisfying $-4\Delta+\mathop{\mathrm{Scal}}\nolimits \ge1$ as a quadratic-form inequality admits a continuous map to a simplicial complex of dimension at most $n-2$ whose entire fibers have diameter bounded only in terms of n . The bound is measured in the original metric. This extends the uniform Urysohn width theorem from a pointwise scalar-curvature lower bound to a spectral lower bound.
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