Spectral scalar curvature, Urysohn width, and macroscopic dimension

2 · Going further

The precise spectral problem concerns a complete connected smooth Riemannian manifold $M^n$ without boundary satisfying $-4\Delta +\mathrm{Scal}\ge 1$ as a quadratic-form inequality. Here $\Delta$ is the Laplace operator and $\mathrm{Scal}$ is scalar curvature. The desired conclusion is a continuous map $f:M\to K$ to a simplicial complex with $\dim K\le n-2$, such that every entire fiber $f^{-1}(y)$ has diameter bounded by a constant depending only on $n$. The diameter is measured using the original metric on $M$.

Manuscript 1 claims this conclusion for $n\ge 4$. Manuscript 2 supplies the three-dimensional case under the more general normalization $-4\Delta +\mathrm{Scal}\ge \lambda >0$: the target is a graph, and every entire fiber has diameter at most $500/\sqrt{\lambda }$. Its abstract explicitly states that orientability, spin, compactness, and bounded geometry are not required. Together, these two manuscripts give the spectral statement for all $n\ge 3$ described in the repository summary.

Manuscript 3 addresses the pointwise hypothesis $\mathrm{Scal}\ge 1$ for $n\ge 4$, calling its result the quantitative continuous form of Gromov's scalar-curvature conjecture. It also claims that universal covers of closed manifolds with positive scalar curvature have continuous macroscopic dimension at most $n-2$ for every $n\ge 2$. The central relationship is therefore an extension from pointwise curvature control to spectral control while retaining uniform bounds on entire fibers. The supplied abstracts do not describe the argument's construction or intermediate steps, so no more specific proof strategy can be reported from this source.

Useful notions

• Smooth Riemannian manifolds and metric completeness
• Scalar curvature
• The Laplace operator and quadratic-form inequalities
• Continuous maps, fibers, and diameter
• Graphs, simplicial complexes, and dimension

What was known before / what is new

Limits and assumptions

The spectral claims require a complete, connected, smooth Riemannian manifold without boundary and a positive quadratic-form lower bound. Manuscript 1 treats dimensions at least four, while Manuscript 2 treats dimension three; the source supplies no spectral width statement in dimension two. The universal-cover conclusion concerns closed manifolds with positive scalar curvature and is stated for continuous macroscopic dimension. No explicit higher-dimensional width constant or argument details are provided in the abstracts. The main result is not formalized in Lean and has not been formally verified. The source does not identify further remaining open problems.

Mini glossary

Spectral scalar-curvature lower bound

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In these manuscripts, this is a lower bound on the operator $-4\Delta +\mathrm{Scal}$ interpreted as a quadratic-form inequality.

Fiber

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The fiber over a target point is the set of all points that a map sends to that point.

Diameter

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The diameter of a set is the supremum of the distances between pairs of its points.

Uniform Urysohn width bound

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Here this means the existence of a continuous map to a complex of a specified dimension with a uniform bound on the diameters of all entire fibers.

Codimension two

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In these conclusions, this means that the target has dimension at most $n-2$ when the original manifold has dimension $n$.

Keywords : Spectral scalar curvature · Urysohn width · Macroscopic dimension

AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.

3 · Technical details and sources

Manuscripts in this family