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Math-2233 · mathematics · Geometry

Scalar Curvature Question [?34]: Bounds on Width and on the Macroscopic Dimension

Status Partially solved · unsolvedmath.comLLM status Open unverified

Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.
Statement recorded in the supplied source.
For every dimension $n$, consider complete Riemannian $n$-manifolds $X$ with scalar curvature bounded below by $\sigma>0$. Must the macroscopic dimension satisfy $\dim_{\mathrm{mc}}X\leq n-2$? Is there a dimension-dependent constant bounding the stated $(n-2)$-dimensional width by $C_n\sigma^{-1/2}$? The question also asks for the weaker $(n-1)$-dimensional width bound at this same scale. Width measures the size of fibers of a map to a lower-dimensional polyhedron.

Formulation references: [S1], [S2].

Short English statement

For every dimension $n$, consider complete Riemannian $n$-manifolds $X$ with scalar curvature bounded below by $\sigma>0$. Must the macroscopic dimension satisfy $\dim_{\mathrm{mc}}X\leq n-2$? Is there a dimension-dependent constant bounding the stated $(n-2)$-dimensional width by $C_n\sigma^{-1/2}$? The question also asks for the weaker $(n-1)$-dimensional width bound at this same scale. Width measures the size of fibers of a map to a lower-dimensional polyhedron.

Sources

Catalogue entry

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2233-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2233

Pointwise scalar curvature and width: The paper proves codimension-two Urysohn width in dimensions at least four and a continuous macroscopic-dimension corollary for universal covers. The extracted complete-manifold target has all dimensions and m2 width conventions that are not identified with the paper Urysohn convention by the supplied text.

Spectral width in dimension three: The conclusion supplies a uniform codimension-two continuous width estimate in dimension three. The source item has all dimensions and specific m1/m2 width conventions not completely identified by its transcription; this is not marked fully covered.

Higher-dimensional spectral width: The conclusion supplies a uniform codimension-two continuous width estimate in dimensions at least four. The source item has all dimensions and specific m1/m2 width conventions not completely identified by its transcription; this is not marked fully covered.

This page records partial support. Complete resolution of the catalogue question is not established by this record.

Scalar Curvature Question [?34]: Bounds on Width and on the Macroscopic Dimension

OpenAI's claim

Pointwise scalar curvature and width: The paper claims dimension-only control of the full fibers of a continuous map from any complete scalar-positive $n$-manifold to an $(n-2)$-complex, for $n\ge4$; it also gives the stated macroscopic-dimension consequence on universal covering spaces when the closed base admits a positive-scalar-curvature metric.

Spectral width in dimension three: A complete three-manifold satisfying $-4\Delta+\mathrm{Scal}\ge\lambda>0$ is claimed to map continuously to a graph with full fiber diameter at most $500/\sqrt\lambda$; no global bounded-geometry condition is assumed.

Higher-dimensional spectral width: The spectral lower bound $-4\Delta+\mathrm{Scal}\ge1$ is claimed to imply uniform continuous $(n-2)$-width for every complete $n$-manifold with $n\ge4$; the bound uses the initial metric rather than an auxiliary conformal distance.

Mathematical statement of the claimed result

Mathematical claim 1.
For every $n\ge4$ there is $C_n<\infty$ such that a connected complete smooth boundaryless $n$-manifold with $\operatorname{Scal}_g\ge1$ has a continuous map $f:M\to K$, where $K$ is a simplicial complex with the usual polyhedral topology and $\dim K\le n-2$, with \[\sup_{y\in K}\operatorname{diam}_g f^{-1}(y)\le C_n.\] Entire fibers, rather than their individual components, are bounded. For smooth Riemannian $(M,g)$ of dimension $n\ge4$, assume $M$ is closed and connected and $\operatorname{Scal}\ge\sigma^2>0$, the corollary gives $\operatorname{FillRad}(M,g)\le C_n/(2\sigma)$. A separate stated corollary covers all closed positive-scalar-curvature dimensions $n\ge2$: the universal-cover geometry obtained by lifting that metric has continuous macroscopic dimension at most $n-2$.

The filling radius uses the integral fundamental class when the manifold is orientable and its mod-two fundamental class otherwise.

Mathematical claim 2.
Let $(M^3,g)$ be connected, complete, smooth and without boundary. If $\lambda>0$ and \[\int_M(4|\nabla\phi|^2+\operatorname{Scal}_g\phi^2)\,dV_g\ge\lambda\int_M\phi^2\,dV_g\quad(\phi\in C_c^\infty(M;\mathbb R)),\] the claim supplies a simplicial complex $K$ satisfying $\dim K\le1$ and a map $f:M\to K$ which is continuous with $\operatorname{diam}_g f^{-1}(y)\le500/\sqrt\lambda$ for every fiber. Compactness, orientability, spin and bounded geometry are not required.
Mathematical claim 3.
For $n\ge4$, a connected complete smooth boundaryless $(M^n,g)$ satisfying \[\int_M(4|\nabla\phi|^2+\operatorname{Scal}_g\phi^2)\,dV_g\ge\int_M\phi^2\,dV_g\quad(\phi\in C_c^\infty(M;\mathbb R))\] is claimed to have a continuous $f:M\to K$ with $\dim K\le n-2$ and all fiber diameters at most $C_n$, depending only on $n$. Thus its Urysohn $(n-2)$-width, the infimum of these uniform fiber-diameter bounds, is at most $C_n$.

Outline of the argument

Pointwise scalar curvature and width: A separated net supplies labeled inner and outer balls in the original metric. Compatible walls and inserted cylinders are constructed while a nearby collapse map keeps track of those original positions. The final walls have no intersection of $n-1$ different labels, so their sign coordinates define an $(n-2)$-dimensional map; shared coordinates place the entire fiber in one uniformly bounded original ball.

Spectral width in dimension three: A positive supersolution is converted into a drift-weighted scalar inequality without changing the original distance. Weighted stable surfaces inherit a two-dimensional inequality, whose geodesic index estimate bounds their intrinsic diameter. Local minimization and smoothing in compact regions then organize these surfaces into a graph-valued decomposition with a uniform bound on each complete fiber.

Higher-dimensional spectral width: A positive supersolution produces a signed drift with a weighted curvature bound. An auxiliary graph metric and updated drift add a quadratic reserve even when the original drift is unbounded. The subsequent stretching and wall construction uses that reserve to preserve curvature while reducing intersection multiplicity, yielding fibers bounded in the original geometry.

Formal verification

Pointwise scalar curvature and width: The pinned catalogue lists no Lean formalization for this result.

Spectral width in dimension three: The pinned catalogue lists no Lean formalization for this result.

Higher-dimensional spectral width: The pinned catalogue lists no Lean formalization for this result.

This site has not run the Lean development. These links describe OpenAI’s selected formal statements and should be read with their stated scope.

Status caveat

OpenAI reports that an internal model produced the manuscripts. Their arguments have not been independently verified as proofs by this site. Any portion outside the documented formal scope may contain mistakes, and the catalogue label reports the announced mathematical claim.

Sources

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