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Math-2234 · mathematics · Geometry

Scalar Curvature Question [?35]: Bound on the Filling Radius forSc ≥σ > 0

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.
Let $X$ be a complete Riemannian $n$-manifold with scalar curvature bounded below by a positive constant. Does there exist a constant $C_n$, depending only on dimension, such that its filling radius satisfies \[\operatorname{FillRad}(X)\leq C_n\left(\inf_{x\in X}\operatorname{Sc}(x)\right)^{-2}?\] The filling radius measures the neighborhood size needed to fill the fundamental cycle after the metric-space embedding used in its definition. The catalogue transcription gives exponent $-2$; the cited original is needed to determine whether this exponent is correct.

Formulation references: [S1], [S2].

Short English statement

Let $X$ be a complete Riemannian $n$-manifold with scalar curvature bounded below by a positive constant. Does there exist a constant $C_n$, depending only on dimension, such that its filling radius satisfies \[\operatorname{FillRad}(X)\leq C_n\left(\inf_{x\in X}\operatorname{Sc}(x)\right)^{-2}?\] The filling radius measures the neighborhood size needed to fill the fundamental cycle after the metric-space embedding used in its definition.

Sources

Catalogue entry

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2234-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2234

Pointwise scalar curvature and width: The closed-manifold filling-radius corollary has dimensions at least four and scaling proportional to the inverse square root of scalar curvature.

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

Scalar Curvature Question [?35]: Bound on the Filling Radius forSc $\geq$$\sigma$ > 0

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.

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.

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.

Formal verification

Pointwise scalar curvature and 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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