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Math-2235 · mathematics · Geometry

Scalar Curvature Question [?91]: Prove that there is a dimension-dependent constant $c_n$ such that every compact Riemannian $n$-manifold $X$ w

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.
Prove that there is a dimension-dependent constant $c_n$ such that every compact Riemannian $n$-manifold $X$ with $\operatorname{Sc}(X)\geq-\sigma^2$ satisfies $\lVert X\rVert\leq c_n\sigma^n\operatorname{vol}(X)$, where $\lVert X\rVert$ is the simplicial volume.

Formulation references: [S1], [S2].

Short English statement

Prove that there is a dimension-dependent constant $c_n$ such that every compact Riemannian $n$-manifold $X$ with $\operatorname{Sc}(X)\geq-\sigma^2$ satisfies $\lVert X\rVert\leq c_n\sigma^n\operatorname{vol}(X)$, where $\lVert X\rVert$ is the simplicial volume.

Sources

Catalogue entry

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2235-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2235

Integral negative scalar curvature: The integral estimate implies the recorded scalar-lower-bound volume estimate for closed oriented manifolds in dimensions at least three. The extracted source asks for compact manifolds without this dimension/orientation qualification; the manuscript does not state the whole source scope, so the match remains partial.

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

Scalar Curvature Question [?91]: Prove that there is a dimension-dependent constant \$c_n\$ such that every compact Riemannian \$n\$-manifold \$X\$ w

OpenAI's claim

Integral negative scalar curvature: For each $n\ge3$, the manuscript claims a dimension-only positive lower bound on the integral of $(\mathrm{Scal}^-)^{n/2}$ in terms of real simplicial volume. The bound is uniform over closed oriented manifolds and their metrics.

Mathematical statement of the claimed result

Mathematical claim 1.
For each integer $n\ge3$, some dimension-dependent $a_n>0$ is claimed to satisfy \[\int_M\left(\max\{0,-\operatorname{Scal}_g\}\right)^{n/2}\,dV_g\ge a_n\|M\|\] for each smooth metric $g$ on a closed smooth $n$-manifold $M$ which is oriented and connected. Here $\|M\|$ is the infimum of the sum of absolute coefficients over finite real singular cycles representing its fundamental class. Thus nonnegative scalar curvature forces $\|M\|=0$, but this implication alone does not treat all aspherical manifolds with vanishing simplicial volume.

Outline of the argument

Integral negative scalar curvature: The companion vanishing result forces a negative Yamabe minimizer, which is normalized to scalar curvature minus one; Hölder then reduces the desired integral estimate to a volume bound. A bounded cocycle is represented through an equivariant probability map from the cover. Weighted graph deformations increase map energy while retaining the volume budget, and a coordinate-by-coordinate degree estimate pays for the detecting cocycle, yielding the uniform comparison.

Formal verification

Integral negative scalar curvature: 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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