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Math-2679 · mathematics · Geometry

Spectral scalar-curvature control of width in dimension three

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
If a connected complete smooth three-manifold without boundary satisfies $\int (4|\nabla \varphi |^{2}+Scal\cdot \varphi ^{2})\geq \lambda \int \varphi ^{2}$ for some $\lambda >0$ and every compactly supported real smooth $\varphi $, must there be a continuous map to a one-dimensional simplicial complex whose every fiber has diameter at most $500/\surd \lambda $?

Short English statement

If a connected complete smooth three-manifold without boundary satisfies $\int (4|\nabla \varphi |^{2}+Scal\cdot \varphi ^{2})\geq \lambda \int \varphi ^{2}$ for some $\lambda >0$ and every compactly supported real smooth $\varphi $, must there be a continuous map to a one-dimensional simplicial complex whose every fiber has diameter at most $500/\surd \lambda $?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2679-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2679

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. The parameters, regularity assumptions and normalization in this question are retained in applying that assertion.

With the hypotheses and conventions above, the claimed result addresses the whole mathematical assertion on this page.

Spectral scalar-curvature control of width in dimension three

OpenAI's claim

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.

Mathematical statement of the claimed result

Mathematical claim 1.
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.

Outline of the argument

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.

Formal verification

Spectral width in dimension three: 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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