we-solve (beta)

Math-2673 · mathematics · Geometry

Euclidean sharp filling bounds in proper CAT(0) spaces

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For every proper CAT(0) space X and integer $n\geq 2$, does every compactly supported Ambrosio–Kirchheim integral n-current T with $\partial T=0$ bound a compactly supported integral (n+1)-current S satisfying $M(S)\leq [(n+1)^((n+1)/n)\omega _{n+1}^{1/n}]^{-1}$ $M(T)^((n+1)/n)$, where $\omega _j$ is unit Euclidean j-ball volume?

Short English statement

For every proper CAT(0) space X and integer $n\geq 2$, does every compactly supported Ambrosio–Kirchheim integral n-current T with $\partial T=0$ bound a compactly supported integral (n+1)-current S satisfying $M(S)\leq [(n+1)^((n+1)/n)\omega _{n+1}^{1/n}]^{-1}$ $M(T)^((n+1)/n)$, where $\omega _j$ is unit Euclidean j-ball volume?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2673-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2673

Sharp fillings in CAT(0): The paper claims the Euclidean-optimal integral filling inequality for compactly supported integer cycles of dimension at least two in proper CAT(0) spaces. Arbitrary integer multiplicities and ambient dimensions are permitted. 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.

Euclidean sharp filling bounds in proper CAT$0$ spaces

OpenAI's claim

Sharp fillings in CAT$0$: The paper claims the Euclidean-optimal integral filling inequality for compactly supported integer cycles of dimension at least two in proper CAT$0$ spaces. Arbitrary integer multiplicities and ambient dimensions are permitted.

Mathematical statement of the claimed result

Mathematical claim 1.
For every proper CAT$(0)$ metric space $X$ and integer $n\ge2$, each compactly supported integral $n$-current $T$ with $\partial T=0$ is claimed to bound a compactly supported integral $(n+1)$-current $S$ with \[\partial S=T,\qquad\mathbf M(S)\le\frac{\mathbf M(T)^{(n+1)/n}}{(n+1)^{(n+1)/n}\omega_{n+1}^{1/n}}.\] No ambient-dimension or multiplicity restriction is imposed, and the constant is claimed optimal. Its manifold corollary gives $\operatorname{area}(\partial\Omega)\ge d\omega_d^{1/d}\operatorname{vol}(\Omega)^{(d-1)/d}$ for smooth bounded domains in a smooth Cartan--Hadamard $d$-manifold, $d\ge3$. For relatively compact measurable $E$ of finite perimeter the assertion instead uses the ambient variation $P_g(E)=|D\mathbf1_E|_g(M)$: \[P_g(E)\ge d\omega_d^{1/d}\operatorname{vol}_g(E)^{(d-1)/d}.\] For smooth $\Omega$, $P_g(\Omega)=\operatorname{area}_g(\partial\Omega)$.

Outline of the argument

Sharp fillings in CAT$0$: Induction reduces a failing inequality to a compact CAT$0$ space and an extremizing cycle of small normalized mass. Inward geodesic variations estimate its curvature-like behavior in current charts. The resulting sharp mass comparison contradicts extremality, and Euclidean examples show that the coefficient cannot be improved.

Formal verification

Family 337 describes Lean coverage for the sharp integral-current CAT$0$ filling estimate and its optimal coefficient; the curved-model perimeter theorem is separate.

Sharp fillings in CAT$0$: The manuscript links FillingCoefficient.lean, SharpCAT0Filling.lean.

The documented selection includes this manuscript’s main assertion; that assertion retains the restrictions stated above.

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

Go to contribution →