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Math-2433 · mathematics · Topology

Cartan–Hadamard Isoperimetric Conjecture

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.
Does every Cartan–Hadamard manifold $(M^{n+1},g)$ satisfy $$ |\Omega|\leq n^{-(n+1)/n}\omega_{n+1}^{-1/n}|\partial\Omega|^{(n+1)/n} $$ for every bounded sufficiently regular open set $\Omega\subset M$?

Formulation references: [S1], [S2].

Short English statement

Does every Cartan–Hadamard manifold $(M^{n+1},g)$ satisfy $$ |\Omega|\leq n^{-(n+1)/n}\omega_{n+1}^{-1/n}|\partial\Omega|^{(n+1)/n} $$ for every bounded sufficiently regular open set $\Omega\subset M$?

Sources

Catalogue entry

Statement licence: CC-BY-SA-4.0 · Source file

Contributions 1

Math-2433-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2433

Model isoperimetry and equality rigidity: In ambient dimension n plus one the sharp coefficient uses n plus one rather than the extracted record denominator n. The paper sharp Euclidean inequality therefore implies even the weaker displayed coefficient in this record, and its model comparison has further conclusions.

Sharp fillings in CAT(0): The current filling theorem implies the Euclidean domain inequality in every smooth ambient dimension, with the classical two-dimensional case explicitly included in Section 08. Arbitrary CAT(0) current fillings extend the domain-only target; the displayed n-versus-n-plus-one coefficient in 30003543 is weaker than the sharp coefficient.

The claimed result implies the stated question and also supplies the additional strength described above.

Cartan–Hadamard Isoperimetric Conjecture

OpenAI's claim

Model isoperimetry and equality rigidity: The manuscript claims sharp comparison with the constant-curvature model ball for every finite-volume finite-perimeter set in a simply connected complete manifold whose sectional curvatures do not exceed $\kappa\le0$; in the Euclidean comparison, bounded equality sets are precisely regions isometric to round Euclidean balls, up to null sets.

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.
Let $n\ge2$, $\kappa\le0$, and assume $(M^n,g)$ is a smooth Riemannian manifold, simply connected and complete, satisfying $\sec_g\le\kappa$. For a measurable set of finite volume and finite ambient perimeter, the claim is \[P_g(E)\ge\mathcal I_{\sqrt{-\kappa}}(\operatorname{vol}_g E).\] Here $s_b(r)=r$ for $b=0$ and $s_b(r)=\sinh(br)/b$ otherwise, $A_b(r)=n\omega_ns_b(r)^{n-1}$, $V_b(r)=n\omega_n\int_0^rs_b(t)^{n-1}\,dt$, and $\mathcal I_b=A_b\circ V_b^{-1}$; $\omega_n$ is unit-ball volume. When $\kappa=0$, equality $P_g(E)=n\omega_n^{1/n}\operatorname{vol}(E)^{(n-1)/n}$ for bounded positive-volume $E$ holds exactly when, up to a volume-zero set, $E$ is an open region isometric to a round Euclidean ball.
Mathematical claim 2.
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

Model isoperimetry and equality rigidity: A sharp critical Sobolev inequality is established on immersed hypersurfaces using radial sphere maps and positive weights compatible with Jacobi comparison. This controls the boundary geometry needed for the isoperimetric estimate. Approximation extends the bound to finite-perimeter sets, and the equality analysis forces the geometric comparison to be exact, identifying a Euclidean ball region.

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.

Model isoperimetry and equality rigidity: No manuscript-specific Comparator is listed in the pinned inventory.

The documented selection does not cover the entire manuscript assertion relevant here.

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

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