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Math-2195 · mathematics · Geometry

The Cartan-Hadamard Conjecture

Status Open · 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.
A Cartan--Hadamard manifold is complete, simply connected, and has nonpositive sectional curvature. In dimension $n$, must every bounded sufficiently regular domain $\Omega$ in such a manifold satisfy the sharp Euclidean isoperimetric bound \[|\partial\Omega|\geq n\omega_n^{1/n}|\Omega|^{(n-1)/n},\] where $\omega_n$ is the volume of the Euclidean unit ball?

Formulation references: [S1], [S2].

Short English statement

A Cartan--Hadamard manifold is complete, simply connected, and has nonpositive sectional curvature. In dimension $n$, must every bounded sufficiently regular domain $\Omega$ in such a manifold satisfy the sharp Euclidean isoperimetric bound \[|\partial\Omega|\geq n\omega_n^{1/n}|\Omega|^{(n-1)/n},\] where $\omega_n$ is the volume of the Euclidean unit ball?

Sources

Catalogue entry

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2195-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2195

Model isoperimetry and equality rigidity: The sharp model theorem contains the entire Euclidean Cartan-Hadamard inequality and additionally supplies negative-curvature-model comparisons and Euclidean equality rigidity.

Sharp fillings in CAT(0): Sharp integral fillings in proper CAT(0) spaces imply the smooth Hadamard-domain isoperimetric bound for ambient dimensions at least three; the two-dimensional bound is the classical case explicitly included in the paper. Equality rigidity is not part of these B statements.

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

The Cartan-Hadamard 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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