Cartan–Hadamard isoperimetry and sharp integral fillings

Source and verification

This article explains family 337 of the October 6, 2026 OpenAI collection. Research conclusions are attributed to its manuscripts. The review covers statements, introductions, and selected proof mechanisms; it is not an independent verification of the complete proofs.

🏷️ Preliminaries: curvature, perimeter, and model balls

A Cartan–Hadamard manifold is a complete simply connected Riemannian manifold with nonpositive sectional curvature. Its exponential map at each point identifies the underlying smooth manifold with Euclidean space. Distances and volumes, however, need not be Euclidean.

For a smooth bounded region, perimeter is boundary area. The definition that also accommodates rough sets is

$$\begin{array}{c}{P}_g(E)=|D{1}_E{|}_g(M),\\ P_g(E)=\underset{|X{|}_g\le 1}{\sup }{\int }_E{\mathrm{div}}_{g}Xd{V}_g,\end{array}$$

where the supremum runs over smooth compactly supported vector fields. This is ambient perimeter: if a proof confines a set to a ball, boundary lying on that ball still counts. Sets differing on a set of volume zero are identified.

Let $d\ge 2$ be the ambient dimension, put $m=d-1$, and write ${\omega }_d$ for the Euclidean unit-ball volume and $s_m=d{\omega }_d$ for the unit-sphere area. In constant curvature $-b^2$, define

$${\mathrm{sn}}_b(r)=\{\begin{array}{ll}r,& b=0,\\ b^{-1}\sinh (br),& b>0.\end{array}$$

The model ball has area and volume

$$\begin{array}{rl}{\mathcal{A}}_b(r)& =s_m{\mathrm{sn}}_b(r{)}^m,\\ {\mathcal{V}}_b(r)& =s_m{\int }_0^r{\mathrm{sn}}_b(t{)}^{m}dt.\end{array}$$

Its isoperimetric profile is ${\mathcal{I}}_b={\mathcal{A}}_b\circ {\mathcal{V}}_b^{-1}$. It gives the boundary area of the model ball with a prescribed volume.

Smooth-manifold manuscript result

If ${\mathrm{Sec}}_g\le -b^2$ on a Cartan–Hadamard manifold, every measurable set of finite volume and finite perimeter satisfies

$$P_g(E)\ge {\mathcal{I}}_b(|E|).$$

The set need not be bounded or smooth. For $b=0$, this becomes the sharp Euclidean inequality in every dimension. (OpenAI, 2026)

For a Euclidean radius-$r$ ball, $V={\omega }_d{r}^d$ and $P=d{\omega }_d{r}^{d-1}$, so eliminating $r$ gives

$$P\ge d{\omega }_d^{1/d}{V}^{(d-1)/d}.$$

Comparison of geodesic-ball volumes alone does not prove this statement for arbitrary sets. A domain could have a complicated boundary unrelated to any distance sphere.

🏷️ Singular spaces and integral fillings

The second manuscript replaces a region by a filling of an oriented cycle. An integral current is a generalized oriented surface with integer multiplicity, finite mass, and a compatible finite-mass boundary. In a smooth example, its action is integration of differential forms. Metric current theory defines the same operations using Lipschitz test functions.

Write $\mathbf{M}(T)$ for mass and $\partial T$ for boundary. A cycle satisfies $\partial T=0$, and a filling satisfies $\partial S=T$. Mass counts absolute multiplicity; oppositely oriented pieces can cancel as currents.

A geodesic space is CAT(0) if distances across a geodesic triangle are no larger than those in its Euclidean comparison triangle. It is proper if closed bounded balls are compact. Proper CAT(0) spaces can have singularities and need not have a fixed finite ambient dimension.

Integral-filling manuscript result

In a proper CAT(0) space, every compactly supported integral $k$-cycle, $k\ge 2$, has a compactly supported integral filling with

$$\begin{array}{c}\partial S=T,\\ \mathbf{M}(S)\le c_k\mathbf{M}(T{)}^{(k+1)/k},\\ c_k=\frac{1}{(k+1)s_k^{1/k}},\\ s_k=(k+1){\omega }_{k+1}.\end{array}$$

The coefficient is optimal, with no restriction on integer multiplicity or ambient dimension. (OpenAI, 2026b)

Here $k$ is the cycle dimension, so a filled region has dimension $k+1$. For example, $c_2=1/(6\sqrt{\pi })$. The statement does not prescribe a ball as the filling’s parameter domain or specify its topology.

Manuscript	Setting	Additional conclusion
Generalized isoperimetry	Smooth manifolds, sectional curvature at most $-b^2$	Model comparison for all finite-volume sets; bounded Euclidean equality rigidity
Integral fillings	Proper CAT(0) spaces, compact integral cycles	Sharp Euclidean filling coefficient in arbitrary codimension

The prerequisites for the mechanisms below are Sobolev inequalities, variational arguments, coarea, and basic Riemannian geometry. The current-theoretic compactness and slicing results are used as stated analytic inputs.

🏷️ A sharp hypersurface inequality

The smooth proof begins with a hypersurface ${\Sigma }^m$ for $m\ge 3$. Write $h$ for mean curvature divided by $m$, with the convention $h=1/r$ on an outward-oriented Euclidean sphere. Set

$$q=\frac{2m}{m-2},Y_m=m{s}_m^{2/m}.$$

The central estimate is

$$\begin{array}{rl}{Q}_{\Sigma }(v)& =\frac{4}{m-2}{\int }_{\Sigma }|\nabla v{|}^{2}d\sigma \\ & +m{\int }_{\Sigma }(h^2-b^2)v^{2}d\sigma ,\\ Q_{\Sigma }(v)& \ge Y_m{\left({\int }_{\Sigma }|v{|}^{q}d\sigma \right)}^{2/q}.\end{array}$$

It applies to Lipschitz tests supported compactly in the interior of a locally $C^{1,1}$ immersed hypersurface. Completeness, connectedness, and a global unit normal are unnecessary; $h^2$ does not change when the normal is reversed.

The manuscript constructs maps from the hypersurface into a unit sphere, together with positive weights. Curvature comparison controls both their differentials and the logarithmic Hessians of the weights. Integration by parts combines these controls into a completed square in $Q_{\Sigma }$.

If the critical quotient lay below the Euclidean threshold, compactness and splitting of critical mass would produce a Dirichlet minimizer. Choosing the center and scale of the sphere map balances every coordinate against that minimizer’s critical mass. Coordinate variations then force a first-order equality. A weighted zero extension would have zero gradient and nonzero mass, contradicting its Dirichlet condition. The estimate is therefore a substantive analytic theorem, not a direct consequence of pointwise curvature comparison.

🏷️ Confined minimizers and the profile equation

Choose a large geodesic ball $B$ and minimize full ambient perimeter among sets inside $B$ with volume $v$. Let $I(v)$ be this confined profile. At a differentiability volume, free boundary variations and inward variations at the obstacle give

$$0\le h\le \frac{I^{\prime }(v)}{m}$$

on the regular boundary. The inequality at obstacle contact is needed because contact area is included in $I(v)$.

One would like to insert the constant function into the hypersurface inequality. A compact perimeter support may have a noncompact regular part after singular points are removed, so this requires justification.

Deleting a small ball near a singular point creates at most order $r^m$ new boundary and removes at most order $r^{m+1}$ volume. A deformation on a fixed smooth free patch repairs that volume with proportional perimeter cost. Minimality consequently gives

$$\sigma (\Sigma \cap B_r(x))\le C{r}^m.$$

The singular set has Hausdorff dimension at most $m-7$. Cover it by balls whose sum of $r^{m-2}$ is arbitrarily small. Cutoffs with gradients of size $r^{-1}$ then have arbitrarily small total Dirichlet energy, because each ball costs at most $C{r}^{m-2}$. They justify the constant test.

Combining that test with the mean-curvature bound yields

$$I^{\prime }(v)\ge m\sqrt{b^2+{\left(\frac{s_m}{I(v)}\right)}^{2/m}}.$$

This differential inequality has exactly the model profile as its equality solution. To integrate it without assuming an initial profile asymptotic, define

$$\begin{array}{c}{\mathcal{W}}_b(A)={\mathcal{V}}_b({\mathcal{A}}_b^{-1}(A)),\\ {\mathcal{W}}_b^{\prime }(A)=\frac{1}{m\sqrt{b^2+(s_m/A{)}^{2/m}}}.\end{array}$$

The chain rule gives $({\mathcal{W}}_b\circ I{)}^{\prime }\ge 1$ almost everywhere. Integrating from $\epsilon$ to $v$ and using ${\mathcal{W}}_b(I(\epsilon ))\ge 0$ gives ${\mathcal{W}}_b(I(v))\ge v$ as $\epsilon \downarrow 0$. Monotonicity gives $I(v)\ge {\mathcal{I}}_b(v)$.

Every bounded set is a competitor in a sufficiently large ball. Classical two- and three-dimensional inputs handle the dimensions outside the critical Sobolev argument. Finally, coarea chooses large truncation spheres whose additional boundary inside a finite-volume set tends to zero. This extends comparison to unbounded finite-volume sets.

🏷️ Equality determines the whole occupied region

For $b=0$, the first manuscript also claims that a bounded positive-volume equality set agrees almost everywhere with an open region isometric to a round Euclidean ball.

Regularity of equality sets is established before applying boundary identities. In dimensions at least four, equality in the hypersurface inequality supplies a sharp spectral bound. Logarithm coordinates centered at a squared-distance barycenter give an upper radial second-moment bound; a radial flux identity gives the matching lower bound. Equality forces a radial outward normal and one radius.

Low dimensions use separate arguments: a periodic Wirtinger estimate in dimension two, and a punctured flux identity in dimension three. Boundary containment by itself is insufficient. Constancy of the indicator on the complementary regions identifies the occupied interior. The exponential-map metric dominates the Euclidean metric there; equality of volume forces every metric eigenvalue to equal one throughout the region.

This proves rigidity of the entire region, not only its boundary. The ambient manifold outside the region need not be flat. The article does not extend this classification to unbounded equality sets or assert a negative-curvature equality theorem.

🏷️ The current proof: radial variation and critical mass

For the filling argument, put the compact support inside a closed convex ball, which is a compact CAT(0) space $Y$. Let $V(B)$ be the least filling mass of a cycle $B$. If the sharp bound fails, compactness produces a nonzero maximizer of

$$V(B)-c\mathbf{M}(B{)}^{(k+1)/k}$$

for some $c>c_k$. Rescaling normalizes its mass to

$$\mathbf{M}(A)=((k+1)c{)}^{-k}<s_k.$$

Write $\mu =\Vert A\Vert$ for its mass measure. Radial deformation toward $y\in Y$, with $r=d(y,\cdot )$, yields

$$\begin{array}{rl}& {\int }_Y(kg+rDr\cdot Dg)d\mu \\ & \le k{\int }_{Y}rg\sqrt{1-|Dr{|}^2}d\mu \end{array}$$

for nonnegative Lipschitz $g$. Derivatives are taken in the Euclidean tangent norms of rectifiable current charts. Triangle comparison supplies a quadratic bound for the full space-time deformation. Its determinant retains the transverse factor $\sqrt{1-|Dr{|}^2}$. No ambient normal bundle is assumed.

For $k=2$, an inverse-square cutoff and the Euclidean lower density force $\mathbf{M}(A)\ge 4\pi$, contradicting normalization. For $k>2$, the preceding filling dimension supplies small-set replacement estimates. Coarea and rearrangement give an almost sharp critical Sobolev inequality.

After closing the chart gradient in $L^2(\mu )$ and proving compactness, minimization gives a nonnegative function $v$ with

$$\begin{array}{c}\beta =\frac{1}{k-2},p=\frac{2k}{k-2},\\ 0<W={\int }_Y{v}^{p}d\mu <s_k,\\ d\nu =\frac{v^p}{W}d\mu .\end{array}$$

The last comparison uses the weighted chord

$$\ell (x,y)=v(y{)}^{\beta }d(y,x)v(x{)}^{\beta }.$$

The radial inequality, the minimizer equation, and tangent density give

$$\begin{array}{rl}& {\int }_Y\ell (x,y{)}^{2}d\nu (x)\\ & \le 2(W/s_k{)}^{2/k}<2\end{array}$$

for almost every center $y$. The composite tests are justified at $v=0$; the argument does not assume Sobolev regularity of $v^{\beta }$ when $\beta <1$.

🏷️ The variance contradiction

The opposite estimate has a useful explicit derivation. Define

$$\begin{array}{rl}{B}_{\ast }& ={\int }_Y{v}^{p+2\beta }d\mu ,\\ I(y)& ={\int }_{Y}d(y,x{)}^{2}v(x{)}^{p+2\beta }d\mu (x).\end{array}$$

Radial variation tested with $v^p$, followed by Cauchy–Schwarz and the minimizer equation tested with $v^{1+2\beta }$, gives

$$W^2\le B_{\ast }I(y)$$

for every center. All required moments and power tests are established first.

Let $o$ minimize $I$. The CAT(0) squared-distance inequality along the segment from $o$ to $y$ implies

$$I(y)\ge I(o)+B_{\ast }d(o,y{)}^2.$$

Indeed, the convexity inequality for the point at fraction $t$ on that segment includes the additional term $-t(1-t)d(o,y{)}^2$. Integrate against $v^{p+2\beta }\mu$, use minimality of $o$, divide by $t$, and let $t\downarrow 0$.

Now averaging the squared chords gives

$$\begin{array}{rl}& \iint \ell (x,y{)}^{2}d\nu (x)d\nu (y)\\ & =\frac{1}{W^2}\int I(y)v(y{)}^{p+2\beta }d\mu (y)\\ & \ge \frac{2B_{\ast }I(o)}{W^2}\ge 2.\end{array}$$

This contradicts the strict upper bound. The factor two comes from the variance inequality, not from a rough diameter estimate.

🏷️ Returning from currents to domains

For a bounded domain $\Omega$ in a $d$-dimensional Cartan–Hadamard manifold, apply the filling result to $T=\partial [[\Omega ]]$, taking $k=d-1$. The resulting filling must equal the integration current of $\Omega$.

To see this, their difference is a compactly supported top-dimensional cycle. Locally it is integration against an integer-valued function whose distributional derivatives vanish, hence a constant. Connectedness makes that constant global, and compact support in a noncompact manifold makes it zero. The filling mass bound is therefore a volume bound for the original domain.

Euclidean spheres attain the coefficient by the same uniqueness argument. The generalized smooth comparison also yields model-ball spectral and torsional comparisons for smooth relatively compact domains and $1<p<\infty$. The first Dirichlet $p$-eigenvalue is the infimum of the ratio of gradient $L^p$ energy to function $L^p$ mass among nonzero functions with zero boundary trace. The torsion quantity used here is the supremum of the $p$th power of the $L^1$ norm divided by that gradient energy.

Equimeasurable rearrangement preserves the relevant function norms and decreases gradient energy. Thus the eigenvalue is at least that of the equal-volume model ball, while this quotient-normalized torsional rigidity is at most its model value.

🏷️ Key steps and proof outlines

The hypersurface constant is produced by a completed square

For the smooth comparison argument, let $f=e^u$ be the positive weight associated with a sphere-valued comparison map $\Phi$, and let $\beta =Nu$ for a local normal $N$. Curvature comparison and restriction of the ambient Hessian give

$$m(h^2-b^2)\ge m{f}^2+m(h+\beta {)}^2+(m-2)|\nabla u{|}^2+2{\Delta }_{\Sigma }u.$$

Multiply by $v^2$ and integrate by parts. With $c=4/(m-2)$ and $\alpha =(m-2)/2$, the gradient terms are exactly

$$c|\nabla v{|}^2-4v⟨\nabla v,\nabla u⟩+(m-2)v^2|\nabla u{|}^2=c|\nabla v-\alpha v\nabla u{|}^2.$$

The differential estimate $|d\Phi {|}^2\le m{f}^2$ then supplies the geometric square used in the critical minimization argument. Reversing the local normal changes both $h$ and $\beta$, leaving every expression invariant. No global orientation is hidden in the estimate. (OpenAI, 2026a)

Why a strict critical deficit has a minimizer

In small charts the metric differs little from Euclidean space, so the sharp Euclidean Sobolev inequality holds with constant $Y_m-\epsilon$ and an added lower-order $L^2$ term. A partition with ${\sum }_i{\chi }_i^2=1$ gives the exact localization identity

$$\sum _i|\nabla ({\chi }_{i}v){|}^2=|\nabla v{|}^2+v^2\sum _i|\nabla {\chi }_i{|}^2.$$

Compactness in $L^2$ controls the lower-order term. Splitting the critical mass of a minimizing sequence shows that lost concentration costs at least the Euclidean threshold. A quotient strictly below that threshold therefore cannot disappear entirely into concentration. The resulting Dirichlet minimizer can be balanced against all coordinates of $\Phi$. Its coordinate variations and the completed square force a zero-gradient weighted extension incompatible with nonzero Dirichlet mass. This is the mechanism that rules out a strict deficit.

Weighted chords must be tested through regular composites

For the current-theoretic companion, retain the cycle dimension $k>2$, put $\beta =1/(k-2)$, and fix a center $y$ with $v(y)>0$. Write $\kappa =v(y{)}^{\beta }$, $r=d(y,\cdot )$, and $s=\kappa r{v}^{\beta }$. When $\beta <1$, the function $v^{\beta }$ is not automatically an admissible Sobolev test at its zeros.

The source instead uses the composites $vw(s)$ and $vw(s{)}^2$, where $w(t)=(1+a{t}^2{)}^{-(k-2)/2}$. As functions of $(r,z)$, these have expansion $z+O(z^{1+2\beta })$ near $z=0$ and extend as $C^1$ functions. Their derivatives are controlled by the proved moments and weighted energies of $v$. This justifies the product rules through the zero set without an unsupported chain rule for $v^{\beta }$ alone. (OpenAI, 2026b)

The radial inequality is completed using

$$1-rb-|Dr{|}^2+r^2/4=(b-r/2{)}^2\ge 0,b=\sqrt{1-|Dr{|}^2}.$$

Combining the resulting estimate with quotient minimality controls $J_y(a)=\int (1+a{s}^2{)}^{-k}d\nu$ by $(1+2a{L}_y{)}^{-k/2}$, where $L_y=\int s^{2}d\nu$. At almost every center, Euclidean tangent density supplies the opposite large-$a$ asymptotic and forces $L_y\le 2(W/s_k{)}^{2/k}<2$. The variance computation above then gives an average at least two. The contradiction depends on both local tangent density and global CAT(0) convexity; neither alone supplies the sharp coefficient.

See Also

Spectral scalar curvature and width concerns a weaker curvature invariant and a different geometric conclusion. Its width bounds do not imply the sharp perimeter or current-filling estimates discussed here.

🏷️ Manuscripts and verification scope

• Generalized Cartan–Hadamard isoperimetry and Euclidean equality rigidity — pinned manuscript and companion source.
• Sharp integral fillings in CAT(0) spaces — pinned manuscript and companion source.

The family scope page describes selected formalization coverage. Despite its broader heading, the documented selection is the sharp integral-current filling theorem and optimality of its coefficient. It does not list the generalized negative-curvature model comparison or the smooth manuscript’s Euclidean equality-region classification. No Lean build, axiom audit, or manuscript-to-formal-statement equivalence check was performed for these notes.

OpenAI collection provides prerequisites and the reading sequence; coverage register records completed and pending families.

🐻  OpenAI 2026a. Generalized Cartan–Hadamard isoperimetry and Euclidean equality rigidity.

🐻  OpenAI 2026b. Sharp integral fillings in CAT(0) spaces.