Negative Kähler curvature: bounded coordinates and the holomorphic Liouville property

Source and verification

This article explains family 359 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.

🏷️ Kähler potentials and bounded holomorphic functions

Negative curvature constrains geodesics, while a bounded holomorphic coordinate system is a global analytic object. Family 359 contains two independent constructions separating these kinds of information. One retains two-sided negative sectional-curvature bounds but excludes a full tuple of bounded coordinates. The other excludes every nonconstant bounded holomorphic function while retaining a uniform negative upper curvature bound.

On a complex manifold, a smooth real function $\Phi$ is strictly plurisubharmonic when its complex Hessian

$${\Phi }_{i\bar{j}}=\frac{{\partial }^2\Phi }{\partial z_i\partial {\bar{z}}_j}$$

is positive definite. This Hermitian matrix defines a Kähler metric: a Riemannian metric compatible with the complex structure whose associated two-form is closed. Such a $\Phi$ is a Kähler potential. Adding the real part of a holomorphic function does not change its complex Hessian.

Sectional curvature here means curvature on every real two-plane. Holomorphic sectional curvature examines only planes invariant under the complex structure, so checking those planes alone would not prove either manuscript’s claimed result. A two-sided negative pinching bound has the form

$$\begin{array}{c}-B\le {\mathrm{Sec}}_g\le -A<0,\\ 0<A\le B<\infty .\end{array}$$

For a complex manifold $M$, let $H^{\infty }(M)$ denote its bounded holomorphic functions. The holomorphic Liouville property means $H^{\infty }(M)=\mathbb{C}$: every such function is constant. A bounded coordinate tuple is much more demanding than a single nonconstant bounded function. In dimension $m$, it is a bounded holomorphic map to ${\mathbb{C}}^m$ with nonzero Jacobian determinant everywhere. Such a map is locally biholomorphic; global injectivity is an additional condition.

🏷️ Two constructions with different curvature hypotheses

The pinched threefold stated in the first manuscript

There is a contractible domain $M\subset {\mathbb{C}}^3$ with a smooth complete Kähler metric satisfying two-sided negative real sectional pinching. No bounded holomorphic map $M\to {\mathbb{C}}^3$ has everywhere nonzero Jacobian. In particular, $M$ is not biholomorphic to a bounded domain. It nevertheless has two nonconstant bounded holomorphic base-coordinate functions. (OpenAI, 2026)

The Liouville example stated in the companion

In some sufficiently large fixed finite complex dimension $m$, there is a domain $\mathcal{T}\subset {\mathbb{C}}^m$, diffeomorphic to ${\mathbb{R}}^{2m}$, with a complete Kähler metric satisfying ${\mathrm{Sec}}_g\le -1$ and $H^{\infty }(\mathcal{T})=\mathbb{C}$. Its sectional curvatures are unbounded below. (OpenAI, 2026b)

Feature	Pinched threefold	Liouville tube
Complex dimension	Three	Some sufficiently large fixed finite dimension
Curvature control	Two finite negative bounds	Uniform negative upper bound; no finite lower bound
Bounded functions	Two nonconstant examples remain	Every bounded holomorphic function is constant
Excluded analytic object	Everywhere independent bounded coordinate triple	Any nonconstant bounded holomorphic function

Neither example decides whether finite two-sided negative real sectional pinching forces at least one nonconstant bounded holomorphic function. The proofs are independent, and the stronger analytic conclusion of the companion comes with a weaker curvature hypothesis.

The papers also use different metric normalizations. For the same complex Hessian, the first takes twice the real metric used by the second. Consequently the ball potential $-\log (1-|z{|}^2)$ has sectional range $[-2,-1/2]$ in the first convention and $[-4,-1]$ in the second. We retain their respective constants below.

🏷️ A disk family and a Jacobian obstruction

For the threefold, start with the unit ball $\mathbb{B}\subset {\mathbb{C}}^2$ and a smooth real function $\varphi$. Define the Hartogs domain

$$\begin{array}{c}{M}_{\varphi }=\{(z,w)\in \mathbb{B}\times \mathbb{C}:t<1\},\\ t(z,w)=e^{\varphi (z)}|w{|}^2.\end{array}$$

The fiber over $z$ is a disk of radius $e^{-\varphi (z)/2}$. The map $(z,w)\mapsto (z,e^{\varphi (z)/2}w)$ is a smooth real diffeomorphism to $\mathbb{B}\times \mathbb{D}$, proving contractibility; it is not asserted to be holomorphic.

The analytic construction arranges that no holomorphic $H$ on $\mathbb{B}$ can satisfy

$$\begin{array}{rl}\mathrm{Re}H(z)& \le C_H+4\log \frac{1}{1-|z|}\\ & +\varphi (z).\end{array}$$

Here $C_H$ can be any real constant. The real part of a holomorphic function is called pluriharmonic. Thus this is a precisely weighted obstruction to a global pluriharmonic minorant.

To see why it excludes bounded coordinates, suppose $F=(F_1,F_2,F_3)$ is bounded by $B_0\ge 1$ componentwise. Along the zero section, define $d(z)=\det DF(z,0)$. Cauchy’s estimates in base disks and the fiber disk give, for $\delta =1-|z|$,

$$\begin{array}{rl}|{\partial }_{z_j}{F}_{\ell }(z,0)|& \le B_0{\delta }^{-1},\\ |{\partial }_w{F}_{\ell }(z,0)|& \le B_0{e}^{\varphi (z)/2}.\end{array}$$

The base estimates take place entirely on the zero section and require no uniform lower fiber radius nearby. Expanding the determinant into six terms yields

$$\begin{array}{rl}|d(z)|& \le 6B_0^3{\delta }^{-2}{e}^{\varphi (z)/2},\\ \log |d(z){|}^2& \le C+4\log \frac{1}{1-|z|}\\ & +\varphi (z).\end{array}$$

If $d$ were nowhere zero, simple connectivity of the ball would give a holomorphic logarithm $\ell$ with $e^{\ell }=d$. Taking $H=2\ell$ contradicts the minorant obstruction. This excludes an everywhere nonsingular bounded map, even without assuming injectivity. It does not show that every bounded map’s determinant vanishes identically. (OpenAI, 2026a)

🏷️ Representing probabilities and negative logarithmic averages

The potential must satisfy the minorant obstruction while maintaining derivative bounds strong enough to control curvature. The first manuscript builds it from lacunary homogeneous polynomial peaks.

Let $S$ be the unit sphere in ${\mathbb{C}}^2$. A positive probability measure $\mu$ on $S$ represents evaluation at zero if

$${\int }_{S}pd\mu =p(0)$$

for every holomorphic polynomial $p$. Taylor approximation then gives the same identity for $H(r\cdot )$ whenever $H$ is holomorphic on the ball and $r<1$.

If positive representing probabilities ${\mu }_N$ and radii $r_N\uparrow 1$ satisfy

$$\begin{array}{c}{\int }_S\varphi (r_N\xi )d{\mu }_N+4\log \frac{1}{1-r_N}\\ ⟶-\infty ,\end{array}$$

integrating a proposed minorant is impossible: its left side always averages to the fixed number $\mathrm{Re}H(0)$. Positivity of the measures is needed to integrate the inequality.

A representing measure need not be a Jensen measure, which would also impose the appropriate logarithmic mean inequality. The construction exploits exactly this distinction. There is no contradiction between preserving holomorphic means and obtaining negative averages of carefully chosen plurisubharmonic logarithmic tests.

At degree $k$, choose separated unit directions $p$ modulo phase and form

$$P_k(z)=\sum _p⟨z,p{⟩}^k.$$

The peaks have width comparable to $k^{-1/2}$. Packing estimates give a uniform bound $|P_k(z)|\le C|z{|}^k$ and Gaussian decay away from the chosen directions. A fixed regularization produces smooth plurisubharmonic functions

$$U_k(z)=\log \frac{|1-a{P}_k(z){|}^2+{ϵ}^2}{1+{ϵ}^2},$$

with $U_k(0)=0$. Their positive complex Hessian follows by differentiating the regularized logarithm.

The representing densities are built recursively at degrees $k_j=Q^j$. Local oscillatory corrections annihilate every holomorphic moment exactly through a radial integration-by-parts identity. The correction includes a derivative of the preceding density to compensate for its existing phase frequencies. Large fixed degree separation preserves positivity. Each new logarithmic test has a fixed negative average, and that bound survives all later density corrections. These persistence and positivity estimates are substantive parts of the source proof.

Set $\psi =-\log (1-|z{|}^2)$ and

$$\varphi =\psi +L\sum _{j\ge 1}{U}_{Q^j}.$$

The series converges smoothly on compact subsets because the high-degree terms decay exponentially there. Its complex Hessian dominates that of $\psi$. At $r_N=1-\eta /Q^N$, the first $N$ tests each contribute at most a fixed negative number $-d_3$. The later terms have a uniformly bounded tail. Including the base potential and the four logarithms from the Jacobian estimate gives an upper bound

$$N(5\log Q-L{d}_3)+O(1).$$

Choose one finite $L>5\log Q/d_3$. This proves the required negative-average limit. Meanwhile, in ball automorphisms centered near the boundary, low-degree terms vary slowly and high-degree terms decay rapidly. Splitting at $k(1-|z|)\approx 1$ yields uniform derivatives through order four in those centered charts. Values of $\varphi$ need not be bounded. (OpenAI, 2026a)

🏷️ Completeness and the all-real-plane pinching estimate

The Kähler potential on $M_{\varphi }$ is

$${\mathcal{K}}_{\lambda }=\lambda \psi -\log (1-t).$$

Write $h=\partial \bar{\partial }\varphi$, $I=\partial \bar{\partial }\psi$, $x=t/(1-t)$, and $\vartheta =dw+w\partial \varphi$. Its Hermitian metric has the useful decomposition

$$\hat{g}=\lambda I+xh+\frac{e^{\varphi }}{(1-t{)}^2}|\vartheta {|}^2.$$

This formula is smooth and positive even at $w=0$. In the first paper’s real-metric convention, direct inversion gives

$$\begin{array}{rl}|\nabla [-\log (1-t)]{|}_g^2& =2t<2,\\ |\nabla \psi {|}_g^2& \le 2/\lambda .\end{array}$$

A curve of bounded length therefore cannot approach either the base boundary or a fiber boundary. Once the base stays in a compact subset, its fiber radius is bounded as well. This gives compact containment of metric balls and geodesic completeness. The potential is also a strictly plurisubharmonic exhaustion.

Curvature requires more than positivity of the complex Hessian. The source proves a transfer statement assuming $cI\le h\le CI$ and bounded component derivatives of $h$ through second order in centered charts. No negative-curvature assumption on the metric $h$ itself is inserted. For one sufficiently large finite $\lambda$, the resulting metric has negative curvature on every real plane.

The delicate case consists of planes whose horizontal and vertical projections have opposite complex orientations. After balancing these projections, the leading curvature numerator is nonnegative but can vanish. The largest error has three horizontal indices, so its contraction contains a mixed block associated with the plane. The source identifies the equality cases of the leading expression and shows that this mixed block vanishes there too.

Quantitatively, the remaining positive gap is bounded below by a constant times

$$\frac{\lambda }{x(1+x)}+\frac{x}{\lambda +x},$$

up to smaller errors in the relevant large-$x$ regime. Splitting at $\lambda =x^{3/2}$ bounds this expression below by a multiple of $x^{-1/2}$. The four-horizontal-index error is $O(x^{-1})$; the mixed-block cancellation improves the potentially larger error to $o(x^{-1/2})$. Thus it cannot cancel the gap.

After fixing $\lambda$, compactness handles bounded $x$. At the fiber boundary $x\to \infty$, rescaled metrics and curvature tensors converge uniformly to a complex-hyperbolic model. This last uniform limit supplies both a finite lower curvature bound and a strictly negative upper bound. A pointwise negativity statement alone would not establish the claimed pinching. (OpenAI, 2026a)

🏷️ The companion’s half-plane tube

The second construction starts with a smooth strictly plurisubharmonic function $\ell$ on ${\mathbb{C}}^{n+1}$ whose potential metric is complete and has nonpositive real sectional curvature. Write the base coordinates as $(s,y)\in {\mathbb{C}}^n\times \mathbb{C}$ and introduce $w=u+iv$. Set

$$\begin{array}{c}b=v-\ell (s,y),\mathcal{T}=\{b>0\},\\ g_{\mathcal{T}}=g_{-\log b}.\end{array}$$

The map $(s,y,u,b)\mapsto (s,y,u+i(\ell (s,y)+b))$ identifies this tube smoothly with Euclidean space. With $d^c\ell =-d\ell \circ J$, the metric decomposes as

$$g_{\mathcal{T}}=b^{-1}{g}_{\ell }+\frac{d{b}^2+(du+d^c\ell {)}^2}{4b^2}.$$

Finite length bounds the total variation of $\log b$, keeping $b$ between positive finite constants. The base projection then has finite length and converges by base completeness. On its compact closure, $d^c\ell$ is bounded, so the remaining real coordinate $u$ also converges. This proves tube completeness.

In suitable holomorphic coordinates at a point, the curvature numerator is the complex-hyperbolic numerator plus $b^{-1}$ times the base curvature numerator on the projected vectors. The extra term is nonpositive on every real plane. Since the model has sectional range $[-4,-1]$ in this paper’s convention, the tube satisfies $\mathrm{Sec}\le -1$.

If the base has a strictly negatively curved plane $\sigma$ with orthonormal basis $X,Y$, its horizontal lifts give the exact formula

$$\begin{array}{rl}{\mathrm{Sec}}_{\mathcal{T}}(\hat{\sigma })& =-1-3g_{\ell }(JX,Y{)}^2\\ & +b{\mathrm{Sec}}_{g_{\ell }}(\sigma ).\end{array}$$

Letting $b\to \infty$ proves that the curvature is unbounded below. The source supplies such a negative base plane. This loss is explicit, not merely an absent estimate. (OpenAI, 2026b)

🏷️ Large disks, phases, and vanishing Taylor coefficients

The remaining task is to choose the base potential so that every bounded holomorphic function on the tube is constant. The analytic argument asks for large disks at carefully chosen points, with prescribed shears in the $(y,w)$ variables.

At a testing stage, fix a fiber center $\zeta$, a Taylor degree $h$, and a large polynomial degree $N>h$. A polynomial sampling construction gives roughly $N^{n-1}$ directions with affine lifts $v_{\xi }$. Each lift comes with the cycle $e^{2\pi iq/N}{v}_{\xi }$. Assign one unit phase $b_{\xi }$ to each entire cycle. The required disks have the form

$$\begin{array}{c}t⟼(Rv,\zeta +t,w+i\alpha b_{\xi }t),\\ |t|<e^{m_{0}c}.\end{array}$$

For $|f|\le 1$, write $A=f_y$ and $I=i{f}_w$, both evaluated at $(Rv,\zeta ,w)$. Cauchy’s estimate on such a disk gives

$$|A+\alpha b_{\xi }I|\le e^{-m_{0}c}.$$

Root-of-unity averaging selects homogeneous Taylor degrees congruent to $h$ modulo $N$. Explicitly, if $A={\sum }_r{A}_r$ is its homogeneous expansion,

$$\begin{array}{c}\frac{1}{N}\sum _{q=0}^{N-1}{e}^{-2\pi ihq/N}A(R{e}^{2\pi iq/N}v)\\ =\sum _{j\ge 0}{A}_{h+jN}(Rv).\end{array}$$

The phases are chosen so that the degree-$h$ contribution from $f_w$ cannot be absorbed into the finite-dimensional space generated by the first few surviving terms from $f_y$. Sufficiently many spatial variables make that unwanted evaluation space occupy a small fraction of the sampled values. The phases work simultaneously for every polynomial of the selected degree, and are fixed before the bounded function is considered.

The quantitative choices are essential. Bounds on a slightly larger spatial ball give a tail of order $e^{-0.99kc}$ after $k$ retained terms, a bound $e^{p_{0}c}$ for $f_y$, and ${\alpha }^{-1}\le e^{p_{1}c}$. Choose

$$m_0>p_1,0.99k>p_0+p_1+1.$$

Only then choose one sufficiently large finite $n$ so that

$$\sum _{r=1}^{k-1}\frac{r^{n-1}}{(n-1)!}<\mathrm{0.01.}$$

The resulting separation estimate makes every tested Taylor coefficient of $f_w$ at most $C_h{e}^{-\epsilon c}$ for a fixed $\epsilon >0$.

Tests occur at increasing heights $D_j$. Exponential smallness there would not alone force vanishing at a fixed height. Each pair $(\zeta ,h)$ is repeated infinitely often with

$$\begin{array}{c}{D}_j\to \infty ,c_j\to \infty ,\\ D_j/c_j\to 0.\end{array}$$

For a fixed Taylor coefficient $g(w)$, Cauchy estimates give a stage-independent bound $B$ above a fixed lower line of height $t_0$. The three-lines theorem on the strip up to $D_j$ gives, at a fixed $t>t_0$,

$$\begin{array}{c}\underset{\mathrm{Im}w=t}{\sup }|g(w)|\le B^{1-{\theta }_j}(C_h{e}^{-\epsilon c_j}{)}^{{\theta }_j},\\ {\theta }_j=\frac{t-t_0}{D_j-t_0}.\end{array}$$

Because $c_j/D_j\to \infty$, the right side tends to zero. All spatial Taylor coefficients of $f_w$ vanish at each scheduled center. A dense set of centers, continuity, and holomorphic uniqueness give $f_w=0$ throughout the connected tube. The function descends to a bounded entire function on ${\mathbb{C}}^{n+1}$ and is constant by Liouville’s theorem. (OpenAI, 2026b)

🏷️ Realizing every test in one complete metric

The geometric construction must install all of those disks while preserving nonpositive curvature of the base. It uses a radial fiber potential obtained by a partial Legendre transform. A positive profile describes fiber size, and its dependence on the spatial variables produces additional curvature terms. The source computes these terms before estimating them.

A real two-plane determines a Hermitian test matrix of rank at most two. Estimating the negative curvature tensor on this class avoids dimension-dependent trace losses. This is needed because the geometric exponent constants must be fixed before choosing the large dimension required by the sampling argument.

Polynomial wells enlarge selected fibers, and small nonholomorphic shear terms give the desired disk directions. Moving a fiber center is also not a holomorphic coordinate change. The paper first aligns the derivatives of two spatial metrics, widens the fiber profile, moves the center in fixed coordinates, and then restores the profile with an allowed constant offset. Curvature margins absorb the resulting perturbations on entire fibers, including their axes.

Each stage reserves bounds for later polynomial wells, whose small tails extend into earlier regions. Rapid spatial growth occurs only on finite intervals and is followed by a capped regime. The stage radii tend to infinity, preventing finite-radius accumulation, and the later tails converge smoothly on every compact set. Thus earlier disks survive in the final potential.

For completeness, the final base metric uniformly controls the Euclidean spatial projection. A finite-length curve therefore stays in a compact spatial region. On that region the fiber part controls logarithmic escape in $|y|$, preventing divergence in the remaining direction. These two estimates complete the base metric, to which the tube transfer applies.

The potential $\ell$ is necessarily unbounded below; the testing disks show this directly. Hence $e^{ikw}$ for $k>0$ is unbounded on the tube and supplies no counterexample to its Liouville property. The construction does not assume all points of the tube lie above a fixed horizontal line. (OpenAI, 2026b)

🏷️ Key steps and proof outlines

A negative logarithmic average from exact cancellation

For the first construction, the distinction between a representing measure and a Jensen measure is made quantitative. The regularized test is

$$U_k(z)=\log \frac{|1-a{P}_k(z){|}^2+{ϵ}^2}{1+{ϵ}^2},$$

where $P_k$ is homogeneous of degree $k$ and $a=e^{\alpha }$. Its complex Hessian is a positive scalar multiple of $\partial P_k\otimes \overline{\partial P_k}$, so it is smooth and plurisubharmonic. The local peak model replaces $P_k$ by $e^{-y}{e}^{it}$. The elementary Fourier identities behind its negative average are

$$\begin{array}{rl}\int \log |1-q{e}^{it}{|}^2\frac{dt}{2\pi }& =2\max \{\log q,0\},\\ \int \log |1-q{e}^{it}{|}^2\cos t\frac{dt}{2\pi }& =-\min \{q,q^{-1}\}.\end{array}$$

The profile $f$ has the exact cancellation $\int e^{-y}f(y)dy=0$ and equals $-s_1$ on $[0,\alpha ]$. Consequently the model average is

$${\alpha }^2-2s_1(\cosh \alpha -1)=-\Delta <0.$$

First fix the regularization $ϵ$ so that this strict gap survives. Then separate the peaks far enough that their approximation errors are small, and only afterwards increase their degrees. Interchanging that order would leave the logarithm’s derivatives uncontrolled near its zeros.

Later density corrections preserve earlier negative tests

Write $k_j=Q^j$. The test $U_{k_j}$ has phase frequencies in $k_j\mathbb{Z}$, whereas the earlier density has bandwidth less than $k_j$. Phase integration therefore removes every earlier nonconstant frequency exactly. Later corrections have frequencies of magnitude at least $(Q-2)k_j$.

For the regularized one-variable logarithm, two integrations by parts give Fourier coefficients bounded by $B_{ϵ}/m^2$. The part that can interact with the later corrections is therefore a uniformly small tail once $Q$ is large. Positivity and unit mass of both densities bound its integrated effect by its uniform size. The strict negative average persists at every later stage. This is the quantitative reason the eventual potential can contain infinitely many tests without destroying the already established obstruction. (OpenAI, 2026a s “Regularized logarithmic tests” and “Construction of the base potential”)

Infinitely many large disks require a reserved error budget

The companion construction has a different diagonal argument. If $W_j$ is the new polynomial well and $K_j$ the entire previously configured spatial ball, the source imposes

$$\underset{K_j}{\sup }|{\partial }^{[r]}{W}_j|\le 2^{-j}(0\le r\le j+4).$$

Every local curvature estimate is arranged in advance to tolerate an additional allowance of one for the value and first four spatial derivatives of all future wells. Their summed errors fit that allowance. Higher derivative bounds give smooth convergence on each fixed compact spatial set.

This alone would not establish curvature on an unbounded fiber: the local curvature lemmas must tolerate these spatial errors uniformly over the whole fiber. The assembly uses those stronger lemmas before passing to the limit. Capped intervals of the radial construction can be extended to arbitrarily large spatial radius, preventing infinitely many tests from accumulating at a finite radius. This is how the analytic schedule and geometric completeness are made compatible in one finite dimension. (OpenAI, 2026b“Assembly of the complete base”)

🏷️ Connections and review limits

The no-conjugate-points construction separates a geodesic condition from the existence of a nonpositively curved metric. Here the separation concerns global bounded holomorphic functions and complete negatively curved metrics.

The review checked both statements and normalizations, the first paper’s zero-section Jacobian argument, representing-measure mechanism, completeness proof and pinching proof, and the companion’s tube transfer, disk-testing criterion, selected fiber-curvature estimates, parameter ordering and final assembly. It did not independently verify the full representing-density estimates, all curvature component calculations, polynomial sampling and phase estimates, or the long recursive construction. The two research conclusions are attributed to their respective manuscripts.

🏷️ Manuscripts and verification scope

• A negatively pinched Kähler threefold without bounded holomorphic coordinates — pinned manuscript and companion source.
• One-sided negative sectional curvature and the holomorphic Liouville property — pinned manuscript and companion source.

The family scope page describes selected formalization coverage. The documented selection is the pinched threefold construction, including its controlled potential and obstruction to bounded holomorphic coordinates. The companion with the holomorphic Liouville property and only a one-sided curvature bound is outside that documented selection. 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. A negatively pinched Kähler threefold without bounded holomorphic coordinates.

🐻  OpenAI 2026b. One-sided negative sectional curvature and the holomorphic Liouville property.