Negative Kähler curvature does not guarantee bounded holomorphic coordinates

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Family 359 · Differential geometry
Importance 5/5 · Provisional editorial rating
Sources consulted 2026-10-26

1. The question

The unit disk is a basic meeting point of complex analysis and negative curvature. In complex dimension one, a complete simply connected surface with curvature bounded above by a negative constant is biholomorphic to the disk. Its geometry therefore comes with a bounded global holomorphic coordinate.

Does a similar principle hold in several complex variables? If a simply connected complete Kähler manifold has uniformly negative curvature, must it be biholomorphic to some bounded domain? At a more modest level, must it possess even one nonconstant bounded holomorphic function?

Family 359 contains two independent constructions addressing different versions of these questions. M1 claims a complex three-dimensional example with curvature bounded between two finite negative constants, but no global bounded holomorphic coordinate triple with everywhere independent differentials. M2 claims an example in a sufficiently large fixed finite dimension with curvature at most $-1$, but with every bounded holomorphic function constant.

The hypotheses differ critically. M2’s curvature is unbounded below. M1 has two-sided pinching but retains nonconstant bounded functions. Neither paper settles the existence of a single nonconstant bounded holomorphic function under finite two-sided negative pinching.

2. Background and essential definitions

Holomorphic functions, coordinates and domains

A domain in $\mathbb C^m$ is a connected open set. A holomorphic function is complex differentiable in its variables; locally it has a convergent power series. It is bounded if its absolute value has a finite global upper bound.

A biholomorphism is a holomorphic bijection with holomorphic inverse. If $M$ is biholomorphic to a bounded domain in $\mathbb C^m$, the target’s coordinate functions pull back to bounded holomorphic functions $F_1,\ldots,F_m$ on $M$. Their complex Jacobian determinant is nowhere zero:

$$\det\left(\frac{\partial F_i}{\partial z_j}\right)\ne0.$$

A bounded holomorphic map with nowhere-zero Jacobian need not be globally injective, so even that condition is weaker than a biholomorphism. Thus a bounded-domain representation gives bounded holomorphic coordinates, which in turn supply a nonconstant bounded holomorphic function.

Reversing these implications requires extra information.

The notation $H^\infty(M)$ denotes the algebra of bounded holomorphic functions. The holomorphic Liouville property is $H^\infty(M)=\mathbb C$: every such function is constant. It concerns bounded functions, not all holomorphic functions.

Kähler metrics and curvature bounds

A Kähler metric is a Riemannian metric compatible with a complex structure whose associated two-form is closed. A common way to construct one is from a strictly plurisubharmonic potential $\Phi$, whose complex Hessian

$$g_{i\bar j}=\frac{\partial^2\Phi}{\partial z_i\partial\bar z_j}$$

is positive definite. A real-metric normalization must then be chosen; constant rescaling changes numerical curvature values.

The papers bound sectional curvature on every real two-plane, not just on complex lines. The latter, more restricted quantity is holomorphic sectional curvature. A two-sided negative bound means

$$-B\le K_g(\sigma)\le-A<0$$

for fixed finite $0<A\le B$, every point and every real tangent two-plane $\sigma$. A one-sided bound $K_g\le-1$ permits curvature to become arbitrarily negative.

Completeness means that geodesics can continue indefinitely, equivalently that escaping to infinity or to the domain boundary costs infinite metric length. A bounded Euclidean domain can carry a complete metric because that metric becomes large near its boundary.

3. Development history and previous partial results

H. Wu’s work on normal families asked whether complete simply connected Kähler manifolds with nonpositive real sectional curvature and uniformly negative holomorphic sectional curvature must be bounded domains in holomorphic coordinates. Wu and Yau later formulated the version with two-sided negative real-sectional pinching and bounded pseudoconvex conclusion.

Earlier examples showed that the unit ball was too narrow a target. Mostow and Siu constructed compact negatively curved Kähler surfaces not covered by the ball; Deraux obtained related three-dimensional examples. Their universal covers inherit finite negative curvature bounds. But failure to be the ball does not exclude being some other bounded domain. M1 targets that stronger obstruction.

The existence of just one bounded holomorphic function is a separate question, traced in the manuscripts to Yau’s 1982 Problem 38. Seshadri, using radial curvature calculations of Klembeck, exhibited complete strictly negatively curved Kähler metrics on $\mathbb C^n$. Liouville’s theorem makes all bounded entire functions constant there. Some sectional curvatures approach zero, so that example does not provide a uniform negative upper bound such as $-1$.

Greene and Wu, and later Wu and Yau, established strong comparisons between intrinsic complex metrics under two-sided pinching. These include the Bergman metric, built from square-integrable holomorphic top-degree forms, and the Kobayashi–Royden metric, built from holomorphic disks mapping into the manifold. Such comparisons do not automatically provide bounded functions mapping out to the disk.

Shcherbina and Zhang constructed a domain that is complete for Bergman and Kobayashi metrics yet has the holomorphic Liouville property. That supplies another warning against identifying completeness of intrinsic metrics with abundance of bounded functions. The additional issue addressed by M2 is real sectional curvature control.

A Cao–Shaw announcement concerning bounded functions under two-sided pinching was withdrawn in 2016. It is not an established input to either construction here.

4. What the new manuscripts claim

M1: two-sided pinching, but no bounded coordinate triple

M1 constructs a contractible domain $M\subset\mathbb C^3$ with a smooth complete Kähler metric satisfying finite uniform bounds

$$-B\le K_g\le-A<0.$$

Every bounded holomorphic map $F:M\to\mathbb C^3$ has a point where its Jacobian determinant vanishes. Consequently, $M$ is not biholomorphic to any bounded domain in $\mathbb C^3$.

The conclusion does not say that each such determinant vanishes identically. Nor does it say all bounded functions are constant: the construction has two nonconstant bounded base-coordinate functions.

The domain is also Stein, a standard analytic condition expressing strong holomorphic convexity and separation properties. The source obtains this from a smooth strictly plurisubharmonic exhaustion. Thus the obstruction concerns bounded global coordinates, not a failure of the basic analytic structure supplied by Stein theory.

M2: no bounded functions under a one-sided bound

M2 constructs, for some sufficiently large fixed finite integer $m$, a domain $T\subset\mathbb C^m$ diffeomorphic to $\mathbb R^{2m}$ and a complete Kähler metric with

$$K_g\le-1,\qquad H^\infty(T)=\mathbb C.$$

Its sectional curvatures are unbounded below. The dimension is chosen once for the entire construction; the theorem does not give the assertion in every dimension or a specific minimal dimension.

The two examples separate the following conclusions:

Construction	Curvature hypothesis achieved	Bounded holomorphic behavior
M1, complex dimension three	Finite two-sided negative pinching	No everywhere nonsingular bounded coordinate triple; two bounded nonconstant functions remain
M2, a fixed sufficiently large dimension	Uniform negative upper bound, no finite lower bound	Every bounded holomorphic function is constant

Their proofs are independent. The stronger function obstruction in M2 is not combined with M1’s stronger curvature hypothesis.

5. Why the advance matters

Importance: 5/5, conditional on the claims. The papers would give negative answers to central higher-dimensional uniformization and bounded-function questions, while identifying precisely which expectations fail.

M1 would show that even uniform negative curvature on all real planes, completeness and contractibility do not force a bounded-domain realization. M2 would show that a uniform negative upper curvature bound does not even guarantee one nonconstant bounded holomorphic function.

The contrast with one complex dimension is substantial. Several-variable holomorphic geometry can impose global compatibility constraints that strong local curvature inequalities do not resolve.

The examples also distinguish intrinsic metrics from bounded coordinate functions. M1 still falls within the two-sided-pinching comparison theory for several complete intrinsic metrics. In M2, the Carathéodory infinitesimal pseudometric—defined by pulling back the disk metric through bounded holomorphic maps—vanishes identically because every such map is constant. Different analytic constructions of metric geometry need not encode the same information.

6. The main idea

M1: vary disk radii over a bounded base

Let $B\subset\mathbb C^2$ be the unit ball. A smooth real function $\phi$ defines a Hartogs domain

$$M_\phi=\{(z,w)\in B\times\mathbb C:e^{\phi(z)}|w|^2<1\}.$$

The fiber above $z$ is a disk of radius $e^{-\phi(z)/2}$. The base coordinates are bounded, explaining the two surviving bounded functions.

With $\psi(z)=-\log(1-|z|^2)$, the metric comes from the potential

$$\Phi(z,w)=\lambda\psi(z)-\log\bigl(1-e^{\phi(z)}|w|^2\bigr).$$

The analytic task is to choose $\phi$ with controlled complex Hessian and derivatives through the order needed for curvature, while preventing any holomorphic function $H$ on $B$ from satisfying

$$\operatorname{Re}H(z)\le C_H+4\log\frac1{1-|z|}+\phi(z).$$

Why this particular obstruction? If a bounded map $F$ had nonvanishing Jacobian everywhere, restrict its Jacobian to the zero section $w=0$, obtaining a nowhere-zero holomorphic function $d(z)$. Cauchy estimates in base and fiber directions yield

$$\log|d(z)|^2\le C+4\log\frac1{1-|z|}+\phi(z).$$

Because the ball is simply connected, $d$ has a holomorphic logarithm. Its real part produces exactly the forbidden inequality.

The potential is built from separated homogeneous polynomial peaks. Positive representing measures preserve the value at the origin of every holomorphic real part, while selected regularized logarithms have increasingly negative averages. Those averages overwhelm the explicit boundary allowance. Representing holomorphic evaluation is weaker than satisfying every logarithmic mean inequality; the construction uses that distinction.

Meanwhile, widely separated polynomial degrees give derivative control in ball coordinates centered at arbitrary points. For a sufficiently large fixed $\lambda$, a full real-sectional curvature calculation gives strict negativity. Near limiting planes where a leading term degenerates, a cancellation in the largest error term is essential. Compact-range bounds and the limiting tensor near the fiber boundary then supply both finite pinching constants.

M2: large testing disks force derivatives to vanish

The second construction uses a tube above a potential $\ell$ on $\mathbb C^{n+1}$:

$$T=\{(s,y,w):\operatorname{Im}w>\ell(s,y)\},$$

$$\Phi=-\log\bigl(\operatorname{Im}w-\ell(s,y)\bigr).$$

Here $s$ has $n$ complex components, so the final dimension is $m=n+2$. A curvature-transfer lemma says that a complete nonpositively curved Kähler potential metric for $\ell$ gives a complete tube metric with real sectional curvature at most $-1$, in M2’s normalization.

The base must simultaneously permit specially placed large holomorphic disks in the tube. Cauchy’s estimate says that a bounded holomorphic function has small derivative along a sufficiently large disk. The difficulty is arranging enough disk directions and positions to force a global conclusion.

Polynomial systems supply finite grids of spatial points. Their affine lifts occur in root-of-unity cycles; averaging around a cycle separates Taylor degrees modulo the polynomial degree. Carefully selected phases prevent low-order derivative data from hiding among the higher-order contributions. Large-disk estimates then force decay of Taylor coefficients of $\partial f/\partial w$.

Each finite-order test is repeated infinitely often, with controlled heights. A three-lines argument ultimately gives $\partial f/\partial w=0$. The function descends to a bounded entire function on the base $\mathbb C^{n+1}$, so it is constant.

Preserve geometry while installing infinitely many tests

The disks require changing fiber centers and directions. A nonholomorphic change of center can damage curvature, so it cannot be dismissed as a coordinate change.

The manuscript uses a partial Legendre transform to organize the fiber metric and its curvature terms. It temporarily widens a fiber profile, moves the center while sufficient margin is available, and restores the profile with controlled offsets. Estimates on matrices of rank at most two handle all real two-planes while keeping key losses independent of dimension.

The recursive construction reserves bounds for later modifications, so each installed testing disk survives. Growth is capped between stages to prevent breakdown at finite spatial radius, and separate length estimates establish completeness. The potential becomes unbounded below; simple functions such as $e^{ikw}$ are therefore not bounded on the resulting tube.

7. Scope, limitations, and open questions

The principal unresolved distinction is explicit: these examples do not decide whether finite two-sided negative real-sectional pinching forces a nonconstant bounded holomorphic function. M1 has such functions, and M2 lacks the lower curvature bound.

M1 excludes bounded maps to $\mathbb C^3$ with nowhere-zero Jacobian; it does not say that every bounded map has rank deficiency everywhere. M2 concerns some sufficiently large fixed dimension, not every dimension beginning with two.

Both bounds concern real sectional curvature. Conclusions based only on holomorphic sectional curvature have different hypotheses. The papers also use different constant conventions for the real metric associated with a complex Hessian, so their numerical model intervals should not be identified without rescaling.

Questions about minimal dimensions, stronger pinching and the exact structure of bounded-function algebras remain separate. The curvature constructions and analytic obstructions are reported as stated, without proof verification.

8. Sources and further reading

M1 separates its controlled potential and bounded-Jacobian obstruction from the real-sectional pinching argument. M2 separates the large-disk testing criterion from its global geometric realization.

Primary background includes Wu–Yau’s intrinsic-metric comparisons, Seshadri’s strictly negative examples, Hwang–Singer’s momentum construction, and Shcherbina–Zhang’s complete domain with the Liouville property. The Cao–Shaw withdrawal record is included to distinguish an earlier announcement from an available theorem.

Source manuscripts

• M1. OpenAI, A negatively pinched Kähler threefold without bounded holomorphic coordinates (September 25, 2026).
• M2. OpenAI, One-sided negative sectional curvature and the holomorphic Liouville property (September 25, 2026).

Sources consulted on 26 October 2026 at repository revision 1581d4c. Download the manuscript citations.