Negative Kähler curvature without bounded holomorphic coordinates

1 · The essentials

Does uniformly negative curvature guarantee that a complex space can be described inside a bounded region without losing its complex structure? The main manuscript gives a counterexample: a region in three complex dimensions that can be continuously shrunk to a point. It carries a complete geometry, meaning that travel cannot reach a missing edge in finite distance. Its curvature stays between two fixed, finite negative values everywhere. Yet no bounded complex-analytic map to three complex coordinates has a nonzero Jacobian everywhere, so the region cannot be identified with a bounded region while preserving its complex structure. The main result has a Lean formalization. A companion manuscript describes a higher-dimensional example where every bounded complex-analytic function is constant, but its curvature has no finite lower bound.

A picture to help (an imperfect image, not to be taken literally)

As an imperfect image, imagine trying to redraw a large landscape inside a small map while preserving a strict rule for valid coordinates. The main example says that even strong, uniform restrictions on the landscape’s geometry do not guarantee such a redraw; this image does not capture the complex-analytic rules involved.

Why it matters

The main result gives a negative answer to the negatively pinched Kähler uniformization question: the stated curvature and completeness assumptions do not force a bounded-domain description. The companion manuscript separates a related issue, showing in the manuscript that a one-sided negative curvature bound does not guarantee nonconstant bounded holomorphic functions. The distinction between two-sided and one-sided curvature bounds is essential to interpreting the two results.

2 · Going further

The main problem asks whether a contractible domain with a complete, negatively pinched Kähler metric must be biholomorphic to a bounded domain. Here negative pinching means that its real sectional curvature $K$ satisfies $a \leq K \leq b < 0$ for two finite constants $a$ and $b$ . Manuscript 1 constructs a domain $D \subset C^{3}$ satisfying these geometric conditions but admitting no bounded holomorphic map $F : D \to C^{3}$ with nowhere-vanishing Jacobian.

The obstruction is stronger than merely ruling out a particular choice of global coordinates. A biholomorphism from $D$ onto a bounded domain would itself be a bounded holomorphic map with nowhere-vanishing Jacobian, exactly what the construction excludes. Thus the example supplies the negative answer stated in the abstract. The supplied abstract describes the construction through its properties and this obstruction, but does not give the construction’s technical mechanism. The main result has a Lean formalization.

Manuscript 2 addresses a related one-sided question. In some sufficiently large fixed finite complex dimension $m$ , it constructs a domain in $C^{m}$ diffeomorphic to $R^{2 m}$ , equipped with a complete Kähler metric satisfying $K \leq - 1$ , on which every bounded holomorphic function is constant. This is shown in the manuscript, rather than identified here as Lean-formalized. Its sectional curvatures are unbounded below, so this stronger restriction on bounded functions does not supply the same two-sided curvature example as Manuscript 1.

Useful notions

Domains in complex coordinate space
Holomorphic functions and maps
Complete metrics and sectional curvature
Kähler metrics
Jacobians and changes of coordinates

What was known before / what is new

Before

The source frames the main result as answering the negatively pinched Kähler uniformization question. It also identifies a related one-sided bounded-holomorphic-function question. No earlier partial results or historical development are supplied.

New

The main construction combines contractibility, complex dimension three, completeness, and two finite negative curvature bounds with the absence of bounded holomorphic coordinates. It excludes every bounded holomorphic map to $C^{3}$ whose Jacobian is nonzero everywhere. The companion manuscript gives a different construction with only constant bounded holomorphic functions under a one-sided curvature bound, in a sufficiently large fixed finite dimension.

Limits and assumptions

The main example is in complex dimension three and excludes bounded holomorphic maps with nowhere-vanishing Jacobian; the source does not say that all its bounded holomorphic functions are constant. The companion example has that stronger function property, but its dimension is not specified beyond being sufficiently large and finite, and its curvature is unbounded below. The main result has a Lean formalization; no formalization is stated for the companion result, and the supplied material gives no finer breakdown of formalization coverage. The source does not state what further cases remain open.

Mini glossary

Kähler metric: A metric on a complex space whose associated differential two-form is closed, expressing compatibility between its geometry and complex structure.
Negatively pinched sectional curvature: Curvature measured along tangent two-dimensional planes that stays between two finite, strictly negative constants.
Biholomorphic: Related by an invertible holomorphic map whose inverse is also holomorphic.
Nowhere-vanishing Jacobian: A map’s complex derivative has nonzero determinant at every point, so it can serve locally as a change of complex coordinates.
Contractible: Able to be continuously shrunk to a point within the space.
Holomorphic Liouville property: The property that every bounded holomorphic function on the space is constant.

Keywords : Kähler curvature · Bounded holomorphic coordinates · Uniformization

AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.

3 · Technical details and sources

Claim made in the manuscript (source text)

Constructs a contractible domain in ℂ³ with a complete negatively pinched Kähler metric but no bounded holomorphic coordinates, disproving bounded-domain uniformization in this setting. A higher-dimensional example has sectional curvature at most −1 and only constant bounded holomorphic functions; its curvature is not bounded below.

Verified by Lean: the Lean software checked the part of the reasoning described below, sometimes only part of the manuscript. This does not mean independent mathematicians have validated the result.

Scope of the Lean formalization (source text)

…

lean/docs/359.md

Manuscripts in this family

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A negatively pinched Kähler threefold without bounded holomorphic coordinates

25/09/2026

Verified by Lean

We construct a contractible domain in complex dimension three with a complete Kähler metric whose real sectional curvatures lie between two finite negative constants. It admits no bounded holomorphic map to ℂ³ with nowhere-vanishing Jacobian and is therefore not biholomorphic to a bounded domain. This gives a negative answer to the negatively pinched Kähler uniformization question.

PDF GitHub folder Lean formalization

One-sided negative sectional curvature and the holomorphic Liouville property

25/09/2026

To be verified

We construct, in some sufficiently large fixed finite complex dimension m, a domain in ℂ^m diffeomorphic to $R^{2 m}$ that admits a complete Kähler metric with real sectional curvature at most −1 and has only constant bounded holomorphic functions. Its sectional curvatures are unbounded below. The construction gives a negative answer to the one-sided bounded-holomorphic-function question.

PDF GitHub folder