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arxiv: 2604.11220 · v1 · submitted 2026-04-13 · 🧮 math.DG

Uniformisation of complete K\"ahler surfaces with positive sectional curvature

Ved Datar , Vamsi Pritham Pingali , Harish Seshadri This is my paper

Pith reviewed 2026-05-10 15:28 UTC · model grok-4.3

classification 🧮 math.DG
keywords Kähler surfacespositive sectional curvatureuniformisationcomplete metricsplurisubharmonic weightsMonge-Ampère massYau conjecture
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The pith

Any complete non-compact Kähler surface with positive sectional curvature is biholomorphic to ℂ².

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that every complete non-compact Kähler surface with strictly positive sectional curvature at every point must be biholomorphic to the complex Euclidean plane squared. This settles the two-dimensional case of a weaker form of Yau's uniformisation conjecture. The argument works without any hypotheses on the metric's behavior at infinity, which removes a restriction present in all earlier results. The method rests on constructing uniformly Lipschitz plurisubharmonic weight functions that carry finite Monge-Ampère mass and then using them to produce sufficiently many weighted L^p holomorphic functions.

Core claim

We prove that any complete non-compact Kähler surface with positive sectional curvature is biholomorphic to ℂ², establishing the two dimensional case of the weaker form of Yau's uniformisation conjecture. In contrast to all previous results, no assumptions are made on the geometry at infinity. The proof introduces a new approach towards Yau-type uniformisation problems, based on uniformly Lipschitz plurisubharmonic weight functions with finite Monge-Ampère mass, and weighted L^p holomorphic functions. As a consequence of the method, we also obtain Bézout-type intersection and multiplicity estimates in considerable generality. We also prove a new obstruction to the existence of complete Käler

What carries the argument

Uniformly Lipschitz plurisubharmonic weight functions with finite Monge-Ampère mass, together with the weighted L^p spaces of holomorphic functions they generate.

If this is right

  • The surface admits a global system of holomorphic coordinates that realises it as ℂ².
  • Bézout-type intersection and multiplicity estimates hold for holomorphic subvarieties in considerable generality.
  • There exists an obstruction preventing complete Kähler metrics with non-negative bisectional curvature on certain non-compact Kähler manifolds.
  • New examples of non-compact Kähler manifolds that carry no complete metric with non-negative bisectional curvature can be constructed.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The weight-function technique may extend to higher-dimensional Kähler manifolds with positive sectional curvature.
  • Similar weighted L^p constructions could produce multiplicity estimates for holomorphic maps on other classes of positively curved manifolds.
  • The method might be tested on explicit non-compact surfaces with positive curvature to see whether the biholomorphism conclusion persists under slightly weaker curvature hypotheses.

Load-bearing premise

The manifold is a complete non-compact Kähler surface whose sectional curvature is strictly positive at every point.

What would settle it

A single complete non-compact Kähler surface with positive sectional curvature that is not biholomorphic to ℂ², for instance one whose holomorphic function algebra differs from that of ℂ² or whose fundamental group is nontrivial.

read the original abstract

We prove that any complete non-compact K\"ahler surface with positive sectional curvature is biholomorphic to $\mathbb{C}^2$, establishing the two dimensional case of the weaker form of Yau's uniformisation conjecture. In contrast to all previous results, no assumptions are made on the geometry at infinity. The proof introduces a new approach towards Yau-type uniformisation problems, based on uniformly Lipschitz plurisubharmonic weight functions with finite Monge-Amp\`ere mass, and weighted $L^p$ holomorphic functions. A central difficulty is that these weights are neither smooth nor proper. As a consequence of the method, we also obtain B\'ezout-type intersection and multiplicity estimates in considerable generality. In a different direction, we also prove a new obstruction to the existence of complete K\"ahler metrics with non-negative bisectional curvature on non-compact K\"ahler manifolds, and use it to construct new examples admitting no such metrics. We conclude by discussing possible extensions of our methods to higher dimensions and related open problems.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript proves that any complete non-compact Kähler surface with strictly positive sectional curvature is biholomorphic to ℂ². This establishes the two-dimensional case of a weaker form of Yau's uniformisation conjecture without any assumptions on the geometry at infinity. The argument proceeds by constructing uniformly Lipschitz plurisubharmonic weight functions of finite Monge-Ampère mass, then using the resulting weighted L^p spaces of holomorphic functions to produce the biholomorphism. As byproducts, Bézout-type intersection and multiplicity estimates are obtained in considerable generality, and a new obstruction to the existence of complete Kähler metrics with non-negative bisectional curvature is derived, yielding new examples of manifolds admitting no such metrics. Possible extensions to higher dimensions are discussed.

Significance. If the central claim holds, the result is a substantial contribution to complex differential geometry: it resolves an important open case of Yau's conjecture under minimal hypotheses and introduces a new analytic framework based on non-smooth, non-proper weights that may extend beyond dimension two. The derivation of Bézout-type estimates and the curvature obstruction further increase the paper's impact. The absence of decay or properness assumptions at infinity distinguishes the work from earlier results and strengthens its applicability.

minor comments (3)
  1. In the introduction, the comparison with prior uniformisation results would benefit from a short table or explicit list of the geometric assumptions (e.g., curvature decay, volume growth) that are removed in the present work.
  2. The definition of 'uniformly Lipschitz' for the plurisubharmonic weights is given in §2; a single displayed inequality summarizing the Lipschitz constant and the finite-mass condition would improve readability for readers outside the immediate area.
  3. In the section deriving the Bézout-type estimates, the statement of the multiplicity bound could be accompanied by a brief remark on how the finite Monge-Ampère mass replaces the usual properness hypothesis.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript and for recommending acceptance. No major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity; direct analytic proof self-contained

full rationale

The paper establishes its central theorem via a new construction of uniformly Lipschitz plurisubharmonic weights with finite Monge-Ampère mass, followed by weighted L^p holomorphic function estimates and Bézout-type intersection bounds. These steps are derived from the given curvature and completeness assumptions without any reduction of the biholomorphism conclusion to a fitted parameter, self-referential definition, or load-bearing self-citation chain. The argument explicitly handles non-smooth, non-proper weights and derives the result as a consequence of the estimates, remaining independent of the target statement by construction. No equations or steps in the provided abstract or description collapse the claim to its inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on the standard definition of Kähler metrics and sectional curvature together with the new analytic construction of Lipschitz plurisubharmonic weights; no free parameters or invented geometric entities are introduced in the abstract.

axioms (2)
  • standard math A Kähler surface is a complex manifold of complex dimension two equipped with a compatible Kähler metric.
    Invoked as the ambient category for the uniformization statement.
  • domain assumption Positive sectional curvature on a Kähler manifold implies positivity properties of the curvature tensor that can be exploited analytically.
    Used to guarantee the existence of the required weight functions.

pith-pipeline@v0.9.0 · 5488 in / 1243 out tokens · 33914 ms · 2026-05-10T15:28:13.095690+00:00 · methodology

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