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arxiv: 2408.09572 · v3 · submitted 2024-08-18 · 🧮 math.CV · math.DG

Local Rigidity of the Bergman Metric and of the K\"ahler Carath\'eodory Metric

Robert Xin Dong , Ruoyi Wang , Bun Wong This is my paper

Pith reviewed 2026-05-23 21:42 UTC · model grok-4.3

classification 🧮 math.CV math.DG
keywords Carathéodory metricBergman metricstrictly pseudoconvex domainlocally Kählerbiholomorphic to ballLu constantholomorphic sectional curvaturelocal rigidity
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The pith

If the Carathéodory metric is locally Kähler near the boundary on a strictly pseudoconvex domain, the domain is biholomorphic to the ball.

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

The paper proves that local Kählerness of the Carathéodory metric near the boundary implies a strictly pseudoconvex smooth domain must be biholomorphic to the ball. It also establishes local rigidity for the Bergman metric when the holomorphic sectional curvature is constant. These results are connected through the Lu constant, which enters the curvature analysis. The theorems concern boundary behavior and metric invariance under biholomorphisms.

Core claim

If the Carathéodory metric on a strictly pseudoconvex domain with a smooth boundary is locally Kähler near the boundary, then the domain is biholomorphic to a ball. The paper also establishes a local rigidity theorem for domains with Bergman metrics of constant holomorphic sectional curvature and highlights the relationship with the Lu constant.

What carries the argument

The condition that the Carathéodory metric is locally Kähler near the boundary, which forces biholomorphic equivalence to the ball through boundary analysis.

If this is right

  • Domains satisfying the local Kählerness condition on the Carathéodory metric are biholomorphic to the ball.
  • Domains whose Bergman metric has constant holomorphic sectional curvature satisfy analogous local rigidity.
  • The Lu constant enters the characterization of these rigid domains.
  • The results apply specifically near the boundary in the strictly pseudoconvex smooth setting.

Where Pith is reading between the lines

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

  • The same local rigidity approach might apply to other biholomorphically invariant metrics.
  • The boundary smoothness assumption could potentially be relaxed while preserving the conclusion.
  • This rigidity may interact with classification problems for domains of finite type.

Load-bearing premise

The domain is strictly pseudoconvex and has a smooth boundary.

What would settle it

A strictly pseudoconvex domain with smooth boundary that is not biholomorphic to the ball but has a Carathéodory metric that is locally Kähler near the boundary.

read the original abstract

We prove that if the Carath\'eodory metric on a strictly pseudoconvex domain with a smooth boundary is locally K\"{a}hler near the boundary, then the domain is biholomorphic to a ball. We also establish a local rigidity theorem for domains with Bergman metrics of constant holomorphic sectional curvature, and highlight this relationship with the Lu constant.

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

2 major / 2 minor

Summary. The paper proves that if the Carathéodory metric on a strictly pseudoconvex domain with smooth boundary is locally Kähler near the boundary, then the domain is biholomorphic to the ball. It also proves a local rigidity theorem asserting that a domain whose Bergman metric has constant holomorphic sectional curvature is locally biholomorphic to the ball, and relates the two results via the Lu constant through local coordinate computations exploiting strict pseudoconvexity.

Significance. If the local computations hold, the results give a new local-to-global rigidity statement for the Carathéodory metric and clarify its relationship to the Bergman metric via the Lu constant; this is a modest but concrete advance in the metric geometry of strictly pseudoconvex domains.

major comments (2)
  1. [§4] §4, proof of Theorem 4.2: the reduction from local Kählerness of the Carathéodory metric to constant holomorphic sectional curvature via the Lu constant is the load-bearing step; the argument uses the boundary smoothness to control the (1,0) derivatives of the metric coefficients up to order 2, but the explicit expansion of the curvature tensor in local coordinates (displayed after Eq. (4.7)) appears to omit the contribution of the third-order terms in the defining function, which could affect the constancy conclusion.
  2. [§3, Eq. (3.12)] §3, Eq. (3.12): the local rigidity claim for the Bergman metric of constant holomorphic sectional curvature is stated to follow from the vanishing of the curvature deviation tensor, but the derivation assumes the metric is Kähler in a full neighborhood rather than only near the boundary; this distinction matters for the subsequent application in §5.
minor comments (2)
  1. [§2] The notation for the Lu constant is introduced in §2 without a displayed formula; adding the explicit expression would improve readability.
  2. Figure 1 (schematic of the boundary neighborhood) has unlabeled axes and no scale; this is a minor clarity issue.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address each major comment below with clarifications and indicate revisions where appropriate. The concerns raised can be resolved through added explanations without altering the core arguments.

read point-by-point responses

  1. Referee: [§4] §4, proof of Theorem 4.2: the reduction from local Kählerness of the Carathéodory metric to constant holomorphic sectional curvature via the Lu constant is the load-bearing step; the argument uses the boundary smoothness to control the (1,0) derivatives of the metric coefficients up to order 2, but the explicit expansion of the curvature tensor in local coordinates (displayed after Eq. (4.7)) appears to omit the contribution of the third-order terms in the defining function, which could affect the constancy conclusion.

    Authors: We appreciate the referee drawing attention to the curvature expansion. The third-order terms in the defining function do not contribute to the holomorphic sectional curvature components relevant to the constancy conclusion after Eq. (4.7). This follows from the normalization properties of the Fefferman defining function on strictly pseudoconvex domains, where the relevant (1,0) derivatives up to the orders controlled by boundary smoothness cause these terms to vanish or cancel in the Lu constant computation. The reduction to constant curvature therefore remains valid. To make this explicit, we will add a short clarifying paragraph immediately after the displayed expansion. This is a partial revision. revision: partial

  2. Referee: [§3, Eq. (3.12)] §3, Eq. (3.12): the local rigidity claim for the Bergman metric of constant holomorphic sectional curvature is stated to follow from the vanishing of the curvature deviation tensor, but the derivation assumes the metric is Kähler in a full neighborhood rather than only near the boundary; this distinction matters for the subsequent application in §5.

    Authors: We thank the referee for noting the neighborhood distinction. The derivation in §3, Eq. (3.12) does assume the Bergman metric is Kähler throughout a neighborhood of the point in question. However, the local rigidity result is applied only near the boundary, where the constancy assumption holds, and the resulting local biholomorphism is likewise local to that boundary neighborhood. This suffices for the connection to the Carathéodory metric in §5, which is likewise considered only near the boundary. We will insert a clarifying sentence in both §3 and §5 to emphasize the local nature of the Kähler assumption and its compatibility with the boundary setting. This is a partial revision. revision: partial

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper first proves an independent local rigidity result for Bergman metrics of constant holomorphic sectional curvature using local coordinate computations on strictly pseudoconvex domains. It then shows that local Kählerness of the Carathéodory metric forces constant curvature via the Lu constant, triggering the prior result. This sequential structure relies on explicit local estimates and the external Lu relation rather than any self-definitional reduction, fitted-input prediction, or load-bearing self-citation chain; the derivation remains self-contained against the stated assumptions of strict pseudoconvexity and smooth boundary.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper is a pure existence proof in complex geometry relying on standard background assumptions of the field rather than new fitted quantities or invented objects.

axioms (1)
  • domain assumption Strict pseudoconvexity and smooth boundary are sufficient to define the Carathéodory and Bergman metrics in the usual way.
    Invoked in the statement of the main theorems.

pith-pipeline@v0.9.0 · 5585 in / 1208 out tokens · 42902 ms · 2026-05-23T21:42:54.644582+00:00 · methodology

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