Maximal Normal Curvature and Veronese Rigidity

Jingbo Wan; Tsz-Kiu Aaron Chow

arxiv: 2607.00949 · v1 · pith:UUWZIUNAnew · submitted 2026-07-01 · 🧮 math.DG

Maximal Normal Curvature and Veronese Rigidity

Tsz-Kiu Aaron Chow , Jingbo Wan This is my paper

Pith reviewed 2026-07-02 06:16 UTC · model grok-4.3

classification 🧮 math.DG

keywords Veronese rigiditymaximal normal curvaturealmost Hermitian manifoldharmonic formsquaternion-Hermitiansubmanifold immersionBochner curvatureshape operator

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The pith

Almost Hermitian or quaternion-Hermitian submanifolds in the unit ball satisfy κ(F) ≥ √(2n/(n+1)) with equality only for Veronese embeddings.

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

The paper proves a lower bound on maximal normal curvature for closed isometric immersions of almost Hermitian manifolds whose fundamental two-form is harmonic, or almost quaternion-Hermitian manifolds whose fundamental four-form is harmonic. The bound is √(2n/(n+1)) for n at least 2 and is achieved exactly when the immersion is a Veronese embedding of a projective space, up to a totally geodesic inclusion. A reader would care because the result converts an intrinsic harmonicity condition into a sharp extrinsic curvature estimate inside the Euclidean ball. The argument rests on a Bochner-Gauss mechanism that extracts an algebraic inequality for the shape operator directly from the curvature term of the harmonic form.

Core claim

If Σ^{2n} is almost Hermitian with harmonic fundamental two-form, or Σ^{4n} is almost quaternion-Hermitian with harmonic fundamental four-form, n≥2, then for an isometric immersion F into the closed unit ball the maximal normal curvature satisfies κ(F) ≥ √(2n/(n+1)). Equality holds only when the harmonic form is parallel and the immersion is, up to totally geodesic inclusion, the standard complex or quaternionic Veronese embedding of the corresponding projective space. The proof proceeds by turning the Bochner curvature term of the harmonic form into a sharp algebraic lower bound on the second fundamental form via a Gauss-type identity.

What carries the argument

Bochner-Gauss mechanism that converts the Bochner curvature term of the harmonic form into a sharp algebraic estimate for the shape operators.

If this is right

Equality forces the harmonic form to be parallel.
The immersion must coincide with a Veronese embedding up to totally geodesic inclusion in the ball.
The same lower bound and rigidity statement hold in both the complex and quaternionic settings for n≥2.
The bound depends only on dimension and the harmonicity assumption, not on further details of the metric.

Where Pith is reading between the lines

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

The same mechanism might produce analogous lower bounds when the ambient space is a sphere or hyperbolic space instead of the ball.
It could be tested whether the inequality persists if the harmonicity assumption is weakened to closedness of the form.
Explicit non-Veronese examples with strictly larger curvature would confirm that the constant is not achieved elsewhere.

Load-bearing premise

The Bochner-Gauss mechanism produces the claimed sharp constant from the Bochner curvature term under the stated harmonicity assumptions.

What would settle it

An explicit isometric immersion of an almost Hermitian manifold with harmonic fundamental two-form into the unit ball whose maximal normal curvature is strictly smaller than √(2n/(n+1)).

read the original abstract

We prove a sharp Veronese rigidity theorem for closed immersed submanifolds of the Euclidean unit ball under intrinsic harmonic-structure assumptions. For an isometric immersion $F:(\Sigma,g)\looparrowright\overline B(1)$, define the maximal normal curvature by \[ \kappa(F):= \sup_{x\in\Sigma} \sup_{\substack{v\in T_x\Sigma\\ |v|_g=1}} |A_x(v,v)|. \] If $\Sigma^{2n}$ is almost Hermitian with harmonic fundamental two-form, or $\Sigma^{4n}$ is almost quaternion-Hermitian with harmonic fundamental four-form, $n\ge2$, then \[ \kappa(F)\ge \sqrt{\frac{2n}{n+1}} . \] In the equality case the harmonic form is parallel and the immersion is, up to a totally geodesic inclusion, the standard complex or quaternionic Veronese embedding of projective spaces. The key input is a Bochner--Gauss mechanism that turns the Bochner curvature term of the harmonic form into a sharp algebraic estimate for the shape operators.

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.

Desk Editor's Note private letter to a colleague

The paper gives a sharp lower bound on maximal normal curvature for immersed almost Hermitian or quaternion-Hermitian manifolds with harmonic fundamental forms, with equality only for Veronese cases.

read the letter

The paper proves that for a closed isometric immersion of an almost Hermitian 2n-manifold with harmonic fundamental two-form (or the quaternion-Hermitian 4n case) into the unit ball, the maximal normal curvature is at least sqrt(2n/(n+1)). Equality forces the form to be parallel and the immersion to be a standard Veronese embedding of projective space, up to totally geodesic inclusion.

The new piece is the direct link between the harmonicity assumption and this specific curvature quantity through a Bochner-Gauss mechanism that produces the sharp constant algebraically from the curvature term. The statement is precise, the equality case is fully characterized, and the setup stays within standard submanifold geometry without extra fitting or self-referential bounds.

The argument looks internally consistent on the terms given. The mechanism converts the Bochner term into an estimate on the shape operators, and the abstract indicates it delivers exactly the claimed constant under the stated harmonicity. No circularity or hidden reduction shows up.

The main soft spot is that the algebraic details of that conversion step are the load-bearing part; if they slip, the sharpness fails. The result sits in a narrow corner of differential geometry, so its reach is limited to people already working on rigidity for Hermitian structures or Veronese-type embeddings.

I would bring this to a reading group only if the group already covers submanifold rigidity. It is worth sending to peer review because the claim is concrete, the hypotheses are clear, and the equality case gives something falsifiable to check.

Referee Report

1 major / 0 minor

Summary. The paper proves a sharp Veronese rigidity theorem: for an isometric immersion F of a closed manifold Σ into the unit ball, if Σ^{2n} (n≥2) carries an almost Hermitian structure with harmonic fundamental 2-form, or Σ^{4n} carries an almost quaternion-Hermitian structure with harmonic fundamental 4-form, then the maximal normal curvature satisfies κ(F) ≥ √(2n/(n+1)). Equality holds if and only if the harmonic form is parallel and, up to totally geodesic inclusion, F is the standard complex or quaternionic Veronese embedding of projective space. The proof relies on a Bochner-Gauss mechanism that converts the Bochner curvature term of the harmonic form into a sharp algebraic lower bound on the second fundamental form.

Significance. If the Bochner-Gauss conversion is valid, the result supplies a new, sharp extrinsic curvature bound controlled by intrinsic harmonic conditions on almost Hermitian or quaternion-Hermitian structures. The equality case recovers the classical Veronese embeddings, furnishing a rigidity statement that may be useful for classification problems in submanifold geometry. The algebraic sharpness of the constant is a positive feature when the mechanism is correctly executed.

major comments (1)

[Bochner-Gauss mechanism (abstract and §3)] The central inequality rests entirely on the Bochner-Gauss mechanism (abstract, final sentence) that converts the curvature term of the harmonic form into the claimed algebraic estimate for the shape operators. Without an explicit verification that this conversion produces exactly the constant √(2n/(n+1)) under the stated harmonicity hypotheses, the lower bound does not follow; a detailed expansion of the relevant curvature identities and the resulting quadratic form on the second fundamental form is required.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the need for greater transparency in the central argument. We address the major comment below and will incorporate the requested expansion in the revised manuscript.

read point-by-point responses

Referee: [Bochner-Gauss mechanism (abstract and §3)] The central inequality rests entirely on the Bochner-Gauss mechanism (abstract, final sentence) that converts the curvature term of the harmonic form into the claimed algebraic estimate for the shape operators. Without an explicit verification that this conversion produces exactly the constant √(2n/(n+1)) under the stated harmonicity hypotheses, the lower bound does not follow; a detailed expansion of the relevant curvature identities and the resulting quadratic form on the second fundamental form is required.

Authors: We agree that the current presentation of the Bochner-Gauss mechanism in §3 would benefit from a more explicit step-by-step expansion. In the revised version we will insert a dedicated subsection that (i) recalls the Bochner formula for the harmonic 2-form (resp. 4-form) under the almost Hermitian (resp. quaternion-Hermitian) structure, (ii) substitutes the Gauss equation to express the curvature term in terms of the second fundamental form A, (iii) performs the algebraic minimization of the resulting quadratic form on the shape operators, and (iv) verifies that the minimum is attained precisely at √(2n/(n+1)) when the harmonicity condition is used. This will make the derivation of the constant fully self-contained. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation rests on a Bochner-Gauss mechanism presented as the key external input that algebraically converts the Bochner curvature term of the harmonic form into a sharp bound on maximal normal curvature. No self-definitional reductions, fitted inputs renamed as predictions, or load-bearing self-citations appear in the abstract or stated assumptions. The claimed inequality follows from this mechanism under the harmonicity hypotheses without the target bound being presupposed by definition or prior self-referential results.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on the standard setup of isometric immersions of Riemannian manifolds into Euclidean space, the existence of an almost Hermitian or quaternion-Hermitian structure whose fundamental form is harmonic, and the validity of the Bochner formula and Gauss equation in this context. No free parameters or invented entities are introduced in the abstract.

axioms (2)

standard math Isometric immersion F: (Σ,g) → closed unit ball in Euclidean space
The definition of κ(F) and the ambient space are taken from classical Riemannian geometry.
domain assumption Existence of almost Hermitian (or quaternion-Hermitian) structure with harmonic fundamental form
The theorem is conditioned on these intrinsic structures; they are not derived.

pith-pipeline@v0.9.1-grok · 5723 in / 1529 out tokens · 36294 ms · 2026-07-02T06:16:51.915711+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

14 extracted references · 5 canonical work pages · 1 internal anchor

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work page arXiv 2024
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A. F. Swann, HyperKähler and quaternionic Kähler geometry,Math. Ann.289(1991), no. 3, 421–450. ↑3 Department of Mathematics, Hong Kong University of Science and Technology, Hong Kong S.A.R., China Email address:chowtka@ust.hk Laboratoire Jacques-Louis Lions de Sorbonne Université, 4 place Jussieu, Paris 75005, France Email address:jingbo.wan@sorbonne-univ...

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[1] [1]

D. V. Alekseevskii, Riemannian spaces with exceptional holonomy groups,Funct. Anal. Appl.2(1968), 97–105.↑22

1968

[2] [2]

Ambrose and I

W. Ambrose and I. M. Singer, A theorem on holonomy,Trans. Amer. Math. Soc.75(1953), 428–443. ↑26

1953

[3] [3]

Chodosh and C

O. Chodosh and C. Li,Immersions with small normal curvature, arXiv:2602.15728, 2026.↑2

work page arXiv 2026

[4] [4]

T.-K. A. Chow and J. Wan,Sharp focal radius estimate and rigidity of hypersurfaces in manifolds with positive curvature, arXiv:2606.02829, 2026.↑5

work page internal anchor Pith review Pith/arXiv arXiv 2026

[5] [5]

Gromov,Isometric immersions with controlled curvatures, arXiv:2212.06122, 2022.↑2

M. Gromov,Isometric immersions with controlled curvatures, arXiv:2212.06122, 2022.↑2

work page arXiv 2022

[6] [6]

Helgason,Differential Geometry, Lie Groups, and Symmetric Spaces, Academic Press, New York, 1978.↑26

S. Helgason,Differential Geometry, Lie Groups, and Symmetric Spaces, Academic Press, New York, 1978.↑26

1978

[7] [7]

Ishihara, Quaternion Kählerian manifolds,J

S. Ishihara, Quaternion Kählerian manifolds,J. Differential Geom.9(1974), 483–500.↑22

1974

[8] [8]

Nash, The imbedding problem for Riemannian manifolds,Ann

J. Nash, The imbedding problem for Riemannian manifolds,Ann. of Math.63(1956), no. 1, 20–63.↑ 3 and 4

1956

[9] [9]

Nomizu, Conditions for constancy of the holomorphic sectional curvature,J

K. Nomizu, Conditions for constancy of the holomorphic sectional curvature,J. Differential Geom.8 (1973), 335–339.↑22

1973

[10] [10]

Petrunin,Gromov’s tori are optimal,Geom

A. Petrunin,Gromov’s tori are optimal,Geom. Funct. Anal.34(2024), no. 1, 202–208; arXiv:2304.00886.↑2

work page arXiv 2024

[11] [11]

Petrunin,Veronese minimizes normal curvatures, arXiv:2408.05909, 2025.↑2

A. Petrunin,Veronese minimizes normal curvatures, arXiv:2408.05909, 2025.↑2

work page arXiv 2025

[12] [12]

Rudin,Function Theory in the Unit Ball ofCn, Grundlehren der mathematischen Wissenschaften, vol

W. Rudin,Function Theory in the Unit Ball ofCn, Grundlehren der mathematischen Wissenschaften, vol. 241, Springer, New York, 1980.↑11

1980

[13] [13]

Savo, The Bochner formula for isometric immersions,Pacific J

A. Savo, The Bochner formula for isometric immersions,Pacific J. Math.272(2014), no. 2, 395–422.↑ 5

2014

[14] [14]

A. F. Swann, HyperKähler and quaternionic Kähler geometry,Math. Ann.289(1991), no. 3, 421–450. ↑3 Department of Mathematics, Hong Kong University of Science and Technology, Hong Kong S.A.R., China Email address:chowtka@ust.hk Laboratoire Jacques-Louis Lions de Sorbonne Université, 4 place Jussieu, Paris 75005, France Email address:jingbo.wan@sorbonne-univ...

1991