Optimal bounds for the Kobayashi distance near $\mathcal C^2$-smooth boundary points

Nikolai Nikolov; Pascal J. Thomas

REVIEW 2 major objections 4 minor 13 references

Optimal bounds for the Kobayashi distance near $\mathcal C^2$-smooth boundary points

T0 review · 2 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper establishes that near every strongly pseudoconvex C^2-smooth boundary point, log(1+cA_D) ≤ k_D ≤ l_D ≤ log(1+CA_D) for some constants 0<c<C, with A_D an explicit Euclidean quantity.

desk verdict A clean regularity improvement plus genuinely new non-semipositive estimates; the main gap is an unstated regularity hypothesis in a cited comparison lemma, fixable without changing the conclusions. read the letter →

arxiv 2506.06507 v1 pith:V35ACWL6 submitted 2025-06-06 math.CV

Nikolai Nikolov , Pascal J. Thomas This is my paper

classification math.CV MSC 32F4532T15

keywords KobayashidistancestrongpseudoconvexityboundaryestimatesC^2-smoothLempertfunctionnon-semipositivepointsinvariantmetricsdistance-to-boundary

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper's central claim is that the optimal two-sided estimates for the Kobayashi distance near strongly pseudoconvex boundary points, which were previously known only under a Hölder condition on second derivatives, hold when the boundary is merely $C^{2}$-smooth. The estimates compare the distance k_D to the explicit Euclidean quantity A_D(z,w) defined by equation (2), and the upper bound is extended further to $C^{{1,1}}$-smooth boundary points. The same methods give two-sided estimates near non-semipositive boundary points, where a single negative eigenvalue of the Levi form changes the growth order of the distance. A reader should care because invariant distances are almost never computable, and sharp local estimates are what determine whether domains behave like hyperbolic metric spaces near their boundaries.

What carries the argument

The carrying object is the Euclidean scale A_D(z,w)=B_D(z,w)/(√δ_D(z)δ_D(w)), with B_D(z,w)=|(z−w)_z|+|z−w|^2+|z−w|√δ_D(z); the main argument shows that the Kobayashi distance and the Lempert function stay within multiplicative constants of log(1+A_D). Three mechanisms do the work: a biholomorphic reduction to a strongly convex domain, where Lempert's theorem makes k_D, l_D, and the Carathéodory distance coincide; tangent Euclidean balls E_{q,1} contained in the domain, whose explicit Kobayashi geometry gives both upper and lower bounds; and the signed-distance second-order expansion (Proposition 13, quoted from [12]) that controls how δ_D(z)−δ_D(w) changes along the segment from z to w. For non-semipositive points the comparable object is H_D(z,w)=|(z−w)_z|/(√(|(z−w)_z|+|z−w|)+√δ_D(z))+|z−w|.

What would settle it

Take a $C^{2}$ strongly pseudoconvex domain whose boundary contains a point where the boundary is $C^{2}$ but not $C^{{2,α}}$ (for example a $C^{2}$ bump superposed on the unit sphere), and compute the ratio k_D(z,w)/log(1+A_D(z,w)) for sequences z,w approaching that point, one with z−w tangential and one normal; Theorem 5 predicts this ratio stays within a fixed interval [c,C], so a ratio escaping that interval—tending to 0 or ∞ along one of the sequences—would disprove the claim.

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Extended reading notes

Core claim

Set δ_D(z) = dist(z,∂D) and let (z−w)_z be the component of z−w along the complex normal at the closest boundary point. Define B_D(z,w) = |(z−w)_z| + |z−w|^2 + |z−w|√δ_D(z) and A_D(z,w) = B_D(z,w)/(√(δ_D(z)δ_D(w))). The paper proves that if p∈∂D is strongly pseudoconvex and ∂D is $C^{2}$ near p, then there exist 0<c<C such that log(1+cA_D(z,w)) ≤ k_D(z,w) ≤ l_D(z,w) ≤ log(1+CA_D(z,w)) for all z,w close enough to p. The lower bound is proved by a comparison with two tangent balls and the upper bound follows from a more general $C^{{1,1}}$ statement (Theorem 6) in which the quantity A_{D,p} includes the boundary-normal Hölder ratio χ_{D,p}. For non-semipositive points Theorem 9 gives k_D(z,w) ≍ H_D(z,w) with H_D defined in (8), so the distance is comparable to a much smaller explicit Euclidean expression.

Load-bearing premise

The proof leans on a second-order expansion of the signed distance function (quoted from [12]) that must hold at every boundary point and direction used; if that expansion fails at merely $C^{2}$ points, the upper bounds and the comparability H_D ≍ H^r_D no longer have a basis in this proof.

Editorial extensions

If this is right

At every strongly pseudoconvex C^2-smooth boundary point, the sharp two-sided estimate log(1+cA_D) ≤ k_D ≤ l_D ≤ log(1+CA_D) holds; the C^{2,α} smoothness required in previous work is unnecessary.
The upper bound extends to C^{1,1}-smooth boundary points: k_D(z,w) ≤ log(1+C_0 A_{D,p}(z,w)) whenever the normal-direction ratio χ_{D,p} is positive, and any ε>0 may replace χ_{D,p} when it is zero.
A global version follows for any strongly pseudoconvex domain: log(1+cA_D) ≤ c_D ≤ k_D ≤ l_D ≤ log(1+CA_D).
Near a non-semipositive boundary point, the Kobayashi distance is comparable to H_D(z,w); in particular k_D(z,w) ≲ √(|(z−w)_z|+|z−w|), and k_D(z,w) ≳ |z−w| + |√δ_D(z)−√δ_D(w)|.
The same log(1+A_D) scale governs the integrated distances of a broad class of Finsler metrics with integrable perturbations of the model metric, as stated in Propositions 18 and 19.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Beyond the paper, the tangent-ball proofs suggest the constants c and C depend only on the C^2 modulus of the boundary; extracting them explicitly is not attempted here.
An extension not claimed by the authors is that the same upper-bound argument should work for Dini-smooth strongly pseudoconvex boundaries if the signed-distance expansion is replaced by a Dini-type second-order condition.
The non-semipositive comparability k_D ≍ H_D points toward a testable connection with visibility: a single negative Levi eigenvalue changes the metric's growth order, so visibility in such domains should exhibit a different quantitative behaviour than in the strongly pseudoconvex case.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Plainly: this is a solid paper that cleans up a recent result and adds a genuinely new case. Theorem 5 removes the C^{2,α} assumption from [4]'s optimal bounds, and Theorem 6 pushes the upper bound to C^{1,1}. The non-semipositive estimates in Theorems 8–9 are the real novelty; getting k_D ≍ H_D at points where the Levi form has a negative eigenvalue is new and connects to the visibility-pseudoconvexity program. The main arguments hold up. The reduction to convex domains via Lempert's theorem is standard, the localization lemmas are carefully stated, and the proofs of Theorems 6 and 9 have the right shape.

The soft spot is the one the stress-test flagged: the lower bound in Theorem 5 imports [4, Prop. 22] to get \hat A_D ≍ B_D/δ_D, and the paper never states under what regularity that proposition is proved or why it survives at C^2. That is an expositional gap, not (I think) a mathematical one. The needed inequality |X|^2 ≲ |X_z| + |X|√δ is a pointwise second-order fact that follows from strong pseudoconvexity by Taylor expansion with Peano remainder; C^{2,α} is more than you need. But the authors need to write that down or quote the proposition with its exact hypotheses. As it stands, a reader cannot check the main advertised extension without going into [4]. Also, Proposition 7 is stated with a skipped proof—fine for a short paper, but a referee may want a few lines. The case analysis in Proposition 17 is long but follows the same pattern as the convex case; I didn't find a real gap.

Verdict: send to a good journal. The paper deserves a serious referee; the gap is patchable and the new results are worth having. If I worked on Kobayashi geometry I'd cite it.

Referee Report

2 major / 4 minor

Summary. The paper establishes sharp two-sided estimates for the Kobayashi distance near boundary points of a domain in C^n. The main result, Theorem 5, states that at any C^2-smooth strongly pseudoconvex boundary point p there are constants 0<c<C such that, for z,w close to p, log(1+c A_D(z,w)) ≤ k_D(z,w) ≤ l_D(z,w) ≤ log(1+C A_D(z,w)), where A_D is the explicit Euclidean quantity defined in equation (2). The upper bound is proved in the more general C^{1,1} setting as Theorem 6, and the lower bound is obtained by reducing to strongly convex domains and combining a directional-distance lower bound from [13] with a comparison imported from [4, Proposition 22]. The paper also states upper and lower bounds near non-semipositive boundary points (Theorems 8 and 9) in terms of a second quantity H_D defined in (8), and it gives a converse statement (Proposition 4) showing that a lower bound of the form (5) forces strong pseudoconvexity.

Significance. If the identified gaps are repaired, this is a useful and natural improvement: it replaces the C^{2,α} regularity required in [4] by merely C^2 for the optimal lower and upper bounds, and it extends the upper bound to C^{1,1} boundaries. The proof strategy is largely elementary and avoids the dilation methods of [4], which is valuable. The non-semipositive results, Theorems 8 and 9, are a genuine addition and are of current interest for the relationship between visibility and pseudoconvexity. The paper is clearly written, carefully distinguishes which results are imported, and does not introduce fitted parameters; its methods are transparent enough that the remaining technical gaps appear fixable.

major comments (2)

[Section 5.1, end of proof of Theorem 9] The proof of the lower bound (5) relies on the comparison \hat A_D(z,w) := |(z-w)_z|/δ_D(z) + |z-w|/√δ_D(z) ≍ B_D(z,w)/δ_D(z), which is imported from [4, Proposition 22]. The manuscript does not state the regularity hypotheses of that proposition, and [4] is a paper in the C^{2,α} category. Since the entire advertised improvement from C^{2,α} to C^2 for the lower bound depends on this comparison, the authors must either prove it directly under C^2 regularity or verify that the proof of [4, Proposition 22] uses only C^2 geometry. In particular, the missing inequality |X|^2 ≲ |X_z| + |X|√δ_D(z) for pairs with δ_D(w) ≥ δ_D(z) is a second-order boundary fact; it is plausible for C^2 boundaries but is not demonstrated here. Without this step, the reduction from the directional-distance bound to the sharp A_D bound is not established.
[Section 5.1, end of proof of Theorem 9] The proof of k_D(z,w) ≲ H_D(w,z) in case (a) contains a false intermediate assertion: the assumption |(z-w)_z| + |z-w|^2 ≤ δ_D(z)/2 does not imply √(|(z-w)_z| + |z-w|) ≤ √δ_D(z), because |z-w| may be of order √δ_D(z). The final inequality B_D(z,w)/√δ_D(z) ≤ 2H_D(z,w) may still be true, but the given justification is invalid and should be replaced by a correct argument.

minor comments (4)

[Section 4, Theorem 6] The reduction to χ_{D,p} ∈ (0,1) by homothety is stated without explaining the scaling: a homothety of factor λ replaces χ_{D,p} by χ_{D,p}/λ. Please make this explicit, since it is not immediately obvious and the text currently says only that a homothety "will modify" the quantity.
[Section 3, proof of Theorem 5] The auxiliary quantity \hat A_D is defined inline in the proof but not numbered or displayed separately. Giving it a numbered display would improve readability, especially because it is the key object in the lower-bound comparison.
[Section 6, Proposition 7] Proposition 7 is stated as a global result but its proof is omitted with the comment that the arguments are the same as in [4]. Since the present paper changes the regularity setting from C^{2,α} to C^2, the authors should either include a proof or explicitly state which parts of [4] carry over unchanged and where the C^2 regularity is used.
[Section 5.2, Proposition 16] In the displayed estimate in the "Otherwise" case, the denominator is typeset ambiguously; it should be clear that the terms are √δ_D(z) + √(|Re(z-w)_z| + |z-w|), not √(δ_D(z) + |Re(z-w)_z|) + |z-w|. Please clarify the parentheses.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the new bounds are derived from elementary geometric estimates and prior lemmas used as tools, not from the conclusion being proved; the main self-citation risk is a regularity gap in the lower-bound proof, not a circular reduction.

full rationale

The paper's central claims are not obtained by defining target quantities in terms of themselves or by fitting parameters. The upper bound (6) is derived from Theorem 6, proved in Section 4 using the C^{1,1} signed-distance expansion (Proposition 13, quoted from [12]) and explicit ball estimates; this is independent of the conclusion. The lower bound (5) in Theorem 5 uses [13, p.633] for a convex-domain ball lower bound and [4, Proposition 22] for the Euclidean comparison A_hat_D ≍ B_D/δ_D. Neither cited result states the target Kobayashi-distance inequality; they are auxiliary lemmas with different content. The paper independently proves the needed biholomorphic invariance (Proposition 11) and symmetry (11), so it does not merely rename or import the target estimate. The skeptic's concern about [4, Proposition 22] is a regularity-margin issue: the paper does not state the hypotheses under which that proposition was proved in [4] (likely C^{2,α}), and if the comparison fails for merely C^2 boundaries, the advertised regularity reduction is not established. That is a correctness risk, not a circularity, because the derivation does not reduce to assuming the theorem being proved. Self-citations to [3], [4], [11], [12], [13] are load-bearing as lemmas, but the present argument contains independent content (C^{1,1} upper bound, non-semipositive distance estimates, symmetric Euclidean comparisons). No fitted parameters or self-definitional normalizations appear, so the circularity burden is low.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central claims rest on several imported results from the prior literature, including results by the same authors. No numerical constants are fitted to data in this proof-based paper; all constants are generic comparison constants. No new physical or mathematical entities are postulated.

assumptions (6)

standard math Lempert's theorem: for bounded convex domains, the Caratheodory, Kobayashi, and Lempert distances coincide.
Used in the proof of Theorem 5 after local biholomorphic reduction to a strongly convex domain, Section 3.
domain assumption Localization theorem [11, Theorem 1.1]: k_D <= k_{D∩U} < k_D + C and k_{D∩U} is comparable to k_D.
Used in the proof of Theorem 5 to replace D by a local piece without changing the distance up to additive constants.
domain assumption Signed-distance expansion [12, Proposition 8]: |r_D(x)-r_D(y)-grad r_D(x)·(x-y)| <= (chi_{D,p}/2 + o(1))(|x-y|^2 - (r_D(x)-r_D(y))^2).
Yields inequality (14), a load-bearing estimate in Sections 4 and 5.
domain assumption C^{2,alpha} auxiliary results from [4]: comparability of A_D under localization and biholomorphism ([4, Lemma 20]) and the comparability Ahat_D roughly B_D/delta_D(z) ([4, Proposition 22]).
Used in the proof of Theorem 5. The paper does not comment on their validity at C^2 regularity.
domain assumption Infinitesimal Kobayashi metric estimates near non-semipositive points [3, Proposition 1].
Used to integrate the metric along curves in the proofs of Theorem 9 and Proposition 15.
domain assumption Lower bound for the Kobayashi distance in C-convex domains [13, p. 633].
Used in the lower-bound half of Theorem 5 after reduction to a strongly convex domain.

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Cite this review

Pith. "Pith review of Optimal bounds for the Kobayashi distance near $\mathcal C^2$-smooth boundary points." pith.science (2026). https://pith.science/paper/V35ACWL6

@misc{pith2026250606507,
  author       = {Pith},
  title        = {Pith review of: Optimal bounds for the Kobayashi distance near $\mathcal C^2$-smooth boundary points},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/V35ACWL6}},
  note         = {Machine review of arXiv:2506.06507}
}

abstract

It is shown that the optimal upper and lower bounds for the Kobayashi distance near $\mathcal C^{2,\alpha}$-smooth strongly pseudoconvex boundary points obtained in L. Kosinski, N. Nikolov, A.Y. Okten: "Precise estimates of invariant distances on strongly pseudoconvex domains", Adv. Math. 478 (2025), 110388, remain true in the general $\mathcal C^2$ strongly pseudoconvex setting. In fact, the upper bound is extended to the general $\mathcal C^{1,1}$-smooth case. We also give upper and lower bounds for the Kobayashi distance near non-semipositive boundary points.

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Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages

[4]

Kosi´ nski, N

L. Kosi´ nski, N. Nikolov, A.Y.¨Okten,Precise estimates of invariant distances on strongly pseudoconvex domains, Adv. Math. 478 (2025), 110388

work page 2025
[13]

Nikolov, M

N. Nikolov, M. Trybu la,The Kobayashi balls of (C-)convex domains, Monatsh. Math. 177 (2015), 627–635. N. Nikolov, Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G. Bonchev 8, 1113 Sofia, Bulgaria F aculty of Information Sciences, State University of Library Studies and Information Technologies, Shipchenski prohod 69A, 1574...

work page 2015
[1]

Balogh, M

Z.M. Balogh, M. Bonk,Gromov hyperbolicity and the Kobayashi metric on strictly pseudoconvex do- mains, Comment. Math. Helv. 75 (2000), 504–533

work page 2000
[2]

Bracci, N

F. Bracci, N. Nikolov, P.J. Thomas,Visibility of Kobayashi geodesics in convex domains and related properties, Math. Z. 301 (2022), 2, 2011–2035

work page 2022
[3]

N.Q. Dieu, N. Nikolov, P.J. Thomas,Estimates for invariant metrics near non-semipositive boundary, J. Geom. Anal. 23 (2013), 598–610

work page 2013
[5]

Kosi´ nski, N

L. Kosi´ nski, N. Nikolov, P.J. Thomas,A Gehring-Hayman inequality for strongly pseudoconvex domains, Int. Math. Res. Not. IMRN, no. 11 (2024), 9165–9177

work page 2024
[6]

J. Liu, X. Pu, H. Wang,Bi-H¨ older extensions of quasi-isometries on pseudoconvex domains of finite type inC 2, J. Geom. Anal., 33:152 (2023)

work page 2023
[7]

H. Li, X. Pu, H. Wang,The Gehring-Hayman type theorem on pseudoconvex domains of finite type in C2, Ann. Mat. Pura Appl. 203 (2024), 2785–2799

work page 2024

Show all 13 references

[8]

Nikolov, L

N. Nikolov, L. Andreev,Estimates of the Kobayashi and quasi-hyperbolic distances, Ann. Mat. Pura Appl. 196 (2017), 43–50

2017
[9]

Nikolov, A.Y

N. Nikolov, A.Y. ¨Okten, P.J. Thomas,VisibleC 2-smooth domains are pseudoconvex, Bull. Sci. Math. 197 (2024), 103525

2024
[10]

Nikolov, P

N. Nikolov, P. Pflug,On the derivatives of the Lempert functions, Ann. Mat. Pura Appl. 187 (2008), 547–553

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[11]

Nikolov, P.J

N. Nikolov, P.J. Thomas,Quantitative localization and comparison of invariant distances of domains inC n, J. Geom. Anal. 33 (2023), article 35

2023
[12]

Nikolov, P.J

N. Nikolov, P.J. Thomas,Boundary regularity for the distance functions, and the eikonal equation, arXiv:2409.01774, to appear in the Journal of Geometric Analysis

arXiv

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [4]

Kosi´ nski, N

L. Kosi´ nski, N. Nikolov, A.Y.¨Okten,Precise estimates of invariant distances on strongly pseudoconvex domains, Adv. Math. 478 (2025), 110388

work page 2025

[2] [13]

Nikolov, M

N. Nikolov, M. Trybu la,The Kobayashi balls of (C-)convex domains, Monatsh. Math. 177 (2015), 627–635. N. Nikolov, Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G. Bonchev 8, 1113 Sofia, Bulgaria F aculty of Information Sciences, State University of Library Studies and Information Technologies, Shipchenski prohod 69A, 1574...

work page 2015

[3] [1]

Balogh, M

Z.M. Balogh, M. Bonk,Gromov hyperbolicity and the Kobayashi metric on strictly pseudoconvex do- mains, Comment. Math. Helv. 75 (2000), 504–533

work page 2000

[4] [2]

Bracci, N

F. Bracci, N. Nikolov, P.J. Thomas,Visibility of Kobayashi geodesics in convex domains and related properties, Math. Z. 301 (2022), 2, 2011–2035

work page 2022

[5] [3]

N.Q. Dieu, N. Nikolov, P.J. Thomas,Estimates for invariant metrics near non-semipositive boundary, J. Geom. Anal. 23 (2013), 598–610

work page 2013

[6] [5]

Kosi´ nski, N

L. Kosi´ nski, N. Nikolov, P.J. Thomas,A Gehring-Hayman inequality for strongly pseudoconvex domains, Int. Math. Res. Not. IMRN, no. 11 (2024), 9165–9177

work page 2024

[7] [6]

J. Liu, X. Pu, H. Wang,Bi-H¨ older extensions of quasi-isometries on pseudoconvex domains of finite type inC 2, J. Geom. Anal., 33:152 (2023)

work page 2023

[8] [7]

H. Li, X. Pu, H. Wang,The Gehring-Hayman type theorem on pseudoconvex domains of finite type in C2, Ann. Mat. Pura Appl. 203 (2024), 2785–2799

work page 2024