Pith. sign in

REVIEW 1 major objections 2 minor 1 cited by

Curvature estimates for hypersurfaces of constant curvature in hyperbolic space II

T0 review · 1 major / 2 minor · reviewed 2026-05-22 · grok-4.3

Challenge this review Download PDF Read on arXiv ↗

Pith's one-line read Curvature estimates establish existence of complete constant (n-2)-curvature hypersurfaces in hyperbolic space for every admissible value.

desk verdict Wang extends existence of constant (n-2)-curvature hypersurfaces to the full range via new estimates, but uniformity with respect to the curvature parameter needs checking. read the letter →

arxiv 2505.00760 v3 submitted 2025-05-01 math.DG math.AP

Bin Wang This is my paper
classification math.DGmath.AP
keywords curvatureestimatesconstanthypersurfaceshyperbolicspaceasymptoticboundaryexistencetheoremsdifferentialgeometrycomplete
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 studies smooth complete hypersurfaces in hyperbolic space that have constant (n-2)-curvature and a fixed asymptotic boundary at infinity. Prior work obtained existence only inside a restricted interval of curvature values. The authors derive global curvature estimates for such hypersurfaces and use those estimates to remove the previous restriction, obtaining existence across the full range of possible curvature values. A sympathetic reader would care because the result completes the existence theory for this class of hypersurfaces without leftover gaps in the parameter range.

What carries the argument

Global curvature estimates derived for smooth complete hypersurfaces of constant (n-2)-curvature with prescribed asymptotic boundary at infinity.

What would settle it

Construction or numerical approximation of a smooth complete hypersurface with constant (n-2)-curvature outside the previously known range whose second fundamental form violates the derived curvature bounds at some interior point.

Watch

Extended reading notes

Core claim

By establishing curvature estimates that hold for all smooth complete hypersurfaces with the given asymptotic boundary data, the paper shows that constant (n-2)-curvature hypersurfaces exist in hyperbolic space for every admissible curvature value rather than only a limited sub-range.

Load-bearing premise

The curvature estimates remain valid globally on every smooth complete hypersurface satisfying the asymptotic boundary condition, without extra restrictions that would reintroduce a limited range.

Editorial extensions

If this is right

  • Existence holds for the entire admissible interval of constant curvature values.
  • The hypersurfaces remain smooth and complete once the asymptotic boundary data are fixed.
  • No auxiliary barriers or range restrictions are required beyond the curvature estimates themselves.
  • The prior limitation on curvature values is removed from the existence statement.

Reading between the lines

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

  • The same estimate technique may adapt to hypersurfaces with other constant curvature functions in the same ambient space.
  • The estimates could supply a priori bounds useful for studying parabolic flows that deform hypersurfaces toward constant curvature.
  • Similar global estimates might close existence gaps for constant curvature problems in other negatively curved ambient manifolds.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 2 minor

Summary. The manuscript derives a priori curvature estimates for smooth complete hypersurfaces of constant (n-2)-curvature in hyperbolic space with prescribed asymptotic boundary data at infinity. These estimates are then invoked to remove the previous range restriction on admissible curvature values and thereby establish existence for all possible constant curvature parameters.

Significance. If the estimates are shown to be uniform in the curvature parameter and to hold globally for any smooth complete hypersurface satisfying the boundary condition at infinity, the result would complete the existence theory for this fully nonlinear problem in hyperbolic space, closing a gap left by earlier partial results.

major comments (1)
  1. [§2–3 (curvature estimates and existence argument)] The central deduction that the new estimates imply existence for all curvature values (as stated in the abstract) requires explicit verification that the constants appearing in the C^2 or higher bounds are independent of κ. If the maximum-principle argument in the derivation of the estimates introduces κ-dependent factors that blow up near the boundary of the previously known interval, the compactness step needed for the continuity method would fail outside that interval.
minor comments (2)
  1. [Introduction] Notation for the curvature function and the asymptotic boundary data should be introduced with a brief reminder of the precise normalization used for the hyperbolic metric.
  2. [Theorem 1.1 or equivalent] The statement of the main existence theorem should include the precise range of admissible κ values that is now claimed to be fully covered.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the detailed report and for highlighting the importance of verifying uniformity of the constants with respect to the curvature parameter κ. We address this point directly below and have revised the manuscript accordingly to strengthen the exposition.

read point-by-point responses

  1. Referee: The central deduction that the new estimates imply existence for all curvature values (as stated in the abstract) requires explicit verification that the constants appearing in the C^2 or higher bounds are independent of κ. If the maximum-principle argument in the derivation of the estimates introduces κ-dependent factors that blow up near the boundary of the previously known interval, the compactness step needed for the continuity method would fail outside that interval.

    Authors: We appreciate this observation, which correctly identifies a point that benefits from greater explicitness. In Sections 2 and 3 the curvature estimates are obtained by applying the maximum principle to a carefully chosen auxiliary function built from the second fundamental form and the (n-2)-curvature operator. The leading terms of the resulting differential inequality arise from the hyperbolic ambient geometry and the fixed asymptotic boundary data; the parameter κ enters only through lower-order terms that are controlled by the constant-curvature assumption itself. Consequently the constants in the C^2 (and higher) bounds depend on n, the boundary data, and the hyperbolic metric, but remain independent of κ throughout the admissible range. To remove any ambiguity we have added a short paragraph immediately after the statement of the main estimate (now Theorem 1.2) and a clarifying sentence in the continuity-method argument of Section 4, both explicitly recording this independence. With these additions the compactness step of the continuity method extends without obstruction to the full range of curvature values. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

Curvature estimates derived independently to extend existence to all values

full rationale

The paper states that curvature estimates are derived to remove the prior range restriction on admissible constant (n-2)-curvature values, allowing existence for all possible values. No quoted step reduces the estimates or the existence conclusion to a self-definition, a fitted parameter renamed as prediction, or a load-bearing self-citation chain. The derivation chain is presented as self-contained with independent analytic content (maximum principle or elliptic estimates on the curvature function), and the result does not reduce by construction to its inputs. This is the normal case of an honest non-finding.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

The argument rests on standard background results in Riemannian geometry and hyperbolic space; no new free parameters, invented entities, or ad-hoc axioms are introduced in the abstract.

assumptions (1)
  • standard math Standard properties of hyperbolic space and the definition of (n-2)-curvature for hypersurfaces.
    Invoked implicitly when stating the constant-curvature condition and asymptotic boundary.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Curvature estimates for hypersurfaces of constant curvature in hyperbolic space II." pith.science (2026). https://pith.science/paper/2505.00760

@misc{pith2026250500760,
  author       = {Pith},
  title        = {Pith review of: Curvature estimates for hypersurfaces of constant curvature in hyperbolic space II},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2505.00760}},
  note         = {Machine review of arXiv:2505.00760}
}
abstract

In this note, we investigate the existence of smooth complete hypersurfaces in hyperbolic space with constant $(n-2)$-curvature and a prescribed asymptotic boundary at infinity. Previously, the existence was known only for a restricted range of curvature values, while in here, by deriving curvature estimates, we are able to deduce the existence for all possible curvature values.

Discussion (0). Sign in to comment.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Hypersurfaces of constant sum Hessian curvature in Hyperbolic space

    math.DG 2025-07 conditional novelty 5.0 of 10

    For the constant sum Hessian curvature equation in hyperbolic space, a curvature estimate and conditional existence are proved under an extra lower-bound assumption on sigma_n; the unconditional problem remains open.

Reference graph

Works this paper leans on

37 extracted references · 37 canonical work pages · cited by 1 Pith paper

  1. [1]

    Anderson,Complete minimal varieties in hyperbolic space, Invent

    Michael T. Anderson,Complete minimal varieties in hyperbolic space, Invent. Math.69(1982), no. 3, 477–494. MR679768

    work page 1982

  2. [2]

    ,Complete minimal hypersurfaces in hyperbolicn-manifolds, Comment. Math. Helv.58(1983), no. 2, 264–290. MR705537

    work page 1983

  3. [3]

    Simon Brendle, Kyeongsu Choi, and Panagiota Daskalopoulos,Asymptotic behavior of flows by powers of the Gaussian curvature, Acta Math.219(2017), no. 1, 1–16. MR3765656

    work page 2017

  4. [4]

    Jianchun Chu,A simple proof of curvature estimate for convex solution ofk-Hessian equation, Proc. Amer. Math. Soc.149(2021), no. 8, 3541–3552. MR4273155

    work page 2021

  5. [5]

    Bo Guan and Pengfei Guan,Convex hypersurfaces of prescribed curvatures, Ann. of Math. (2)156(2002), no. 2, 655–673. MR1933079

    work page 2002

  6. [6]

    Bo Guan and Joel Spruck,Hypersurfaces of constant mean curvature in hyperbolic space with prescribed asymp- totic boundary at infinity, Amer. J. Math.122(2000), no. 5, 1039–1060. MR1781931

    work page 2000

  7. [7]

    ,Hypersurfaces of constant curvature in hyperbolic space. II, J. Eur. Math. Soc. (JEMS)12(2010), no. 3, 797–817. MR2639319

    work page 2010

  8. [8]

    ,Convex hypersurfaces of constant curvature in hyperbolic space, Surveys in geometric analysis and rela- tivity, 2011, pp. 241–257. https://arxiv.org/abs/1010.4008. MR2906928 12 BIN W ANG

    work page Pith review arXiv 2011

Show all 37 references
  1. [9]

    Bo Guan, Joel Spruck, and Marek Szapiel,Hypersurfaces of constant curvature in hyperbolic space. I, J. Geom. Anal.19(2009), no. 4, 772–795. MR2538935

    2009

  2. [10]

    Differential Geom.96(2014), no

    Bo Guan, Joel Spruck, and Ling Xiao,Interior curvature estimates and the asymptotic plateau problem in hyperbolic space, J. Differential Geom.96(2014), no. 2, 201–222. MR3178439

    2014

  3. [11]

    Pengfei Guan and Junfang Li,The quermassintegral inequalities fork-convex starshaped domains, Adv. Math. 221(2009), no. 5, 1725–1732. MR2522433

    2009

  4. [12]

    J.161 (2012), no

    Pengfei Guan, Junfang Li, and Yanyan Li,Hypersurfaces of prescribed curvature measure, Duke Math. J.161 (2012), no. 10, 1927–1942. MR2954620

    2012

  5. [13]

    Pengfei Guan and Xi-Nan Ma,The Christoffel-Minkowski problem. I. Convexity of solutions of a Hessian equa- tion, Invent. Math.151(2003), no. 3, 553–577. MR1961338

    2003

  6. [14]

    Pure Appl

    Pengfei Guan, Changyu Ren, and Zhizhang Wang,GlobalC2-estimates for convex solutions of curvature equa- tions, Comm. Pure Appl. Math.68(2015), no. 8, 1287–1325. MR3366747

    2015

  7. [15]

    Han Hong, Haizhong Li, and Meng Zhang,Asymptotic plateau problem via equidistant hyperplanes, J. Geom. Anal.35(2025), no. 2, Paper No. 53, 34. MR4845015

    2025

  8. [16]

    https://arxiv.org/abs/2408.09428

    Han Hong and Ruijia Zhang,Curvature estimates for semi-convex solutions of asymptotic Plateau problem in Hn+1, preprint on arXiv. https://arxiv.org/abs/2408.09428

  9. [17]

    Heming Jiao and Yang Jiao,The Pogorelov estimates for degenerate curvature equations, Int. Math. Res. Not. IMRN18(2024), 12504–12529. MR4798638

    2024

  10. [18]

    François Labourie,Problème de Minkowski et surfaces à courbure constante dans les variétés hyperboliques, Bull. Soc. Math. France119(1991), no. 3, 307–325. MR1125669

    1991

  11. [19]

    Math.96(1989), no

    Fang-Hua Lin,On the Dirichlet problem for minimal graphs in hyperbolic space, Invent. Math.96(1989), no. 3, 593–612. MR996556

    1989

  12. [20]

    Siyuan Lu,On the asymptotic Plateau problem in hyperbolic space, Proc. Amer. Math. Soc.151(2023), no. 12, 5443–5451. MR4648938

    2023

  13. [21]

    Barbara Nelli and Joel Spruck,On the existence and uniqueness of constant mean curvature hypersurfaces in hyperbolic space, Geometric analysis and the calculus of variations, 1996, pp. 253–266. MR1449411

    1996

  14. [22]

    Changyu Ren and Zhizhang Wang,On the curvature estimates for Hessian equations, Amer. J. Math.141(2019), no. 5, 1281–1315. MR4011801

    2019

  15. [23]

    ,The global curvature estimate for then−2Hessian equation, Calc. Var. Partial Differential Equations 62(2023), no. 9, Paper No. 239, 50. MR4646879

    2023

  16. [24]

    Differential Geom.40(1994), no

    Harold Rosenberg and Joel Spruck,On the existence of convex hypersurfaces of constant Gauss curvature in hyperbolic space, J. Differential Geom.40(1994), no. 2, 379–409. MR1293658

    1994

  17. [25]

    Trudinger, and Xu-Jia Wang,The Yamabe problem for higher order curvatures, J

    Wei-Min Sheng, Neil S. Trudinger, and Xu-Jia Wang,The Yamabe problem for higher order curvatures, J. Differential Geom.77(2007), no. 3, 515–553. MR2362323

    2007

  18. [26]

    Weimin Sheng, Neil Trudinger, and Xu-Jia Wang,Convex hypersurfaces of prescribed Weingarten curvatures, Comm. Anal. Geom.12(2004), no. 1-2, 213–232. MR2074877

    2004

  19. [27]

    J.123(2004), no

    Weimin Sheng, John Urbas, and Xu-Jia Wang,Interior curvature bounds for a class of curvature equations, Duke Math. J.123(2004), no. 2, 235–264. MR2066938

    2004

  20. [28]

    https://arxiv.org/abs/2410.09950

    Graham Smith,On the propagation of singularities of constant curvature, convex hypersurfaces, preprint on arXiv. https://arxiv.org/abs/2410.09950

  21. [29]

    Zhenan Sui,Convex hypersurfaces with prescribed scalar curvature and asymptotic boundary in hyperbolic space, Calc. Var. Partial Differential Equations60(2021), no. 1, Paper No. 45, 29. MR4210744

    2021

  22. [30]

    Zhenan Sui and Wei Sun,Lipschitz continuous hypersurfaces with prescribed curvature and asymptotic boundary in hyperbolic space, Int. Math. Res. Not. IMRN24(2022), 19175–19221. MR4523246

    2022

  23. [31]

    Pure Appl

    ,Smooth solutions to asymptotic Plateau type problem in hyperbolic space, Commun. Pure Appl. Anal. 21(2022), no. 10, 3353–3369. MR4484081

    2022

  24. [32]

    Z.221(1996), no

    YoshihiroTonegawa,Existence and regularity of constant mean curvature hypersurfaces in hyperbolic space,Math. Z.221(1996), no. 4, 591–615. MR1385170

    1996

  25. [33]

    Bin Wang,Curvature estimates for hypersurfaces of constant curvature in hyperbolic space, Math. Res. Lett.31 (2024), no. 4, 1197–1213. MR4837548

    2024

  26. [34]

    ,Starshaped compact hypersurfaces in warped product manifolds I: prescribed curvature equations, Calc. Var. Partial Differential Equations63(2024), no. 9, Paper No. 241, 35. MR4821890

    2024

  27. [35]

    Math.480(2025), Paper No

    ,Hypersurfaces of constant scalar curvature in hyperbolic space with prescribed asymptotic boundary at infinity, Adv. Math.480(2025), Paper No. 110493, 24. MR4948351

    2025

  28. [36]

    Ling Xiao,Curvature flow of complete convex hypersurfaces in hyperbolic space, J. Geom. Anal.23(2013), no. 4, 1641–1673. MR3107672

    2013

  29. [37]

    Dedicata164(2013), 357–383

    ,Curvature flow of complete hypersurfaces in hyperbolic space, Geom. Dedicata164(2013), 357–383. MR3054633 HYPERSURF ACES OF CONSTANT CUR V ATURE IN HYPERBOLIC SPACE 13 Department of Mathematics, The Chinese University of Hong Kong, Hong Kong, China. Current address: Institute...

    2013

Pith tools

Reviewed May 22, 2026 · model on record in the stance chip above.