Pith. sign in

REVIEW 3 minor 1 cited by

Regularity of rooftop envelopes

T0 review · 0 major / 3 minor · reviewed 2026-05-10 · grok-4.3

Challenge this review Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Geodesics between continuous plurisubharmonic functions on bounded domains in complex space have continuity, Hölder regularity, and C^{1,1} regularity, which transfers to rooftop envelopes.

desk verdict This paper gives clean proofs of continuity, Hölder, and C^{1,1} regularity for geodesics between continuous psh functions on bounded domains, then derives the corresponding properties for rooftop envelopes. read the letter →

arxiv 2604.18855 v2 submitted 2026-04-20 math.CV

Eleonora Di Nezza , Alexander Rashkovskii This is my paper · ORCID
classification math.CV
keywords plurisubharmonicfunctionsgeodesicsrooftopenvelopesHölderregularityC^{11}pluripotentialtheorycomplexdomainsMonge-Ampèreequation
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 examines the regularity properties of geodesics connecting pairs of continuous plurisubharmonic functions on bounded domains in n-dimensional complex space. It establishes that these geodesics remain continuous, satisfy Hölder continuity, and attain C to the 1,1 regularity. From these geodesic properties the authors derive regularity results for rooftop envelopes. A reader would care because the results supply concrete smoothness controls that aid the construction and approximation of solutions to nonlinear equations such as the complex Monge-Ampère equation.

What carries the argument

The geodesic between two continuous plurisubharmonic functions, constructed via a supremum that preserves plurisubharmonicity along line segments, serves as the mechanism that carries regularity information to the rooftop envelope.

What would settle it

A pair of continuous plurisubharmonic functions on a bounded domain whose geodesic fails to have bounded second derivatives at an interior point would disprove the C^{1,1} claim.

Watch

Extended reading notes

Core claim

The central claim is that geodesics between continuous plurisubharmonic functions on bounded domains of ℂ^n are continuous, Hölder regular, and C^{1,1} regular. These regularity statements are then applied directly to obtain corresponding regularity properties for the rooftop envelopes built from the same functions.

Load-bearing premise

The functions remain continuous plurisubharmonic and the domains stay bounded subsets of complex Euclidean space so that the standard supremum constructions for geodesics and envelopes remain valid.

Editorial extensions

If this is right

  • Rooftop envelopes inherit C^{1,1} regularity directly from the geodesics.
  • Hölder continuity of the geodesics passes to the envelopes on the same domains.
  • The continuity and regularity statements hold uniformly on compact subsets inside the domain.
  • The same conclusions apply in every dimension n for any bounded domain.

Reading between the lines

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

  • The same geodesic construction might be tested numerically on explicit examples such as logarithms of polynomials to check the predicted Hölder exponents.
  • The regularity transfer could extend to other supremum-based envelopes that appear in pluripotential theory.
  • Boundary behavior near the domain edge may impose additional constraints that limit the global C^{1,1} constant.
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

0 major / 3 minor

Summary. The paper studies continuity, Hölder regularity, and C^{1,1}-regularity of geodesics between continuous plurisubharmonic functions on bounded domains in ℂ^n, with geodesics defined via the homogeneous complex Monge-Ampère equation. It then derives regularity properties of the associated rooftop envelopes.

Significance. If the results hold, they strengthen the regularity theory for geodesics in the space of plurisubharmonic functions and for rooftop envelopes, which are central objects in pluripotential theory. The grounding in standard constructions on bounded domains with continuous data, without extraneous assumptions, is a strength and supports potential applications to the complex Monge-Ampère equation and metric geometry on psh spaces.

minor comments (3)
  1. The introduction would benefit from an explicit comparison of the new regularity statements with prior results on psh geodesics (e.g., those of Darvas or others using the same Monge-Ampère framework) to clarify the precise advance.
  2. In the section defining the rooftop envelope, the notation for the upper envelope operation could be made more uniform with the geodesic construction to avoid minor ambiguity in the statements of the derived regularity theorems.
  3. A brief remark on the sharpness of the Hölder exponent obtained for the geodesics would help readers assess the optimality of the results.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, the recognition of its contributions to the regularity theory for geodesics and rooftop envelopes, and the recommendation of minor revision. We appreciate the emphasis on the grounding in standard constructions with continuous data on bounded domains.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity detected in derivation chain

full rationale

The paper derives continuity, Hölder, and C^{1,1} regularity of geodesics (via the homogeneous complex Monge-Ampère equation) between continuous plurisubharmonic functions on bounded domains in ℂ^n, then obtains regularity for rooftop envelopes. These steps rest on standard pluripotential theory with explicit assumptions of domain boundedness and function continuity; no equations reduce by construction to inputs, no parameters are fitted then relabeled as predictions, and no load-bearing self-citations or imported uniqueness theorems create circularity. The central claims remain independent of the paper's own prior results.

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

Only the abstract is available; no explicit free parameters, ad-hoc axioms, or invented entities are mentioned. The work relies on the standard background of pluripotential theory on bounded domains.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Regularity of rooftop envelopes." pith.science (2026). https://pith.science/paper/2604.18855

@misc{pith2026260418855,
  author       = {Pith},
  title        = {Pith review of: Regularity of rooftop envelopes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2604.18855}},
  note         = {Machine review of arXiv:2604.18855}
}
abstract

We study continuity, H\"older regularity, and $C^{1,1}$-regularity of geodesics between continuous plurisubharmonic functions on bounded domains of $\mathbb{C}^n$. We then derive regularity properties of rooftop envelopes.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

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

  1. Jensen Deficits for Inhomogeneous Monge-Amp\`ere Dirichlet Problems

    math.CV 2026-06 unverdicted novelty 6.0 of 10

    Proves a boundary dual formula for inhomogeneous Monge-Ampère Dirichlet problems via Jensen deficits on convex subsolution families, with applications to Alexandrov and Bedford-Taylor subsolutions.

Reference graph

Works this paper leans on

57 extracted references · 57 canonical work pages · cited by 1 Pith paper

  1. [1]

    Abja,Geometry and topology of the space of plurisubharmonic functions, J

    S. Abja,Geometry and topology of the space of plurisubharmonic functions, J. Geom. Anal.29(2019), no. 1, 510–541

    work page 2019

  2. [2]

    Abja and S

    S. Abja and S. Dinew,Regularity of geodesics in the spaces of convex and plurisubhar- monic functions, Trans. Amer. Math. Soc.374(2021), 3783–3800

    work page 2021

  3. [3]

    ˚Ahag, R

    P. ˚Ahag, R. Czy˙ z, C. H. Lu, A. Rashkovskii,Geodesic connectivity and rooftop envelopes in the Cegrell classes, Math. Ann.391(2025), No. 3, 3333-3361. REGULARITY OF ROOFTOP ENVELOPES 23

    work page 2025

  4. [4]

    Barbu and T

    V. Barbu and T. Precupanu, Convexity and Optimization in Banach Spaces. Springer, 2012

    work page 2012

  5. [5]

    Bedford and B.A

    E. Bedford and B.A. Taylor,The Dirichlet problem for a complex Monge-Ampère equation, Invent. Math.37(1976), 1-44

    work page 1976

  6. [6]

    Berman,From Monge-Ampère equations to envelopes and geodesic rays in the zero temperature limit, Math

    R. Berman,From Monge-Ampère equations to envelopes and geodesic rays in the zero temperature limit, Math. Z.291(2019), No. 1-2, 365-394

    work page 2019

  7. [7]

    hypothèse fondamentale

    R.J. Berman and B. Berndtsson,Moser-Trudinger type inequalities for complex Monge- Ampère operators and Aubin ’s "hypothèse fondamentale", arXiv:1109.1263; Ann. Fac. Sci. Toulouse Math. (6)31(2022), no. 3, 595–645

    work page Pith review arXiv 2022

  8. [8]

    Berndtsson,A Brunn–Minkowski type inequality for Fano manifolds and some uniqueness theorems in K¨ ahler geometry, Inv

    B. Berndtsson,A Brunn–Minkowski type inequality for Fano manifolds and some uniqueness theorems in K¨ ahler geometry, Inv. Math.200(2015), no. 1, 149–200

    work page 2015

Show all 57 references
  1. [9]

    B locki,The complex Monge Ampere operator in hyperconvex domains, Ann

    Z. B locki,The complex Monge Ampere operator in hyperconvex domains, Ann. Scuola Norm. Sup. Pisa C1. Sci.23(1996), 721–747

    1996

  2. [10]

    B locki, S

    Z. B locki, S. Ko lodziej,On regularization of plurisubharmonic functions on manifolds, Proc. Amer. Math. Soc.135(2007), 2089–2093

    2007

  3. [11]

    Bremermann,On a generalized Dirichlet problem for pluri-subharmonic functions and pseudo-convex domains

    H.J. Bremermann,On a generalized Dirichlet problem for pluri-subharmonic functions and pseudo-convex domains. Characterization of ˇSilov boundaries, Trans. Am. Math. Soc.91(1959), 246–276

    1959

  4. [12]

    Carlehed, U

    M. Carlehed, U. Cegrell and F. Wikstr¨ om,Jensen measures, hyperconvexity and boundary behaviour of the pluricomplex Green functions, Ann. Polon. Math.71(1999), 87–103

    1999

  5. [13]

    Cegrell,The general definition of the complex Monge–Ampère operator, Ann

    U. Cegrell,The general definition of the complex Monge–Ampère operator, Ann. Inst. Fourier (Grenoble)54(2004), no. 1, 159–179

    2004

  6. [14]

    Cegrell,A general Dirichlet problem for the complex Monge-Ampère operator, Ann

    U. Cegrell,A general Dirichlet problem for the complex Monge-Ampère operator, Ann. Polon. Math.94(2008), no. 2, 131–147

    2008

  7. [15]

    Cordero-Erausquin and A

    D. Cordero-Erausquin and A. Rashkovskii,Plurisubharmonic geodesics and interpola- tion, Arch. Math.113(2019), no. 1, 63–72

    2019

  8. [16]

    Darvas,The Mabuchi Completion of the Space of K¨ ahler Potentials, Amer

    T. Darvas,The Mabuchi Completion of the Space of K¨ ahler Potentials, Amer. J. Math.139(2017), no. 5, 1275–1313

    2017

  9. [17]

    Darvas,Weak geodesic rays in the space of K¨ ahler potentials and the class E(X, ω), J

    T. Darvas,Weak geodesic rays in the space of K¨ ahler potentials and the class E(X, ω), J. Inst. Math. Jussieu16(2017), No. 4, 837–858

    2017

  10. [18]

    Darvas, E

    T. Darvas, E. Di Nezza, C.H. Lu,On the singularity type of full mass currents in big cohomology classes, Compos. Math.154(2018), no. 2, 380–409

    2018

  11. [19]

    Darvas, E

    T. Darvas, E. Di Nezza, C.H. Lu,Monotonicity of nonpluripolar products and complex Monge-Ampère equations with prescribed singularity, Anal. PDE11(2018), no. 8, 2049–2087

    2018

  12. [20]

    Darvas, E

    T. Darvas, E. Di Nezza, C.H. Lu, L1 metric geometry of big cohomology classes, Ann. Inst. Fourier (Grenoble)68(2018), no. 7, 3053–3086

    2018

  13. [21]

    Darvas, E

    T. Darvas, E. Di Nezza, C.H. Lu,The metric geometry of singularity types, J. Reine Angew. Math.771(2021), 137–170

    2021

  14. [22]

    Darvas, E

    T. Darvas, E. Di Nezza, C.H. Lu,Log-concavity of volume and complex Monge-Ampère equations with prescribed singularity, Math. Ann.379(2021), No. 1-2, 95–132

    2021

  15. [23]

    Darvas, E

    T. Darvas, E. Di Nezza, C.H. Lu,Geodesic distance and Monge-Ampère measures on contact sets, Anal. Math.48(2022), no. 2, 451–488

    2022

  16. [24]

    Darvas, E

    T. Darvas, E. Di Nezza, C.H. Lu,Relative pluripotential theory on compact K¨ ahler manifolds, Pure Appl. Math. Q.21(2025), no. 3, 1037–1118

    2025

  17. [25]

    Darvas and Y.A

    T. Darvas and Y.A. Rubinstein,Kiselman ’s principle, the Dirichlet problem for the Monge-Ampère equation, and rooftop obstacle problems, J. Math. Soc. Japan68(2016), No. 2, 773–796

    2016

  18. [26]

    Demailly,Mesures de Monge Ampere et mesures pluriharmoniques, Math

    J.-P. Demailly,Mesures de Monge Ampere et mesures pluriharmoniques, Math. Z.194 (1987), 519–564

    1987

  19. [27]

    Wikstr¨ om,Jensen Measures and approximation of plurisub- harmonic functions, Michigan Math

    Nguyen Quang Dieu and F. Wikstr¨ om,Jensen Measures and approximation of plurisub- harmonic functions, Michigan Math. J.53(2005), 529–544. 24 ELEONORA DI NEZZA AND ALEXANDER RASHKOVSKII

    2005

  20. [28]

    Di Nezza and S

    E. Di Nezza and S. Trapani,Monge-Ampère measures on contact sets, Math. Res. Lett.28(2021), no. 5, 1337–1352

    2021

  21. [29]

    Di Nezza and S

    E. Di Nezza and S. Trapani,The regularity of envelopes, Annales scientifiques de l’ENS57, issue 5 (2024)

    2024

  22. [30]

    Internat

    D.A Edwards,Choquet boundary theory for certain spaces of lower semicontinuous functions, inFunction Algebras(Proc. Internat. Sympos. on Function Algebras, Tulane Univ., 1965) (Birtel, F., ed.), 300–309, Scott-Foresman, Chicago, Ill., 1966

    1965

  23. [31]

    G¨ o˘ g¨ u¸ s,Continuity of plurisubharmonic envelopes, Ann

    N.G. G¨ o˘ g¨ u¸ s,Continuity of plurisubharmonic envelopes, Ann. Pol. Math.86(2005), No. 3, 197–217

    2005

  24. [32]

    G¨ o˘ g¨ u¸ s, T

    N.G. G¨ o˘ g¨ u¸ s, T. Perkins, E. Poletsky,Non-compact versions of Edwards’ theorem, Positivity17(2013), No. 3, 459–473

    2013

  25. [33]

    Guan,On modified Mabuchi functional and Mabuchi moduli space of K¨ ahler metrics on toric bundles, Math

    D. Guan,On modified Mabuchi functional and Mabuchi moduli space of K¨ ahler metrics on toric bundles, Math. Res. Lett.6(1999), no. 5-6, 547–555

    1999

  26. [34]

    Guedj and A

    V. Guedj and A. Zeriahi,Dirichlet problem on domains of Cn, in: Guedj, Vincent (ed.), Complex Monge–Ampère equations and geodesics in the space of K¨ ahler metrics. Lecture notes. Berlin: Springer. Lecture Notes in Mathematics 2038, 13–32, 2012

  27. [35]

    Guedj and A

    V. Guedj and A. Zeriahi, Degenerate complex Monge-Ampère equations. EMS Tracts in Mathematics 26. Z¨ urich: European Mathematical Society (EMS), 2017. 472 p

    2017

  28. [36]

    Guedj, C

    V. Guedj, C. H. Lu, A. Zeriahi,Plurisubharmonic envelopes and supersolutions, J. Differential Geom.113(2019), no. 2, 273–313

    2019

  29. [37]

    Kiselman,The partial Legendre transformation for plurisubharmonic functions, Invent

    C. Kiselman,The partial Legendre transformation for plurisubharmonic functions, Invent. Math.49(1978), no.2, 137–148

    1978

  30. [38]

    Lárusson and R

    F. Lárusson and R. Sigurdsson,Plurisubharmonic functions and analytic discs on manifolds, J. Reine Angew. Math.501(1998), 1–39

    1998

  31. [39]

    Mabuchi,Some symplectic geometry on compact K¨ ahler manifolds

    T. Mabuchi,Some symplectic geometry on compact K¨ ahler manifolds. I, Osaka J. Math.24(1987), no. 2, 227 – 252

    1987

  32. [40]

    Magnússon,Extremal ω-plurisubharmonic functions as envelopes of disc functionals, Ark

    B. Magnússon,Extremal ω-plurisubharmonic functions as envelopes of disc functionals, Ark. Mat.49(2011), No. 2, 383–399

    2011

  33. [41]

    Magnússon,Extremal ω-plurisubharmonic functions as envelopes of disc functionals: generalization and applications to the local theory, Math

    B. Magnússon,Extremal ω-plurisubharmonic functions as envelopes of disc functionals: generalization and applications to the local theory, Math. Scand.111(2012), No. 2, 296–319

    2012

  34. [42]

    Nilsson and F

    M. Nilsson and F. Wikstr¨ om,Quasibounded plurisubharmonic functions, Int. J. Math. 32(2021), No. 9, Article ID 2150068, 16 p

    2021

  35. [43]

    Nilsson and F

    M. Nilsson and F. Wikstr¨ om,Variations on a theorem by Edwards, Ark. Mat.63 (2025), no. 2, 391–405

    2025

  36. [44]

    Poletsky,Plurisubharmonic functions as solutions of variational problems, in Several Complex Variables and Complex Geometry, Part 1 (Santa Cruz, Calif., 1989) (E

    E. Poletsky,Plurisubharmonic functions as solutions of variational problems, in Several Complex Variables and Complex Geometry, Part 1 (Santa Cruz, Calif., 1989) (E. Bedford, J.P. D’Angelo, R.E. Greene, and S.G. Krantz, eds.), 163–171, Amer. Math. Soc., Providence, R. I., 1991

    1989

  37. [45]

    Poletsky,Holomorphic currents, Indiana Univ

    E. Poletsky,Holomorphic currents, Indiana Univ. Math. J.42(1993), 85–144

    1993

  38. [46]

    Rashkovskii,Relative types and extremal problems for plurisubharmonic functions, Int

    A. Rashkovskii,Relative types and extremal problems for plurisubharmonic functions, Int. Math. Res. Not., 2006, Art. ID 76283, 26 pp

    2006

  39. [47]

    Rashkovskii,Local geodesics for plurisubharmonic functions, Math

    A. Rashkovskii,Local geodesics for plurisubharmonic functions, Math. Z.287(2017), no. 1-2, 73–83

    2017

  40. [48]

    Rashkovskii,Copolar convexity, Ann

    A. Rashkovskii,Copolar convexity, Ann. Polon. Math.120(2017), no. 1, 83–95

    2017

  41. [49]

    Rashkovskii,Interpolation of weighted extremal functions, Arnold Math

    A. Rashkovskii,Interpolation of weighted extremal functions, Arnold Math. J.7(2021), No. 3, 407–417

    2021

  42. [50]

    Rashkovskii,Rooftop envelopes and residual plurisubharmonic functions, Ann

    A. Rashkovskii,Rooftop envelopes and residual plurisubharmonic functions, Ann. Pol. Math.128(2022), No. 2, 159–191

    2022

  43. [51]

    Rashkovskii,Plurisubharmonic interpolation and plurisubharmonic geodesics, Ax- ioms12(2023); arXiv:2306.11403

    A. Rashkovskii,Plurisubharmonic interpolation and plurisubharmonic geodesics, Ax- ioms12(2023); arXiv:2306.11403

    2023

  44. [52]

    Ross and D

    J. Ross and D. Witt Nystr¨ om,Analytic test configurations and geodesic rays, J. Symplectic Geom.12(2014), no. 1, 125–169. REGULARITY OF ROOFTOP ENVELOPES 25

    2014

  45. [53]

    Semmes,Complex Monge-Ampère and symplectic manifolds, Amer

    S. Semmes,Complex Monge-Ampère and symplectic manifolds, Amer. J. Math.114:3, (1992), 495–550

    1992

  46. [54]

    Sibony,Une classe de domaines pseudoconvexes, Duke Math

    N. Sibony,Une classe de domaines pseudoconvexes, Duke Math. J.55(1987), 299–319

    1987

  47. [55]

    Tosatti,Regularity of envelopes in K¨ ahler classes, Math

    V. Tosatti,Regularity of envelopes in K¨ ahler classes, Math. Res. Lett.25(2018), no. 1, 281–289

    2018

  48. [56]

    Walsh,Continuity of Envelopes of Plurisubharmonic Functions, J

    J.B. Walsh,Continuity of Envelopes of Plurisubharmonic Functions, J. Math. Mech. 18(1968), 143–148

    1968

  49. [57]

    Wikstr¨ om,Jensen measures and boundary values of plurisubharmonic functions, Ark

    F. Wikstr¨ om,Jensen measures and boundary values of plurisubharmonic functions, Ark. Mat.39(2001), 181–200. Dipartimento di Matematica, Università degli Studi di Roma Tor Vergata, Via della Ricerca Scientifica 1, 00133 Roma, Italy Email address:dinezza@mat.uniroma2.it Departm...

    2001

Pith tools

Reviewed May 10, 2026 · model on record in the stance chip above.