arxiv: 2604.18855 · v2 · submitted 2026-04-20 · 🧮 math.CV

Recognition: unknown

Regularity of rooftop envelopes

Alexander Rashkovskii, Eleonora Di Nezza

Pith reviewed 2026-05-10 02:32 UTC · model grok-4.3

classification 🧮 math.CV

keywords plurisubharmonic functionsgeodesicsrooftop envelopesHölder regularityC^{1,1} regularitypluripotential theorycomplex domainsMonge-Ampère equation

0 comments

The pith

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.

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

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.

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.

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.

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.

Where Pith is reading between the lines

These are 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.

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.

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.

read the original 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.

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

read the letter

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. The geodesic is defined in the usual way via the homogeneous complex Monge-Ampère equation, and the arguments stay within the standard pluripotential framework on bounded sets in C^n with continuous data. The step from geodesic regularity to the rooftop envelope results is the clearest new piece, and it is carried out directly without extra machinery. The proofs look internally consistent, with assumptions stated explicitly and no sign of circular reasoning or unverified boundary extensions. The work does what it sets out to do under the stated conditions. A minor limitation is that everything is restricted to bounded domains and continuous functions, which is the natural setting here but keeps the results from applying immediately to unbounded or less regular cases. The citation pattern is appropriate and does not rely on self-referential shortcuts. This is for people already working in pluripotential theory or Kähler geometry who need precise regularity statements for these objects. A reader in that subfield will find the explicit theorems and proofs useful. The paper shows clear thinking on its own terms and deserves a serious referee.

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)

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.
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.
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

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.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 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.

pith-pipeline@v0.9.0 · 5313 in / 1070 out tokens · 26107 ms · 2026-05-10T02:32:39.737529+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

57 extracted references · 2 canonical work pages

[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

2019
[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

2021
[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

2025
[4]

Barbu and T

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

2012
[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

1976
[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

2019
[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 arXiv 2022
[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

2015
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[29]

Di Nezza and S

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

2024
[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
[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
[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
[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
[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

2038
[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
[36]

Guedj, C

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

2019
[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
[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
[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
[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
[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
[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
[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
[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
[45]

Poletsky,Holomorphic currents, Indiana Univ

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

1993
[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
[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
[48]

Rashkovskii,Copolar convexity, Ann

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

2017
[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
[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
[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

work page arXiv 2023
[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
[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
[54]

Sibony,Une classe de domaines pseudoconvexes, Duke Math

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

1987
[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
[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
[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 Department of Mathematics and Physics, University of Sta v anger, 4036 Sta v anger, Nor...

2001