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

Regularity of rooftop envelopes

Eleonora Di Nezza , Alexander Rashkovskii This is my paper

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
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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.

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

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)

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Reference graph

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