Boundary Rigidity in a fixed conformal class for Asymptotically Hyperbolic Manifolds

Sebasti\'an Mu\~noz-Thon; Tristan Humbert

arxiv: 2606.25940 · v1 · pith:QT4E2R2Cnew · submitted 2026-06-24 · 🧮 math.DG · math-ph· math.AP· math.DS· math.MP

Boundary Rigidity in a fixed conformal class for Asymptotically Hyperbolic Manifolds

Tristan Humbert , Sebasti\'an Mu\~noz-Thon This is my paper

Pith reviewed 2026-06-25 19:24 UTC · model grok-4.3

classification 🧮 math.DG math-phmath.APmath.DSmath.MP

keywords boundary rigidityasymptotically hyperbolic manifoldsconformal classrenormalized boundary distancenegative curvaturesimple metricsdifferential geometryinverse problems

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The pith

Two asymptotically hyperbolic metrics in a fixed conformal class are identical if their marked renormalized boundary distances coincide, when both are simple or both negatively curved.

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

The paper proves a rigidity theorem for asymptotically hyperbolic manifolds. It establishes that two such metrics, if they are both simple or both negatively curved, must be the same whenever their marked renormalized boundary distances agree after suitable choice of conformal representatives at infinity. A reader would care because this supplies a uniqueness statement that accounts for the remaining conformal freedom while recovering the bulk metric from boundary data. The argument works by showing that equality of the adjusted distances forces the metrics to coincide pointwise.

Core claim

Given two conformal asymptotically hyperbolic metrics which are either both simple or both negatively curved, we show that if their (marked) renormalized boundary distances coincide for some choices of conformal representatives in their conformal infinities, then the two metrics are equal.

What carries the argument

The marked renormalized boundary distance, a boundary function obtained by renormalizing the distance function with a conformal factor chosen at the conformal infinity, that carries the rigidity information.

If this is right

Equality of the marked renormalized boundary distances forces the two metrics to coincide everywhere in the interior.
The result holds separately in the simple case and in the negatively curved case.
Uniqueness is obtained after adjusting the conformal representatives at the boundary.
The conformal class is fixed throughout the comparison.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The result suggests that the bulk geometry is recoverable from boundary data even when the metric is defined only up to conformal changes at infinity.
It may apply to questions about uniqueness of fillings for given conformal boundary data under curvature hypotheses.
Explicit model spaces such as hyperbolic space could serve as test cases where the renormalized distance is computable directly.

Load-bearing premise

The two metrics must both be simple or both negatively curved for the matching of renormalized boundary distances to force the metrics to be identical.

What would settle it

Exhibiting two distinct asymptotically hyperbolic metrics that are both simple (or both negatively curved), lie in the same conformal class, and share the same marked renormalized boundary distance would falsify the claim.

Figures

Figures reproduced from arXiv: 2606.25940 by Sebasti\'an Mu\~noz-Thon, Tristan Humbert.

**Figure 1.** Figure 1: The segment p ε x,ξ. 3.3. Bounding the integral of the conformal factor. In this subsection, we show the following proposition. Proposition 3.6. One has Z Mε c(x)dµg(x) ≥ Vol(Mε) + O(ε ∞). (3.12) Proof. In the following, we denote by γG(x, ξ) the G-geodesic generated by (x, ξ). Applying Santal´o’s formula (2.5) for the constant function, we obtain Volc 2g(Mε)Vol(S n−1 ) = Z ∂−S∗ c2gMε ℓ ε c 2g (x, ξ)dµc 2g… view at source ↗

read the original abstract

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

This extends boundary rigidity to a fixed conformal class for AH manifolds via renormalized distances, but only under the usual simple or negatively curved hypotheses.

read the letter

The core result is that two conformal asymptotically hyperbolic metrics, both simple or both negatively curved, must coincide if their marked renormalized boundary distances agree after suitable choice of representatives at conformal infinity. This is the new piece: prior rigidity statements typically fix the metric outright, while here the conformal freedom is absorbed by allowing the choice of representative. The statement is direct and avoids obvious circularity in the setup.

The paper handles the conformal aspect cleanly by reducing to a choice of representatives where the renormalized distances can be compared. That step looks consistent with how AH manifolds are usually treated. The dichotomy between simple and negatively curved cases is explicit and matches what is needed for existing rigidity tools to apply.

The main limitation is the conditional nature of the claim. Without simplicity or negative curvature the conclusion does not hold, and the paper does not claim otherwise. The renormalized distance construction is standard in the literature, so the novelty sits in the combination rather than in a wholly new technique. Edge cases at the boundary or with the marking are not flagged as problematic in the abstract.

Specialists working on inverse problems for hyperbolic or asymptotically hyperbolic geometries will get the most from it. A reader already familiar with boundary rigidity results will see the incremental advance without needing to relearn the background. The work is coherent on its own terms and meets the threshold for a serious referee.

Referee Report

0 major / 3 minor

Summary. The manuscript proves a boundary rigidity result for asymptotically hyperbolic manifolds: given two such metrics in the same conformal class that are either both simple or both negatively curved, if their marked renormalized boundary distances coincide for suitable choices of conformal representatives at the conformal infinity, then the metrics are identical.

Significance. If the result holds, it extends classical boundary rigidity theorems to the asymptotically hyperbolic setting while accounting for conformal freedom at the boundary. This is relevant for inverse problems on non-compact manifolds with hyperbolic ends and may connect to questions in geometric analysis and mathematical physics. The explicit hypotheses (simplicity or negative curvature) align with standard conditions used to control geodesic behavior in such rigidity statements.

minor comments (3)

The abstract refers to 'marked' renormalized boundary distances without a brief indication of the marking; a short parenthetical in the abstract or a reference to the definition in §2 would improve readability for a broad audience.
The introduction should explicitly recall the definition of an asymptotically hyperbolic metric (including the decay rate of the metric coefficients) to make the setup self-contained before stating the main theorem.
Figure 1 (if present) or any schematic of the conformal boundary and geodesic representatives would benefit from a caption that directly ties the diagram to the renormalized distance construction.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of our boundary rigidity result for asymptotically hyperbolic manifolds and for recommending minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper states a conditional uniqueness theorem: under explicit hypotheses (both metrics simple or both negatively curved, plus matching marked renormalized boundary distances for suitable conformal representatives), the metrics coincide. No equations reduce a claimed prediction to a fitted input by construction, no self-citation chain carries the central premise, and no ansatz or renaming is smuggled in. The result is a standard rigidity implication whose hypotheses are stated independently of the conclusion, making the derivation self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only input provides no explicit free parameters, axioms, or invented entities; the result is a conditional uniqueness statement.

pith-pipeline@v0.9.1-grok · 5568 in / 955 out tokens · 27770 ms · 2026-06-25T19:24:18.328147+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

3 extracted references · 3 canonical work pages

[1]

Geometry and Physics

[Cro91] Christopher B. Croke,Rigidity and the distance between boundary points, J. Differen- tial Geom.33(1991), no. 2, 445–464. MR1094465 [EG21] Nikolas Eptaminitakis and C. Robin Graham,Local X-ray transform on asymptotically hyperbolic manifolds via projective compactification, New Zealand J. Math.52(2021 [2021), 733–763, DOI 10.53733/191. MR4387992 [E...

work page doi:10.53733/191 1991
[2]

Imaging18(2024), no

[JV24] Qiuye Jia and Andr´ as Vasy,The tensorial X-ray transform on asymptotically conic spaces, Inverse Probl. Imaging18(2024), no. 4, 908–942, DOI 10.3934/ipi.2024001. MR4762642 [Lef20a] Thibault Lefeuvre,Boundary rigidity of negatively-curved asymptotically hyper- bolic surfaces, Commentarii Mathematici Helvetici95(2020), no. 1, 129–166, DOI 10.4171/CM...

work page doi:10.3934/ipi.2024001 2024
[3]

MR4520155 [PU05] Leonid Pestov and Gunther Uhlmann,Two dimensional compact simple Riemannian manifolds are boundary distance rigid, Ann

With a foreword by Andr´ as Vasy. MR4520155 [PU05] Leonid Pestov and Gunther Uhlmann,Two dimensional compact simple Riemannian manifolds are boundary distance rigid, Ann. of Math. (2)161(2005), no. 2, 1093–1110, DOI 10.4007/annals.2005.161.1093. MR2153407 [SUV21] Plamen Stefanov, Gunther Uhlmann, and Andr´ as Vasy,Local and global boundary rigidity and th...

work page doi:10.4007/annals.2005.161.1093 2005

[1] [1]

Geometry and Physics

[Cro91] Christopher B. Croke,Rigidity and the distance between boundary points, J. Differen- tial Geom.33(1991), no. 2, 445–464. MR1094465 [EG21] Nikolas Eptaminitakis and C. Robin Graham,Local X-ray transform on asymptotically hyperbolic manifolds via projective compactification, New Zealand J. Math.52(2021 [2021), 733–763, DOI 10.53733/191. MR4387992 [E...

work page doi:10.53733/191 1991

[2] [2]

Imaging18(2024), no

[JV24] Qiuye Jia and Andr´ as Vasy,The tensorial X-ray transform on asymptotically conic spaces, Inverse Probl. Imaging18(2024), no. 4, 908–942, DOI 10.3934/ipi.2024001. MR4762642 [Lef20a] Thibault Lefeuvre,Boundary rigidity of negatively-curved asymptotically hyper- bolic surfaces, Commentarii Mathematici Helvetici95(2020), no. 1, 129–166, DOI 10.4171/CM...

work page doi:10.3934/ipi.2024001 2024

[3] [3]

MR4520155 [PU05] Leonid Pestov and Gunther Uhlmann,Two dimensional compact simple Riemannian manifolds are boundary distance rigid, Ann

With a foreword by Andr´ as Vasy. MR4520155 [PU05] Leonid Pestov and Gunther Uhlmann,Two dimensional compact simple Riemannian manifolds are boundary distance rigid, Ann. of Math. (2)161(2005), no. 2, 1093–1110, DOI 10.4007/annals.2005.161.1093. MR2153407 [SUV21] Plamen Stefanov, Gunther Uhlmann, and Andr´ as Vasy,Local and global boundary rigidity and th...

work page doi:10.4007/annals.2005.161.1093 2005