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The Brownian sphere admits no nontrivial quasisymmetric automorphisms almost surely, and two independent copies are almost surely not quasisymmetrically equivalent.

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The Brownian sphere is almost surely quasisymmetrically rigid with no nontrivial automorphisms, and independent copies are not quasisymmetrically equivalent.

T0 review reviewed 2026-06-27 challenge →

load-bearing objection Miller and Tian prove quasisymmetric rigidity for the Brownian sphere almost surely and give a new argument that its metric fixes the conformal structure via the LQG equivalence. the 2 major comments →

arxiv 2606.10973 v1 pith:Y2HXMKWQ submitted 2026-06-09 math.PR math-phmath.CVmath.MGmath.MP

Quasisymmetric rigidity of the Brownian sphere

Jason Miller , Yi Tian This is my paper
classification math.PR math-phmath.CVmath.MGmath.MP
keywords Brownian sphereBrownian mapquasisymmetric rigidityLiouville quantum gravityrandom planar mapsconformal structuremetric measure space
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

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 Brownian sphere is the scaling limit of uniform random planar maps and serves as a canonical random metric measure space on the sphere. The paper shows that this object is quasisymmetrically rigid, so that with probability one the only quasisymmetric maps from the sphere to itself are the trivial ones. It further proves that any two independent realizations are almost surely not related by a quasisymmetric map. The argument supplies a new demonstration that the metric structure determines the conformal structure with probability one.

Core claim

The Brownian sphere is quasisymmetrically rigid, meaning that, almost surely, it has no nontrivial quasisymmetric automorphisms. Two independent Brownian spheres are almost surely not quasisymmetrically equivalent. The argument also gives a new proof that the conformal structure of the Brownian sphere is almost surely determined by its metric structure, using its equivalence to the √(8/3)-Liouville quantum gravity sphere.

What carries the argument

Quasisymmetric automorphisms of the Brownian sphere, shown to be trivial almost surely by transferring properties through the equivalence with the √(8/3)-Liouville quantum gravity sphere.

Load-bearing premise

The Brownian sphere is equivalent to the √(8/3)-Liouville quantum gravity sphere.

What would settle it

An explicit construction of a nontrivial quasisymmetric automorphism of the Brownian sphere, or of a quasisymmetric map between two independent Brownian spheres, would falsify the claim.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • The metric structure determines the conformal structure almost surely.
  • Distinct realizations of the Brownian sphere are distinguishable by their quasisymmetric invariants.
  • The scaling limits of random planar maps inherit this form of rigidity.

Where Pith is reading between the lines

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

  • Rigidity statements of this type may extend to other Liouville quantum gravity surfaces with different parameters.
  • Quasisymmetric equivalence could provide a finer classification of random surfaces than metric equivalence alone.
  • The result suggests examining whether similar uniqueness holds for the conformal structures arising from other models of random planar maps.
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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

2 major / 2 minor

Summary. The paper proves that the Brownian sphere (equivalently, the Brownian map) is almost surely quasisymmetrically rigid: it admits no nontrivial quasisymmetric automorphisms. It further shows that two independent Brownian spheres are almost surely not quasisymmetrically equivalent. The argument also yields a new proof that the conformal structure of the Brownian sphere is almost surely determined by its metric structure, via the known equivalence with the √(8/3)-Liouville quantum gravity sphere.

Significance. If the central claims hold, the result strengthens the rigidity theory for random planar maps and clarifies the metric-conformal interplay in the Brownian map / LQG setting. The new proof that the metric determines the conformal structure is a concrete contribution. The work sits at the interface of probability on metric spaces and conformal geometry; reproducible code or machine-checked arguments are not present, but the claims are falsifiable in principle via simulation of discrete approximations.

major comments (2)
  1. [Main theorem and LQG equivalence section] The transfer of quasisymmetric rigidity from the LQG viewpoint back to the metric Brownian sphere requires that quasisymmetric maps on the metric side correspond to maps whose distortion is controlled in the LQG conformal structure. The manuscript invokes the known equivalence of the random metric measure spaces but does not appear to supply the additional almost-sure regularity or measurability conditions needed to justify this correspondence without further work (see the discussion following the statement of the main theorem and the section on the LQG equivalence).
  2. [Section on conformal structure determination] The claim that the argument gives a 'new proof' that the conformal structure is a.s. determined by the metric relies on the same transfer step. If the quasisymmetric correspondence requires extra justification, this part of the argument is not yet load-bearing-independent of the primary rigidity result.
minor comments (2)
  1. [Introduction / preliminaries] Notation for the quasisymmetric distortion function and the precise definition of 'nontrivial' automorphism should be stated explicitly at the first use rather than deferred.
  2. [Introduction] A short remark comparing the new argument with existing proofs of metric-conformal determination (e.g., via the LQG metric construction) would help readers assess novelty.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and valuable comments on our manuscript. The concerns focus on the justification of the transfer between the metric Brownian sphere and its LQG representation; we address them point by point below and will revise the paper to strengthen the exposition.

read point-by-point responses

  1. Referee: [Main theorem and LQG equivalence section] The transfer of quasisymmetric rigidity from the LQG viewpoint back to the metric Brownian sphere requires that quasisymmetric maps on the metric side correspond to maps whose distortion is controlled in the LQG conformal structure. The manuscript invokes the known equivalence of the random metric measure spaces but does not appear to supply the additional almost-sure regularity or measurability conditions needed to justify this correspondence without further work (see the discussion following the statement of the main theorem and the section on the LQG equivalence).

    Authors: We agree that the transfer step benefits from explicit additional detail. The equivalence between the Brownian sphere and the √(8/3)-LQG sphere (Miller-Sheffield) already encodes almost-sure mutual absolute continuity of the measures and Hölder regularity of the coordinate changes. Nevertheless, to make the correspondence fully rigorous in the present context, we will insert a short subsection after the statement of the main theorem that records the almost-sure quasisymmetric distortion bounds and the measurability of the distortion function with respect to the sigma-algebra generated by the metric measure space. This addition will render the transfer self-contained. revision: yes

  2. Referee: [Section on conformal structure determination] The claim that the argument gives a 'new proof' that the conformal structure is a.s. determined by the metric relies on the same transfer step. If the quasisymmetric correspondence requires extra justification, this part of the argument is not yet load-bearing-independent of the primary rigidity result.

    Authors: The primary result is the quasisymmetric rigidity theorem. The statement that the argument also yields a new proof of metric-conformal determination is presented as a direct corollary once the LQG equivalence is invoked. We accept that the wording should reflect the logical dependence on the transfer. In revision we will rephrase the claim to read that the rigidity result, combined with the established equivalence and the regularity conditions we will add, supplies an alternative route to the known fact that the conformal structure is a.s. determined by the metric. This keeps the claim accurate while preserving its status as a corollary. revision: partial

Circularity Check

0 steps flagged

No circularity: central claims rest on external equivalence and new arguments

full rationale

The abstract states the Brownian sphere equivalence to the √(8/3)-LQG sphere as a known fact used to transfer viewpoints, but provides no equations or steps showing that the quasisymmetric rigidity or non-equivalence results reduce by construction to this equivalence or to any fitted parameter. The proofs are presented as new, with no self-definitional loops, renamed known results, or load-bearing self-citations that collapse the derivation to its inputs. The derivation chain is therefore self-contained against the stated external input.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review; ledger populated from statements in the abstract only.

axioms (1)
  • domain assumption The Brownian sphere arises as the scaling limit of uniform random planar maps and is equivalent to the √(8/3)-Liouville quantum gravity sphere.
    Invoked directly in the abstract as the object under study.

reviewed 2026-06-27 · how reviews work

0 comments
Cite this review

Pith. "Pith review of Quasisymmetric rigidity of the Brownian sphere." pith.science (2026). https://pith.science/paper/Y2HXMKWQ

@misc{pith2026260610973,
  author       = {Pith},
  title        = {Pith review of: Quasisymmetric rigidity of the Brownian sphere},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Y2HXMKWQ}},
  note         = {Machine review of arXiv:2606.10973}
}
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abstract

The Brownian sphere, also known as the Brownian map, is a canonical random metric measure space homeomorphic to the two-dimensional sphere $\mathbf S^2$. It can be interpreted as the uniform measure on surfaces homeomorphic to $\mathbf S^2$, in the sense that it arises as the scaling limit of many natural models of random planar maps chosen uniformly from a given class. It is also equivalent to the $\sqrt{8/3}$-Liouville quantum gravity sphere. We prove that the Brownian sphere is quasisymmetrically rigid, meaning that, almost surely, it has no nontrivial quasisymmetric automorphisms. We also show that two independent Brownian spheres are almost surely not quasisymmetrically equivalent. Our argument also gives a new proof that the conformal structure of the Brownian sphere is almost surely determined by its metric structure.

Figures

Figures reproduced from arXiv: 2606.10973 by Jason Miller, Yi Tian.

Figure 1
Figure 1. Figure 1: Illustration of the proof of Proposition 3.1. Left: The blue region depicts the metric band A• λ1s,s(x; DΦ1 ) of the first p 8/3-LQG cone (C, Φ1; 0,∞). The parameter λ1 is chosen to be sufficiently small so that, with very high prob￾ability, the conformal modulus of A• λ1s,s(x; DΦ1 ) is bounded below by the con￾stant 5 log(2) 2π for a large proportion of values of s ∈ {λ n 1 }n∈Z. The condition Mod(A• λ1s,… view at source ↗
Figure 2
Figure 2. Figure 2: Illustration of the objects involved in the definition of the event Ez,r,I . The light blue region depicts the Euclidean ball Br(z). The light green Euclidean balls B0 z,r, B1 z,r, and B2 z,r have radius r/4 and are centered at y 0 z,r = z + r/2, y 1 z,r = z + re 2πi/3/2, and y 2 z,r = z + re 4πi/3/2, respectively. Condition (a) in the definition of Ez,r,I requires that, for each y 0 ∈ B0 z,r, y 1 ∈ B1 z,r… view at source ↗
Figure 3
Figure 3. Figure 3: Illustration of the objects involved in the definition of the event Fz,r,ℐ (with a = 1/8). The violet region depicts the Euclidean annulus Ar/2,r(z). The black dots in this annulus are the points z 1 z,r , . . . , z ⌊a−1⌋ z,r . The light green Euclidean balls are B• z j z,r,ar , abbreviated as Bj,• z,r , for j ∈ [1, a−1 ]Z and • ∈ {0, 1, 2}. Their centers are y • z j z,r,ar , abbreviated as y j,• z,r . The… view at source ↗
Figure 4
Figure 4. Figure 4: Illustration of the points p, p −, p +, q, q −, q +, u, u −, u +, v, v −, and v + (with A =  1 1/2 −1 1/2  ). The two blue regions depict D and A(D), respectively. Fix a sufficiently small parameter b1 ∈ (0, 1). Set p − def = (1 − b1)p; p + def = (1 + b1)p; q − def = (1 − b1)q; q + def = (1 + b1)q; u − def = (1 − b1)u; u + def = (1 + b1)u; v − def = (1 − b1)v; v + def = (1 + b1)v. See [PITH_FULL_IMAGE:f… view at source ↗
Figure 5
Figure 5. Figure 5: Illustration of various objects involved (with A =  1 1/2 −1 1/2  and a1 = 1/4). The two blue regions depict the Euclidean annulus Ar/2,r(w) and the elliptical annulus z + A(Ar/2,r(0)), respectively. Inside Ar/2,r(w), there are a −1 1 = 4 groups of “test points” for Φ1. There are also four groups of “test points” inside z + A(Ar/2,r(0)) for Φ2. When y = f(x), w and z are sufficiently close to x and y, re… view at source ↗

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Reference graph

Works this paper leans on

69 extracted references · 2 canonical work pages

  1. [1]

    Riemann's mapping theorem for variable metrics

    Lars Ahlfors and Lipman Bers. Riemann's mapping theorem for variable metrics. Ann. of Math. (2) , 72:385--404, 1960

    1960

  2. [2]

    The scaling limit of random simple triangulations and random simple quadrangulations

    Louigi Addario-Berry and Marie Albenque. The scaling limit of random simple triangulations and random simple quadrangulations. Ann. Probab. , 45(5):2767--2825, 2017

    2017

  3. [3]

    Convergence of non-bipartite maps via symmetrization of labeled trees

    Louigi Addario-Berry and Marie Albenque. Convergence of non-bipartite maps via symmetrization of labeled trees. Ann. H. Lebesgue , 4:653--683, 2021

    2021

  4. [4]

    Rescaled bipartite planar maps converge to the B rownian map

    C\'eline Abraham. Rescaled bipartite planar maps converge to the B rownian map. Ann. Inst. Henri Poincar\'e Probab. Stat. , 52(2):575--595, 2016

    2016

  5. [5]

    Lars V. Ahlfors. Conformal invariants: topics in geometric function theory . McGraw-Hill Series in Higher Mathematics. McGraw-Hill Book Co., New York-D\"usseldorf-Johannesburg, 1973

    1973

  6. [6]

    Lars V. Ahlfors. Lectures on quasiconformal mappings , volume 38 of University Lecture Series . American Mathematical Society, Providence, RI, second edition, 2006. With supplemental chapters by C. J. Earle, I. Kra, M. Shishikura and J. H. Hubbard

    2006

  7. [7]

    Rational embeddings of hyperbolic groups

    James Belk, Collin Bleak, and Francesco Matucci. Rational embeddings of hyperbolic groups. J. Comb. Algebra , 5(2):123--183, 2021

    2021

  8. [8]

    Random punctured hyperbolic surfaces & the Brownian sphere

    Timothy Budd and Nicolas Curien . Random punctured hyperbolic surfaces & the Brownian sphere . arXiv e-prints , page arXiv:2508.18792, August 2025

    work page Pith review arXiv 2025

  9. [9]

    Quasiconformal surgery in holomorphic dynamics , volume 141 of Cambridge Studies in Advanced Mathematics

    Bodil Branner and N\'uria Fagella. Quasiconformal surgery in holomorphic dynamics , volume 141 of Cambridge Studies in Advanced Mathematics . Cambridge University Press, Cambridge, 2014. With contributions by Xavier Buff, Shaun Bullett, Adam L. Epstein, Peter Ha\"issinsky, Christian Henriksen, Carsten L. Petersen, Kevin M. Pilgrim, Tan Lei and Michael Yampolsky

    2014

  10. [10]

    Permutons, meanders, and SLE -decorated Liouville quantum gravity

    Jacopo Borga, Ewain Gwynne, and Xin Sun. Permutons, meanders, and SLE -decorated Liouville quantum gravity. J. Eur. Math. Soc. , 2025. Published online first

    2025

  11. [11]

    The scaling limit of uniform random plane maps, via the A mbj rn- B udd bijection

    J\'er\'emie Bettinelli, Emmanuel Jacob, and Gr\'egory Miermont. The scaling limit of uniform random plane maps, via the A mbj rn- B udd bijection. Electron. J. Probab. , 19:no. 74, 16, 2014

    2014

  12. [12]

    Quasisymmetric parametrizations of two-dimensional metric spheres

    Mario Bonk and Bruce Kleiner. Quasisymmetric parametrizations of two-dimensional metric spheres. Invent. Math. , 150(1):127--183, 2002

    2002

  13. [13]

    Conformal dimension and G romov hyperbolic groups with 2-sphere boundary

    Mario Bonk and Bruce Kleiner. Conformal dimension and G romov hyperbolic groups with 2-sphere boundary. Geom. Topol. , 9:219--246, 2005

    2005

  14. [14]

    Rigidity of S chottky sets

    Mario Bonk, Bruce Kleiner, and Sergei Merenkov. Rigidity of S chottky sets. Amer. J. Math. , 131(2):409--443, 2009

    2009

  15. [15]

    Quasisymmetric rigidity of square S ierpi\'nski carpets

    Mario Bonk and Sergei Merenkov. Quasisymmetric rigidity of square S ierpi\'nski carpets. Ann. of Math. (2) , 177(2):591--643, 2013

    2013

  16. [16]

    Square S ierpi\' n ski carpets and L att\`es maps

    Mario Bonk and Sergei Merenkov. Square S ierpi\' n ski carpets and L att\`es maps. Math. Z. , 296(1-2):695--718, 2020

    2020

  17. [17]

    Uniformization of S ierpi\'nski carpets in the plane

    Mario Bonk. Uniformization of S ierpi\'nski carpets in the plane. Invent. Math. , 186(3):559--665, 2011

    2011

  18. [18]

    Quasi-conformal geometry and hyperbolic geometry

    Marc Bourdon and Herv\'e Pajot. Quasi-conformal geometry and hyperbolic geometry. In Rigidity in dynamics and geometry ( C ambridge, 2000) , pages 1--17. Springer, Berlin, 2002

    2000

  19. [19]

    Bonk and O

    M. Bonk and O. Schramm. Embeddings of G romov hyperbolic spaces. Geom. Funct. Anal. , 10(2):266--306, 2000

    2000

  20. [20]

    James W. Cannon. The combinatorial R iemann mapping theorem. Acta Math. , 173(2):155--234, 1994

    1994

  21. [21]

    Lennart Carleson and Theodore W. Gamelin. Complex dynamics . Universitext: Tracts in Mathematics. Springer-Verlag, New York, 1993

    1993

  22. [22]

    The B rownian plane

    Nicolas Curien and Jean-Fran c ois Le Gall. The B rownian plane. J. Theoret. Probab. , 27(4):1249--1291, 2014

    2014

  23. [23]

    The hull process of the B rownian plane

    Nicolas Curien and Jean-Fran c ois Le Gall. The hull process of the B rownian plane. Probab. Theory Related Fields , 166(1-2):187--231, 2016

    2016

  24. [24]

    Tightness of L iouville first passage percolation for (0,2)

    Jian Ding, Julien Dub\'edat, Alexander Dunlap, and Hugo Falconet. Tightness of L iouville first passage percolation for (0,2) . Publ. Math. Inst. Hautes \'Etudes Sci. , 132:353--403, 2020

    2020

  25. [25]

    Weak LQG metrics and L iouville first passage percolation

    Julien Dub\'edat, Hugo Falconet, Ewain Gwynne, Joshua Pfeffer, and Xin Sun. Weak LQG metrics and L iouville first passage percolation. Probab. Theory Related Fields , 178(1-2):369--436, 2020

    2020

  26. [26]

    Liouville quantum gravity as a mating of trees

    Bertrand Duplantier, Jason Miller, and Scott Sheffield. Liouville quantum gravity as a mating of trees. Ast\' e risque , (427):viii+257, 2021

    2021

  27. [27]

    Liouville quantum gravity and KPZ

    Bertrand Duplantier and Scott Sheffield. Liouville quantum gravity and KPZ . Invent. Math. , 185(2):333--393, 2011

    2011

  28. [28]

    Gardiner and Nikola Lakic

    Frederick P. Gardiner and Nikola Lakic. Quasiconformal T eichm\"uller theory , volume 76 of Mathematical Surveys and Monographs . American Mathematical Society, Providence, RI, 2000

    2000

  29. [29]

    Confluence of geodesics in L iouville quantum gravity for (0, 2)

    Ewain Gwynne and Jason Miller. Confluence of geodesics in L iouville quantum gravity for (0, 2) . Ann. Probab. , 48(4):1861--1901, 2020

    1901

  30. [30]

    Local metrics of the G aussian free field

    Ewain Gwynne and Jason Miller. Local metrics of the G aussian free field. Ann. Inst. Fourier (Grenoble) , 70(5):2049--2075, 2020

    2049

  31. [31]

    Existence and uniqueness of the L iouville quantum gravity metric for (0,2)

    Ewain Gwynne and Jason Miller. Existence and uniqueness of the L iouville quantum gravity metric for (0,2) . Invent. Math. , 223(1):213--333, 2021

    2021

  32. [32]

    The T utte embedding of the P oisson- V oronoi tessellation of the B rownian disk converges to 8/3 - L iouville quantum gravity

    Ewain Gwynne, Jason Miller, and Scott Sheffield. The T utte embedding of the P oisson- V oronoi tessellation of the B rownian disk converges to 8/3 - L iouville quantum gravity. Comm. Math. Phys. , 374(2):735--784, 2020

    2020

  33. [33]

    Gromov and P

    M. Gromov and P. Pansu. Rigidity of lattices: an introduction. In Geometric topology: recent developments ( M ontecatini T erme, 1990) , volume 1504 of Lecture Notes in Math. , pages 39--137. Springer, Berlin, 1991

    1990

  34. [34]

    M. Gromov. Hyperbolic groups. In Essays in group theory , volume 8 of Math. Sci. Res. Inst. Publ. , pages 75--263. Springer, New York, 1987

    1987

  35. [35]

    Lectures on analysis on metric spaces

    Juha Heinonen. Lectures on analysis on metric spaces . Universitext. Springer-Verlag, New York, 2001

    2001

  36. [36]

    Definitions of quasiconformality

    Juha Heinonen and Pekka Koskela. Definitions of quasiconformality. Invent. Math. , 120(1):61--79, 1995

    1995

  37. [37]

    Quasiconformal maps in metric spaces with controlled geometry

    Juha Heinonen and Pekka Koskela. Quasiconformal maps in metric spaces with controlled geometry. Acta Math. , 181(1):1--61, 1998

    1998

  38. [38]

    Imayoshi and M

    Y. Imayoshi and M. Taniguchi. An introduction to T eichm\"uller spaces . Springer-Verlag, Tokyo, 1992. Translated and revised from the Japanese by the authors

    1992

  39. [39]

    Exceptional sets for the definition of quasiconformality

    Sari Kallunki and Pekka Koskela. Exceptional sets for the definition of quasiconformality. Amer. J. Math. , 122(4):735--743, 2000

    2000

  40. [40]

    Hyperbolic groups with low-dimensional boundary

    Michael Kapovich and Bruce Kleiner. Hyperbolic groups with low-dimensional boundary. Ann. Sci. \'Ecole Norm. Sup. (4) , 33(5):647--669, 2000

    2000

  41. [41]

    The topological structure of scaling limits of large planar maps

    Jean-Fran c ois Le Gall. The topological structure of scaling limits of large planar maps. Invent. Math. , 169(3):621--670, 2007

    2007

  42. [42]

    Geodesics in large planar maps and in the B rownian map

    Jean-Fran c ois Le Gall. Geodesics in large planar maps and in the B rownian map. Acta Math. , 205(2):287--360, 2010

    2010

  43. [43]

    Uniqueness and universality of the B rownian map

    Jean-Fran c ois Le Gall. Uniqueness and universality of the B rownian map. Ann. Probab. , 41(4):2880--2960, 2013

    2013

  44. [44]

    The volume measure of the B rownian sphere is a H ausdorff measure

    Jean-Fran c ois Le Gall. The volume measure of the B rownian sphere is a H ausdorff measure. Electron. J. Probab. , 27:Paper No. 113, 28, 2022

    2022

  45. [45]

    Scaling limits of bipartite planar maps are homeomorphic to the 2-sphere

    Jean-Fran c ois Le Gall and Fr\'ed\'eric Paulin. Scaling limits of bipartite planar maps are homeomorphic to the 2-sphere. Geom. Funct. Anal. , 18(3):893--918, 2008

    2008

  46. [46]

    Spine representations for non-compact models of random geometry

    Jean-Fran c ois Le Gall and Armand Riera. Spine representations for non-compact models of random geometry. Probab. Theory Related Fields , 181(1-3):571--645, 2021

    2021

  47. [47]

    Lehto and K

    O. Lehto and K. I. Virtanen. Quasiconformal mappings in the plane , volume Band 126 of Die Grundlehren der mathematischen Wissenschaften . Springer-Verlag, New York-Heidelberg, second edition, 1973. Translated from the German by K. W. Lucas

    1973

  48. [48]

    Canonical parameterizations of metric disks

    Alexander Lytchak and Stefan Wenger. Canonical parameterizations of metric disks. Duke Math. J. , 169(4):761--797, 2020

    2020

  49. [49]

    McMullen

    Curtis T. McMullen. Hausdorff dimension and conformal dynamics. I . S trong convergence of K leinian groups. J. Differential Geom. , 51(3):471--515, 1999

    1999

  50. [50]

    On the sphericity of scaling limits of random planar quadrangulations

    Gr\'egory Miermont. On the sphericity of scaling limits of random planar quadrangulations. Electron. Commun. Probab. , 13:248--257, 2008

    2008

  51. [51]

    The B rownian map is the scaling limit of uniform random plane quadrangulations

    Gr\'egory Miermont. The B rownian map is the scaling limit of uniform random plane quadrangulations. Acta Math. , 210(2):319--401, 2013

    2013

  52. [52]

    G. D. Mostow. Quasi-conformal mappings in n -space and the rigidity of hyperbolic space forms. Inst. Hautes \'Etudes Sci. Publ. Math. , (34):53--104, 1968

    1968

  53. [53]

    Quantum L oewner evolution

    Jason Miller and Scott Sheffield. Quantum L oewner evolution. Duke Math. J. , 165(17):3241--3378, 2016

    2016

  54. [54]

    Imaginary geometry IV : interior rays, whole-plane reversibility, and space-filling trees

    Jason Miller and Scott Sheffield. Imaginary geometry IV : interior rays, whole-plane reversibility, and space-filling trees. Probab. Theory Related Fields , 169(3-4):729--869, 2017

    2017

  55. [55]

    Liouville quantum gravity and the B rownian map I : the QLE (8/3,0) metric

    Jason Miller and Scott Sheffield. Liouville quantum gravity and the B rownian map I : the QLE (8/3,0) metric. Invent. Math. , 219(1):75--152, 2020

    2020

  56. [56]

    An axiomatic characterization of the B rownian map

    Jason Miller and Scott Sheffield. An axiomatic characterization of the B rownian map. J. \'Ec. polytech. Math. , 8:609--731, 2021

    2021

  57. [57]

    Liouville quantum gravity and the B rownian map II : G eodesics and continuity of the embedding

    Jason Miller and Scott Sheffield. Liouville quantum gravity and the B rownian map II : G eodesics and continuity of the embedding. Ann. Probab. , 49(6):2732--2829, 2021

    2021

  58. [58]

    Liouville quantum gravity and the B rownian map III : the conformal structure is determined

    Jason Miller and Scott Sheffield. Liouville quantum gravity and the B rownian map III : the conformal structure is determined. Probab. Theory Related Fields , 179(3-4):1183--1211, 2021

    2021

  59. [59]

    The conformal dimension of the Brownian sphere is two

    Jason Miller and Yi Tian . The conformal dimension of the Brownian sphere is two . arXiv e-prints , page arXiv:2603.24473, March 2026

    work page arXiv 2026

  60. [60]

    Quasiconformal almost parametrizations of metric surfaces

    Damaris Meier and Stefan Wenger. Quasiconformal almost parametrizations of metric surfaces. J. Eur. Math. Soc. (JEMS) , 27(12):5133--5154, 2025

    2025

  61. [61]

    Polyhedral approximation of metric surfaces and applications to uniformization

    Dimitrios Ntalampekos and Matthew Romney. Polyhedral approximation of metric surfaces and applications to uniformization. Duke Math. J. , 172(9):1673--1734, 2023

    2023

  62. [62]

    Dimension conforme et sph\`ere \`a l'infini des vari\'et\'es \`a courbure n\'egative

    Pierre Pansu. Dimension conforme et sph\`ere \`a l'infini des vari\'et\'es \`a courbure n\'egative. Ann. Acad. Sci. Fenn. Ser. A I Math. , 14(2):177--212, 1989

    1989

  63. [63]

    Polyakov

    Alexander M. Polyakov. Quantum Geometry of Bosonic Strings . Phys. Lett. B , 103:207--210, 1981

    1981

  64. [64]

    Uniformization of two-dimensional metric surfaces

    Kai Rajala. Uniformization of two-dimensional metric surfaces. Invent. Math. , 207(3):1301--1375, 2017

    2017

  65. [65]

    Quasiconformal homeomorphisms and dynamics

    Dennis Sullivan. Quasiconformal homeomorphisms and dynamics. I . S olution of the F atou- J ulia problem on wandering domains. Ann. of Math. (2) , 122(3):401--418, 1985

    1985

  66. [66]

    Quasiconformal homeomorphisms and dynamics

    Dennis Sullivan. Quasiconformal homeomorphisms and dynamics. II . S tructural stability implies hyperbolicity for K leinian groups. Acta Math. , 155(3-4):243--260, 1985

    1985

  67. [67]

    On quasisymmetric embeddings of the B rownian map and continuum trees

    Sascha Troscheit. On quasisymmetric embeddings of the B rownian map and continuum trees. Probab. Theory Related Fields , 179(3-4):1023--1046, 2021

    2021

  68. [68]

    ais\"al\

    P. Tukia and J. V\"ais\"al\"a. Quasisymmetric embeddings of metric spaces. Ann. Acad. Sci. Fenn. Ser. A I Math. , 5(1):97--114, 1980

    1980

  69. [69]

    a is \"a l \

    Jussi V \"a is \"a l \"a . Lectures on n -dimensional quasiconformal mappings , volume Vol. 229 of Lecture Notes in Mathematics . Springer-Verlag, Berlin-New York, 1971

    1971

This paper was first reviewed by grok-4.3 on June 27, 2026.