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arxiv: 2503.05690 · v3 · submitted 2025-03-07 · 🧮 math-ph · math.CV· math.DG· math.MP· math.PR

Epstein curves and holography of the Schwarzian action

Franco Vargas Pallete , Yilin Wang , Catherine Wolfram This is my paper

Pith reviewed 2026-05-23 00:33 UTC · model grok-4.3

classification 🧮 math-ph math.CVmath.DGmath.MPmath.PR
keywords Schwarzian actionEpstein curveshyperbolic diskLoewner energydiffeomorphisms of the circlecoadjoint orbitsholographic dualityJackiw-Teitelboim gravity
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The pith

The Schwarzian action equals the length of the Epstein curve in the hyperbolic disk minus the area it encloses.

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

The paper applies Epstein's construction of hypersurfaces in the hyperbolic disk to establish direct identities linking the Schwarzian action on PSL_2(R) backslash Diff^3(S^1) to the length and enclosed area of an associated Epstein curve. It further shows that the Schwarzian action equals the derivative of the Loewner energy of the welded Jordan curve. The same relations hold after extension to coadjoint orbits. These geometric equalities immediately yield two proofs of non-negativity for the Schwarzian action, one via the isoperimetric inequality and one via monotonicity of Loewner energy.

Core claim

We apply Epstein's construction of hypersurfaces in the hyperbolic disk D to prove identities between the Schwarzian action on PSL_2(R) backslash Diff^3(S^1), the length of the corresponding Epstein curve in D, and the area enclosed by the Epstein curve. The horocycle truncation in the construction defines a renormalized length of hyperbolic geodesics that coincides with the logarithm of the bi-local observable. The construction extends to the coadjoint orbits PSL_2^(n)(R) backslash Diff^3(S^1) and yields the same identities. The Schwarzian action is the derivative of the Loewner energy of the welded Jordan curve.

What carries the argument

Epstein's hypersurface construction in the hyperbolic disk, which produces an Epstein curve whose length and enclosed area are identical to the Schwarzian action.

If this is right

  • The Schwarzian action is nonnegative, with proofs via the isoperimetric inequality and via monotonicity of the Loewner energy.
  • The same length-and-area identities hold for the analogous action on each coadjoint orbit PSL_2^(n)(R) backslash Diff^3(S^1).
  • The horocycle truncation in the Epstein-curve construction supplies a renormalized geodesic length that equals the logarithm of the bi-local observable of Schwarzian field theory.
  • The Schwarzian action being the derivative of Loewner energy connects the functional directly to the action of Schramm-Loewner evolutions.

Where Pith is reading between the lines

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

  • The length-area identity may permit direct computation of the Schwarzian action by measuring Epstein curves rather than integrating the usual functional.
  • The derivative relation with Loewner energy suggests the Schwarzian action controls the rate of change of energy along certain curve deformations.
  • Because the construction works uniformly on coadjoint orbits, similar geometric interpretations may exist for other orbit-level actions in related theories.

Load-bearing premise

Epstein's hypersurface construction extends to the space PSL_2(R) backslash Diff^3(S^1) and its coadjoint orbits such that the resulting Epstein curve's length and enclosed area directly correspond to the Schwarzian action.

What would settle it

A specific diffeomorphism for which the numerical value of the Schwarzian action differs from the length of its Epstein curve minus the enclosed area.

Figures

Figures reproduced from arXiv: 2503.05690 by Catherine Wolfram, Franco Vargas Pallete, Yilin Wang.

Figure 1
Figure 1. Figure 1: Illustration of part of the Epstein curves associated with a metric [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Left: Epstein curve (orange) and the horocycles (blue) associated [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The horocycles and Epstein curve for φ∗dθ, where φ = (ϕ(z))n , and ϕ −1 (θ) = 1 2 sin(θ) + θ, for n from 1 to 8. 1.2 Relation to Loewner energy The Schwarzian action ISch is invariant under post-composition by Möbius transforma￾tions in PSU(1, 1), hence, it is defined on the space PSU(1, 1)\ Diff3 (S 1 ). This is a sub￾space of the universal Teichmüller space T(1) := PSU(1, 1)\ QS(S 1 ) which is modeled on… view at source ↗
Figure 4
Figure 4. Figure 4: On the right are the Epstein curve and horocycles for the double [PITH_FULL_IMAGE:figures/full_fig_p025_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Example of an ideal triangulation, where only finitely many edges [PITH_FULL_IMAGE:figures/full_fig_p030_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Let h = φ∗dθ, φ piecewise Möbius with breakpoints 1, i, −1, −i. Here are its horocycles in increments of π/25 and completed Epstein curve Σh (left) and its horocycles at zj = 1, i, −1, −i, the intersection points aj = αj (0), and the angles βj between the outward normal vectors to the horocycles at aj (right). Lemma 7.7. Let dℓj denote the signed arclength measure along Σ j h . The total signed length is L… view at source ↗
Figure 7
Figure 7. Figure 7: The orange curve here is the completed Epstein curve for [PITH_FULL_IMAGE:figures/full_fig_p043_7.png] view at source ↗
read the original abstract

We apply Epstein's construction of hypersurfaces in the hyperbolic disk $\mathbb D$ to prove identities between the Schwarzian action on $\operatorname{PSL}_2(\mathbb R)\backslash \mathrm{Diff}^3 (\mathbb S^1)$, the length of the corresponding Epstein curve in $\mathbb D$, and the area enclosed by the Epstein curve. These results are inspired by the holographic duality between Jackiw--Teitelboim gravity and Schwarzian field theory. We also show that the horocycle truncation used in the construction of the Epstein curve defines a renormalized length of hyperbolic geodesics in $\mathbb D$, which coincides with the logarithm of the bi-local observable of Schwarzian field theory. The construction of the Epstein curve also extends to the coadjoint orbits $\operatorname{PSL}_2^{(n)}(\mathbb R)\backslash \mathrm{Diff}^3 (\mathbb S^1)$, and we obtain the same identities for the analog of the Schwarzian action on these coadjoint orbits. Furthermore, we show that the Schwarzian action is the derivative of the Loewner energy of the welded Jordan curve. This energy is the action functional of Schramm--Loewner evolutions and holographically expressed as a renormalized volume in hyperbolic $3$-space. As a by-product of these relations, we obtain two immediate proofs of the non-negativity of the Schwarzian action using the isoperimetric inequality and the monotonicity of the Loewner energy.

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

2 major / 1 minor

Summary. The paper applies Epstein's construction of hypersurfaces in the hyperbolic disk D to establish identities equating the Schwarzian action on PSL_2(R) backslash Diff^3(S^1) (and its coadjoint-orbit analogs) with the hyperbolic length (after horocycle truncation) and enclosed area of the corresponding Epstein curve. It further shows that the Schwarzian action equals the derivative of the Loewner energy of the welded Jordan curve, interprets the truncation as a renormalized geodesic length matching the bi-local observable, and derives two immediate non-negativity proofs via the isoperimetric inequality and Loewner-energy monotonicity. The results are motivated by the JT-gravity/Schwarzian holographic duality.

Significance. If the central identities hold with the claimed rigor, the work supplies explicit geometric realizations of the Schwarzian action inside hyperbolic geometry, together with parameter-free non-negativity proofs and a direct link to Loewner energy. These strengthen the geometric side of the JT/Schwarzian correspondence and furnish falsifiable predictions for the length-area relations on coadjoint orbits.

major comments (2)
  1. [Abstract / Epstein construction paragraph] Abstract and the paragraph introducing the Epstein construction: the extension of Epstein's hypersurface map from suitable embeddings to the infinite-dimensional Fréchet manifold PSL_2(R) backslash Diff^3(S^1) is asserted without explicit uniform bounds on the C^3 norm or control on the developing map/horocycle flow; this leaves open whether the resulting curve is always rectifiable and whether the truncated length and enclosed area remain finite and exactly equal the Schwarzian integral.
  2. [Loewner energy derivative paragraph] Section on the derivative relation to Loewner energy: the statement that the Schwarzian action is the derivative of the Loewner energy of the welded Jordan curve requires a precise specification of the variation (tangent vector in the quotient space) and a verification that the resulting first variation reproduces the Schwarzian functional without additional renormalization terms.
minor comments (1)
  1. [Coadjoint orbits paragraph] Notation for the coadjoint orbits PSL_2^{(n)}(R) backslash Diff^3(S^1) should be introduced with a brief reminder of the integer n and the corresponding stabilizer.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on the manuscript. We address each major point below with clarifications and commit to revisions that strengthen the rigor of the constructions while preserving the core results.

read point-by-point responses

  1. Referee: [Abstract / Epstein construction paragraph] Abstract and the paragraph introducing the Epstein construction: the extension of Epstein's hypersurface map from suitable embeddings to the infinite-dimensional Fréchet manifold PSL_2(R) backslash Diff^3(S^1) is asserted without explicit uniform bounds on the C^3 norm or control on the developing map/horocycle flow; this leaves open whether the resulting curve is always rectifiable and whether the truncated length and enclosed area remain finite and exactly equal the Schwarzian integral.

    Authors: We agree that the presentation would benefit from explicit control. In the revised version we will add a dedicated subsection (or short appendix) establishing uniform C^3 bounds on the developing map for elements of Diff^3(S^1), using the standard Fréchet topology and the fact that the Epstein hypersurface is locally determined by the jet of the diffeomorphism. These bounds guarantee rectifiability of the curve and finiteness of the horocycle-truncated length and enclosed area. The equality with the Schwarzian integral then follows by integrating the local Epstein identity, which we will spell out explicitly. The truncation is already identified with the finite bi-local observable, so no additional convergence issues arise. revision: yes

  2. Referee: [Loewner energy derivative paragraph] Section on the derivative relation to Loewner energy: the statement that the Schwarzian action is the derivative of the Loewner energy of the welded Jordan curve requires a precise specification of the variation (tangent vector in the quotient space) and a verification that the resulting first variation reproduces the Schwarzian functional without additional renormalization terms.

    Authors: We will revise the relevant section to identify the tangent vector explicitly as the class of a vector field on S^1 in the quotient PSL_2(R) backslash Diff^3(S^1). The first variation of the Loewner energy is computed via the chain rule applied to the welding map and the renormalized-volume definition; the renormalization term is independent of the variation and therefore drops out, yielding precisely the Schwarzian action. A self-contained calculation verifying the absence of extra terms will be inserted, together with a remark on the tangent-space identification. revision: yes

Circularity Check

0 steps flagged

No significant circularity; geometric identities derived independently via Epstein construction

full rationale

The paper applies Epstein's pre-existing hypersurface construction in the hyperbolic disk to the quotient space PSL_2(R) backslash Diff^3(S^1) and its coadjoint orbits, then derives identities relating the Schwarzian action to curve length and enclosed area, plus a derivative relation to Loewner energy. These steps rely on the extension of the external construction, horocycle truncation, the isoperimetric inequality, and monotonicity properties, none of which reduce by the paper's own equations to fitted parameters or self-citations. The abstract and described claims contain no self-definitional loops, fitted-input predictions, or load-bearing self-citation chains; the derivations remain self-contained against external geometric benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Based solely on the abstract, the paper relies on standard facts from hyperbolic geometry, diffeomorphism groups, and Loewner energy theory without introducing new free parameters or invented entities; no explicit fitting or ad-hoc constants are mentioned.

axioms (2)
  • standard math Standard properties of hyperbolic geometry, the diffeomorphism group Diff^3(S^1), and Epstein's hypersurface construction hold in this setting.
    Invoked throughout the application of Epstein curves to the Schwarzian action and coadjoint orbits (abstract).
  • domain assumption The horocycle truncation defines a renormalized length coinciding with the logarithm of the bi-local observable.
    Stated as part of the construction relating to Schwarzian field theory observables.

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Reference graph

Works this paper leans on

52 extracted references · 52 canonical work pages · 2 internal anchors

  1. [1]

    Anton Alekseev, Olga Chekeres, and Donald R. Youmans. Towards bosonization of Virasoro coadjoint orbits.Ann. Henri Poincaré, 25(1):5–34, 2024

    work page 2024

  2. [2]

    E. M. Andreev. Convex polyhedra in Lobačevskii spaces. Mat. Sb. (N.S.), 81(123):445–478, 1970

    work page 1970

  3. [3]

    Probabilistic Definition of the Schwarzian Field Theory.Preprint, 2024

    Roland Bauerschmidt, Ilya Losev, and Peter Wildemann. Probabilistic Definition of the Schwarzian Field Theory.Preprint, 2024. arXiv: 2406.17068

    work page arXiv 2024

  4. [4]

    Belokurov and Evgeniy T

    Vladimir V. Belokurov and Evgeniy T. Shavgulidze. Exact solution of the schwarzian theory. Phys. Rev. D, 96:101701, Nov 2017

    work page 2017

  5. [5]

    Belokurov and Evgeniy T

    Vladimir V. Belokurov and Evgeniy T. Shavgulidze. Correlation functions in the Schwarzian theory.J. High Energy Phys., (11):036, front matter+27, 2018

    work page 2018

  6. [6]

    Conformal removability is hard.preprint, 2020

    Christopher J Bishop. Conformal removability is hard.preprint, 2020

    work page 2020

  7. [7]

    Weil–Petersson curves,β-numbers, and minimal surfaces

    Christopher J Bishop. Weil–Petersson curves,β-numbers, and minimal surfaces. Ann. of Math, to appear

    work page

  8. [8]

    Youmans.SL(2, R) Gauge Theory, Hyperbolic Geom- etry and Virasoro Coadjoint Orbits.arXiv:2410.01302, 2024

    Matthias Blau and Donald R. Youmans.SL(2, R) Gauge Theory, Hyperbolic Geom- etry and Virasoro Coadjoint Orbits.arXiv:2410.01302, 2024

    work page arXiv 2024

  9. [9]

    Mertens, and Henri Verschelde

    Andreas Blommaert, Thomas G. Mertens, and Henri Verschelde. The Schwarzian Theory - A Wilson Line Perspective.JHEP, 12:022, 2018. 46

    work page 2018

  10. [10]

    Piecewise geodesic Jordan curves II: Loewner energy, projective structures, and accessory parameters

    Mario Bonk, Janne Junnila, Steffen Rohde, and Yilin Wang. Piecewise geodesic Jordan curves II: Loewner energy, projective structures, and accessory parameters. arXiv:2410.22275, 2024

    work page arXiv 2024

  11. [11]

    Schwarzian derivatives, projective structures, and the Weil–Petersson gradient flow for renormalized volume

    Martin Bridgeman, Jeffrey Brock, and Kenneth Bromberg. Schwarzian derivatives, projective structures, and the Weil–Petersson gradient flow for renormalized volume. Duke Math. J., 168(5):867–896, 2019

    work page 2019

  12. [12]

    Universal Liouville action as a renormalized volume and its gradient flow

    Martin Bridgeman, Kenneth Bromberg, Franco Vargas Pallete, and Yilin Wang. Universal Liouville action as a renormalized volume and its gradient flow. Duke Math. J., to appear

    work page

  13. [13]

    Brock, Richard D

    Jeffrey F. Brock, Richard D. Canary, and Yair N. Minsky. The classification of Kleinian surface groups, II: The ending lamination conjecture.Ann. of Math. (2), 176(1):1–149, 2012

    work page 2012

  14. [14]

    Onsager–Machlup functional forSLEκloop mea- sures

    Marco Carfagnini and Yilin Wang. Onsager–Machlup functional forSLEκloop mea- sures. Comm. Math. Phys., 405(11):Paper No. 258, 14, 2024

    work page 2024

  15. [15]

    Mertens, and Herman Verlinde

    Julius Engelsöy, Thomas G. Mertens, and Herman Verlinde. An investigation of AdS2 backreaction and holography.JHEP, 07:139, 2016

    work page 2016

  16. [16]

    EnvelopesofHorospheresandWeingartenSurfacesinHyperbolic 3-Space

    CharlesL.Epstein. EnvelopesofHorospheresandWeingartenSurfacesinHyperbolic 3-Space. preprint, 1984. Available on arXiv: 2401.12115

    work page arXiv 1984

  17. [17]

    Random disks of constant curvature: the lattice story.Preprint, 2024

    Frank Ferrari. Random disks of constant curvature: the lattice story.Preprint, 2024. arXiv: 2406.06875

    work page arXiv 2024

  18. [18]

    Hodgson and Igor Rivin

    Craig D. Hodgson and Igor Rivin. A characterization of compact convex polyhedra in hyperbolic 3-space. Invent. Math., 111(1):77–111, 1993

    work page 1993

  19. [19]

    Lower dimensional gravity.Nuclear Physics B, 252:343–356, 1985

    Roman Jackiw. Lower dimensional gravity.Nuclear Physics B, 252:343–356, 1985

    work page 1985

  20. [20]

    Chaos in AdS2 Holography

    Kristan Jensen. Chaos in AdS2 Holography. Phys. Rev. Lett., 117(11):111601, 2016

    work page 2016

  21. [21]

    Coulomb gas and the Grunsky operator on a Jordan domain with corners.Preprint, 2023

    Kurt Johansson and Fredrik Viklund. Coulomb gas and the Grunsky operator on a Jordan domain with corners.Preprint, 2023. arXiv: 2309.00308

    work page arXiv 2023

  22. [22]

    Josephine Suh

    Alexei Kitaev and S. Josephine Suh. Statistical mechanics of a two-dimensional black hole. JHEP, 05:198, 2019

    work page 2019

  23. [23]

    Holography and Riemann surfaces

    Kirill Krasnov. Holography and Riemann surfaces. Adv. Theor. Math. Phys., 4(4):929–979, 2000

    work page 2000

  24. [24]

    On the renormalized volume of hyperbolic 3-manifolds

    Kirill Krasnov and Jean-Marc Schlenker. On the renormalized volume of hyperbolic 3-manifolds. Comm. Math. Phys., 279(3):637–668, 2008

    work page 2008

  25. [25]

    Large Deviations of the Schwarzian Field Theory.Preprint, 2024

    Ilya Losev. Large Deviations of the Schwarzian Field Theory.Preprint, 2024. arXiv: 2406.17069

    work page internal anchor Pith review arXiv 2024

  26. [26]

    Probabilistic Correlation Functions of the Schwarzian Field Theory

    Ilya Losev. Probabilistic Correlation Functions of the Schwarzian Field Theory. Preprint, 2024. arXiv: 2406.17071

    work page internal anchor Pith review arXiv 2024

  27. [27]

    The largeN limit of superconformal field theories and supergravity

    Juan Maldacena. The largeN limit of superconformal field theories and supergravity. Adv. Theor. Math. Phys., 2(2):231–252, 1998

    work page 1998

  28. [28]

    Remarks on the Sachdev-Ye-Kitaev model

    Juan Maldacena and Douglas Stanford. Remarks on the Sachdev-Ye-Kitaev model. Phys. Rev. D, 94:106002, Nov 2016. 47

    work page 2016

  29. [29]

    Conformal symmetry and its breaking in two-dimensional nearly anti-de sitter space.Progress of Theoretical and Experimental Physics, 2016(12):12C104, 11 2016

    Juan Maldacena, Douglas Stanford, and Zhenbin Yang. Conformal symmetry and its breaking in two-dimensional nearly anti-de sitter space.Progress of Theoretical and Experimental Physics, 2016(12):12C104, 11 2016

    work page 2016

  30. [30]

    McMullen

    Curtis T. McMullen. The moduli space of Riemann surfaces is Kähler hyperbolic. Ann. of Math. (2), 151(1):327–357, 2000

    work page 2000

  31. [31]

    Thomas G. Mertens. The Schwarzian theory—origins.J. High Energy Phys., (5):036, front matter+44, 2018

    work page 2018

  32. [32]

    Mertens, Gustavo J

    Thomas G. Mertens, Gustavo J. Turiaci, and Herman L. Verlinde. Solving the Schwarzian via the Conformal Bootstrap.JHEP, 08:136, 2017

    work page 2017

  33. [33]

    The Loewner energy via the renormalised energy of moving frames.Arch

    Alexis Michelat and Yilin Wang. The Loewner energy via the renormalised energy of moving frames.Arch. Ration. Mech. Anal., 248(2):Paper No. 15, 60, 2024

    work page 2024

  34. [34]

    R. C. Penner. Universal constructions in Teichmüller theory.Adv. Math., 98(2):143– 215, 1993

    work page 1993

  35. [35]

    Penner.Decorated Teichmüller theory

    Robert C. Penner.Decorated Teichmüller theory. QGM Master Class Series. Euro- pean Mathematical Society (EMS), Zürich, 2012. With a foreword by Yuri I. Manin

    work page 2012

  36. [36]

    Every circle homeomorphism is the composition of two weldings

    Alex Rodriguez. Every circle homeomorphism is the composition of two weldings. Preprint, 2025. arXiv: 2501.06347

    work page arXiv 2025

  37. [37]

    The Loewner energy of loops and regularity of driving functions

    Steffen Rohde and Yilin Wang. The Loewner energy of loops and regularity of driving functions. Int. Math. Res. Not. IMRN, 2021(10):7715–7763, 2021

    work page 2021

  38. [38]

    Circle homeomorphisms with square summable diamond shears

    Dragomir Šarić, Yilin Wang, and Catherine Wolfram. Circle homeomorphisms with square summable diamond shears. International Mathematics Research Notices, 2024(17):12219–12268, 07 2024

    work page 2024

  39. [39]

    Métriques sur les polyèdres hyperboliques convexes.J

    Jean-Marc Schlenker. Métriques sur les polyèdres hyperboliques convexes.J. Dif- ferential Geom., 48(2):323–405, 1998

    work page 1998

  40. [40]

    Weil–Petersson Teichmüller space.Amer

    Yuliang Shen. Weil–Petersson Teichmüller space.Amer. J. Math., 140(4):1041–1074, 2018

    work page 2018

  41. [41]

    Fermionic localization of the Schwarzian theory

    Douglas Stanford and Edward Witten. Fermionic localization of the Schwarzian theory. J. High Energy Phys., (10):008, front matter+27, 2017

    work page 2017

  42. [42]

    Quasiconformal deformation of the chordal Loewner driving function and first variation of the Loewner energy.Math

    Jinwoo Sung and Yilin Wang. Quasiconformal deformation of the chordal Loewner driving function and first variation of the Loewner energy.Math. Ann., 390(3):4789– 4812, 2024

    work page 2024

  43. [43]

    Takhtajan and Lee-Peng Teo

    Leon A. Takhtajan and Lee-Peng Teo. Liouville action and Weil–Petersson metric on deformation spaces, global Kleinian reciprocity and holography.Comm. Math. Phys., 239(1-2):183–240, 2003

    work page 2003

  44. [44]

    Takhtajan and Lee-Peng Teo

    Leon A. Takhtajan and Lee-Peng Teo. Weil–Petersson metric on the universal Te- ichmüller space.Mem. Amer. Math. Soc., 183(861):viii+119, 2006

    work page 2006

  45. [45]

    Gravitation and hamiltonian structure in two spacetime dimen- sions

    Claudio Teitelboim. Gravitation and hamiltonian structure in two spacetime dimen- sions. Physics Letters B, 126:41–45, 1983

    work page 1983

  46. [46]

    Isoperimetric interpretation for the renor- malized volume of convex co-compact hyperbolic 3-manifolds

    Franco Vargas Pallete and Celso Viana. Isoperimetric interpretation for the renor- malized volume of convex co-compact hyperbolic 3-manifolds. Amer. J. Math., 48 147(1), 2025

    work page 2025

  47. [47]

    Interplay between Loewner and Dirichlet energies via conformal welding and flow-lines.Geom

    Fredrik Viklund and Yilin Wang. Interplay between Loewner and Dirichlet energies via conformal welding and flow-lines.Geom. Funct. Anal., 30(1):289–321, 2020

    work page 2020

  48. [48]

    The Loewner–Kufarev energy and foliations by Weil-Petersson quasicircles

    Fredrik Viklund and Yilin Wang. The Loewner–Kufarev energy and foliations by Weil-Petersson quasicircles. Proc. Lond. Math. Soc. (3), 128(2):Paper No. e12582, 62, 2024

    work page 2024

  49. [49]

    Equivalent descriptions of the Loewner energy

    Yilin Wang. Equivalent descriptions of the Loewner energy. Invent. Math., 218(2):573–621, 2019

    work page 2019

  50. [50]

    Two optimization problems for the Loewner energy.J

    Yilin Wang. Two optimization problems for the Loewner energy.J. Math. Phys., 66(2):Paper No. 023502, 2025

    work page 2025

  51. [51]

    Coadjoint orbits of the Virasoro group

    Edward Witten. Coadjoint orbits of the Virasoro group. Comm. Math. Phys., 114(1):1–53, 1988

    work page 1988

  52. [52]

    SLE loop measures.Probab

    Dapeng Zhan. SLE loop measures.Probab. Theory Related Fields, 179(1-2):345–406, 2021. 49

    work page 2021