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Cuspidal Singularities in Collapsing Domain Walls

T0 review · 2 major / 1 minor · reviewed 2026-05-25 · grok-4.3

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Pith's one-line read Collapsing domain walls generically develop cuspidal edge and vertex singularities on their worldvolume from smooth initial conditions.

desk verdict Cuspidal singularities show up in the simulations but the thin-wall approximation is likely invalid exactly at those points. read the letter →

arxiv 2605.22974 v1 pith:WGOUY3L7 submitted 2026-05-21 hep-th astro-ph.COgr-qchep-ph

Jose J. Blanco-Pillado , Matthew Elley , Francesc Ferrer , Alberto García Martín-Caro , Daniel Jiménez-Aguilar , Oriol Pujolàs , Juan S. Valbuena-Bermúdez This is my paper · ORCID
classification hep-thastro-ph.COgr-qchep-ph
keywords domainwallscuspidalsingularitiesNambu-Gotoequationssingularitytheoryfieldsimulationsthinwallapproximationeikonalcollapsing
checked against Cost.FunctionalEquationalexander_duality_circle_linking
verification ladder T0 review T1 audit T2 compute T3 formal

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 paper establishes that individual closed domain walls, when collapsing, form two distinct types of worldvolume singularities: one-dimensional cuspidal edges that travel at light speed for a finite duration and instantaneous cuspidal vertices where the wall briefly reaches light speed. These structures emerge generically and follow the standard classification from singularity theory. The same features appear in Nambu-Goto evolution, an eikonal approximation, and full field-theory simulations with adaptive mesh refinement, showing they are not artifacts of the thin-wall limit. This matters for understanding energy focusing during domain-wall collapse and its possible cosmological effects.

What carries the argument

Cuspidal singularities on the worldvolume, analyzed via singularity theory applied to Nambu-Goto and eikonal dynamics of the collapsing surface.

What would settle it

A high-resolution field theory simulation of a collapsing closed domain wall starting from smooth initial data that fails to produce propagating light-speed edges or instantaneous light-speed spikes would falsify the generic formation claim.

Watch

Extended reading notes

Core claim

Collapsing domain walls generically develop worldvolume singularities of two types: cuspidal edge singularities, consisting of one dimensional singular edges that propagate along the wall surface at the speed of light for a finite time, and cuspidal vertex singularities, which are spike like and instantaneous events where the wall moves momentarily at the speed of light. Both types of features arise generically from smooth initial conditions, and their formation and evolution follow the universal patterns of singularity theory. These structures are captured both by the Nambu-Goto equations and by an eikonal like approximation valid in the relativistic regime, and the same singular structures

Load-bearing premise

The thin-wall approximation together with the Nambu-Goto and eikonal descriptions remain valid through the collapse, and the adaptive mesh refinement field theory simulations faithfully reproduce the thin-wall limit without introducing numerical artifacts at the singular points.

Editorial extensions

If this is right

  • The singularities produce localized high-energy-density regions in the field theory.
  • The same cuspidal structures appear qualitatively in both thin-wall and full simulations.
  • Phenomenological implications arise from the energy focusing during collapse.
  • Formation and evolution follow universal patterns of singularity theory.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Higher-resolution simulations could test whether the cuspidal vertices produce measurable deviations from the thin-wall prediction before the approximation breaks.
  • If similar singularities occur in domain-wall networks rather than isolated walls, they might alter the spectrum of gravitational waves emitted during collapse.
  • The eikonal approximation might allow analytic tracking of singularity formation times for a wider class of initial shapes.
  • Connections to other relativistic membrane or brane systems could reveal whether cuspidal edges are a universal feature of light-speed focusing.
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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 / 1 minor

Summary. The paper claims that collapsing closed domain walls generically develop two types of worldvolume singularities—cuspidal edge singularities (one-dimensional edges propagating at light speed for finite time) and cuspidal vertex singularities (instantaneous spikes where the wall reaches light speed)—from smooth initial conditions. These follow universal patterns from singularity theory, are captured by the Nambu-Goto equations and an eikonal approximation in the relativistic regime, and are qualitatively reproduced in adaptive mesh refinement field-theory simulations, implying they are robust features rather than thin-wall artifacts, with possible implications for localized high-energy-density regions.

Significance. If the thin-wall and simulation results hold, the work identifies a previously under-appreciated generic mechanism for singularity formation and energy focusing in domain-wall networks, extending thin-wall analyses with a direct link to singularity theory and providing a complementary analytic handle on collapse dynamics that large-scale simulations alone cannot easily reveal. The use of the standard Nambu-Goto action without fitted parameters and the independent field-theory evolution are strengths.

major comments (2)
  1. [abstract and simulation comparison section] The central claim that the cuspidal singularities are robust features of realistic domain-wall dynamics (abstract) rests on the thin-wall/Nambu-Goto/eikonal descriptions remaining valid through collapse. At the cuspidal edges and vertices the local curvature radius shrinks to zero while the wall thickness is fixed by the scalar potential; the manuscript provides no controlled study of the vanishing-wall-width limit nor explicit resolution checks at the singular loci in the AMR simulations, so finite-thickness regularization effects cannot be ruled out.
  2. [abstract] The abstract states that the singular structures are 'reproduced qualitatively' in field-theory simulations, yet no quantitative error bars, direct metric comparisons (e.g., position or velocity of the cusps), or convergence tests with respect to wall width are reported; this leaves the agreement open to possible numerical artifacts precisely where the thin-wall approximation is most stressed.
minor comments (1)
  1. [eikonal approximation section] Notation for the eikonal approximation and its relation to the Nambu-Goto equations should be clarified with an explicit equation reference.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thoughtful review and for highlighting important points regarding the robustness of our claims. We address each major comment below, indicating planned revisions where appropriate.

read point-by-point responses

  1. Referee: [abstract and simulation comparison section] The central claim that the cuspidal singularities are robust features of realistic domain-wall dynamics (abstract) rests on the thin-wall/Nambu-Goto/eikonal descriptions remaining valid through collapse. At the cuspidal edges and vertices the local curvature radius shrinks to zero while the wall thickness is fixed by the scalar potential; the manuscript provides no controlled study of the vanishing-wall-width limit nor explicit resolution checks at the singular loci in the AMR simulations, so finite-thickness regularization effects cannot be ruled out.

    Authors: We agree that a controlled exploration of the vanishing-wall-width limit would provide stronger support for robustness against finite-thickness effects. Our AMR simulations maintain resolution of the fixed wall thickness set by the potential, and the cuspidal features appear consistently. In revision we will add explicit resolution checks at the singular loci (reporting local grid spacing relative to wall width) and a dedicated paragraph discussing the limitations of the thin-wall approximation near the cusps. A full parameter scan over wall width is beyond the scope of the present work but will be noted as desirable future work. revision: partial

  2. Referee: [abstract] The abstract states that the singular structures are 'reproduced qualitatively' in field-theory simulations, yet no quantitative error bars, direct metric comparisons (e.g., position or velocity of the cusps), or convergence tests with respect to wall width are reported; this leaves the agreement open to possible numerical artifacts precisely where the thin-wall approximation is most stressed.

    Authors: The abstract deliberately uses 'qualitatively' because the cusps involve diverging curvature, rendering precise quantitative metric comparisons (such as exact cusp trajectories) impractical without matched initial data across all methods. We will revise the manuscript to include convergence tests with respect to numerical resolution and to report error estimates on the timing and location of singularity formation where measurable. These additions will clarify the strength of the numerical evidence while preserving the qualitative nature of the comparison stated in the abstract. revision: partial

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity; derivations from independent Nambu-Goto/eikonal and simulations

full rationale

The paper derives cuspidal singularities from the standard Nambu-Goto action and an eikonal approximation applied to smooth initial data, then verifies qualitatively via independent AMR field-theory simulations. No parameters are fitted to a subset and then relabeled as predictions, no self-definitional loops appear in the equations, and no load-bearing uniqueness theorem or ansatz is imported solely via self-citation. The central claim remains externally falsifiable against the field simulations, consistent with the reader's assessment of minor (score-2) self-citation at most in the phrase 'extending previous thin wall analyses.'

Assumptions & free parameters 0 free parameters · 2 assumptions · 0 invented entities

The central claim rests on the standard Nambu-Goto action for thin walls and the applicability of classical singularity theory; no new free parameters, ad-hoc axioms, or invented entities are introduced.

assumptions (2)
  • domain assumption Dynamics of thin domain walls are governed by the Nambu-Goto action
    Invoked to derive the worldvolume evolution and singularities.
  • standard math Universal patterns from singularity theory govern the formation of cusps on evolving surfaces
    Used to classify the edge and vertex singularities.

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Cite this review

Pith. "Pith review of Cuspidal Singularities in Collapsing Domain Walls." pith.science (2026). https://pith.science/paper/WGOUY3L7

@misc{pith2026260522974,
  author       = {Pith},
  title        = {Pith review of: Cuspidal Singularities in Collapsing Domain Walls},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WGOUY3L7}},
  note         = {Machine review of arXiv:2605.22974}
}
read the original abstract

Domain wall networks have attracted renewed interest, particularly in relation to the dynamics of network collapse. Accurately describing this process is challenging and typically requires large scale numerical simulations. Here we adopt a complementary approach by studying the collapse of individual closed domain walls, extending previous thin wall analyses and comparing them with adaptive mesh refinement field theory simulations. Firstly, we show that collapsing domain walls generically develop worldvolume singularities of two types: cuspidal edge singularities, consisting of one dimensional singular edges that propagate along the wall surface at the speed of light for a finite time, and cuspidal vertex singularities, which are spike like and instantaneous events where the wall moves momentarily at the speed of light. Both types of features arise generically from smooth initial conditions, and their formation and evolution follow the universal patterns of singularity theory. We show that these structures are captured both by the Nambu-Goto equations and by an eikonal like approximation valid in the relativistic regime. Furthermore, we demonstrate that the same singular structures are reproduced qualitatively in full field theory simulations, establishing that they are not artifacts of the thin wall approximation but robust features of realistic domain wall dynamics. Naturally, such focusing effects in the field theory simulations result in localized regions of high energy density. We briefly discuss possible phenomenological implications.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Biased Domain Wall Networks and their Gravitational Waves

    astro-ph.CO 2026-07 conditional novelty 6.0 of 10

    Population-biased domain wall networks annihilate at T_ann ~ T_s B_s^0.8 and emit a single-broken-power-law gravitational-wave spectrum peaking near twice the Hubble scale.

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Works this paper leans on

66 extracted references · 66 canonical work pages · cited by 1 Pith paper

  1. [1]

    T. W. B. Kibble,Topology of Cosmic Domains and Strings,J. Phys. A9(1976) 1387–1398

    work page 1976

  2. [2]

    Vilenkin and E

    A. Vilenkin and E. P. S. Shellard,Cosmic Strings and Other Topological Defects. Cambridge University Press, 7, 2000

    work page 2000

  3. [3]

    Vachaspati,Kinks and Domain Walls : An Introduction to Classical and Quantum Solitons

    T. Vachaspati,Kinks and Domain Walls : An Introduction to Classical and Quantum Solitons. Oxford University Press, 2007, 10.1017/9781009290456

    work page doi:10.1017/9781009290456 2007

  4. [4]

    Y. B. Zeldovich, I. Y. Kobzarev and L. B. Okun,Cosmological Consequences of the Spontaneous Breakdown of Discrete Symmetry,Zh. Eksp. Teor. Fiz.67(1974) 3–11

    work page 1974

  5. [5]

    Vilenkin,Gravitational Field of Vacuum Domain Walls and Strings,Phys

    A. Vilenkin,Gravitational Field of Vacuum Domain Walls and Strings,Phys. Rev. D23(1981) 852–857

    work page 1981

  6. [6]

    Sikivie,Of Axions, Domain Walls and the Early Universe,Phys

    P. Sikivie,Of Axions, Domain Walls and the Early Universe,Phys. Rev. Lett.48(1982) 1156–1159

    work page 1982

  7. [7]

    G. B. Gelmini, M. Gleiser and E. W. Kolb,Cosmology of Biased Discrete Symmetry Breaking,Phys. Rev. D39(1989) 1558

    work page 1989

  8. [8]

    Gravitational Waves from Collapsing Vacuum Domains

    M. Gleiser and R. Roberts,Gravitational waves from collapsing vacuum domains,Phys. Rev. Lett.81 (1998) 5497–5500, [astro-ph/9807260]. – 28 –

    work page Pith review arXiv 1998

Show all 66 references
  1. [9]

    Hiramatsu, M

    T. Hiramatsu, M. Kawasaki and K. Saikawa,Gravitational Waves from Collapsing Domain Walls, JCAP05(2010) 032, [1002.1555]

    2010 arXiv

  2. [10]

    Kawasaki and K

    M. Kawasaki and K. Saikawa,Study of gravitational radiation from cosmic domain walls,JCAP09 (2011) 008, [1102.5628]

    2011 arXiv

  3. [11]

    Hiramatsu, M

    T. Hiramatsu, M. Kawasaki, K. Saikawa and T. Sekiguchi,Axion cosmology with long-lived domain walls,JCAP01(2013) 001, [1207.3166]

    2013 arXiv

  4. [12]

    Krajewski, Z

    T. Krajewski, Z. Lalak, M. Lewicki and P. Olszewski,Domain walls and gravitational waves in the Standard Model,JCAP12(2016) 036, [1608.05719]

    2016 arXiv

  5. [13]

    Krajewski, Z

    T. Krajewski, Z. Lalak, M. Lewicki and P. Olszewski,Domain walls in the extensions of the Standard Model,JCAP05(2018) 007, [1709.10100]

    2018 arXiv

  6. [14]

    Saikawa,A review of gravitational waves from cosmic domain walls,Universe3(2017) 40, [1703.02576]

    K. Saikawa,A review of gravitational waves from cosmic domain walls,Universe3(2017) 40, [1703.02576]

    2017 arXiv

  7. [15]

    R. Z. Ferreira, A. Notari, O. Pujol` as and F. Rompineve,Collapsing domain wall networks: impact on pulsar timing arrays and primordial black holes,JCAP06(2024) 020, [2401.14331]

    2024

  8. [16]

    Notari, F

    A. Notari, F. Rompineve and F. Torrenti,The spectrum of gravitational waves from annihilating domain walls,JCAP07(2025) 049, [2504.03636]

    2025

  9. [17]

    B. Cyr, S. Cotterill and R. Battye,Near-peak spectrum of gravitational waves from collapsing domain walls,Phys. Rev. D113(2026) 043549, [2504.02076]

    2026

  10. [18]

    Gr¨ uber, L

    D. Gr¨ uber, L. Sousa and P. P. Avelino,Stochastic gravitational wave background generated by domain wall networks,Phys. Rev. D110(2024) 023505, [2403.09816]

    2024

  11. [19]

    Dankovsky, E

    I. Dankovsky, E. Babichev, D. Gorbunov, S. Ramazanov and A. Vikman,Revisiting evolution of domain walls and their gravitational radiation with CosmoLattice,JCAP09(2024) 047, [2406.17053]

    2024

  12. [20]

    Dankovsky, S

    I. Dankovsky, S. Ramazanov, E. Babichev, D. Gorbunov and A. Vikman,Numerical analysis of melting domain walls and their gravitational waves,JCAP02(2025) 064, [2410.21971]

    2025

  13. [21]

    Garriga, A

    J. Garriga, A. Vilenkin and J. Zhang,Black holes and the multiverse,JCAP02(2016) 064, [1512.01819]

    2016 arXiv

  14. [22]

    H. Deng, J. Garriga and A. Vilenkin,Primordial black hole and wormhole formation by domain walls, JCAP04(2017) 050, [1612.03753]

    2017 arXiv

  15. [23]

    Ferrer, E

    F. Ferrer, E. Masso, G. Panico, O. Pujolas and F. Rompineve,Primordial Black Holes from the QCD axion,Phys. Rev. Lett.122(2019) 101301, [1807.01707]

    2019 arXiv

  16. [24]

    G. B. Gelmini, A. Simpson and E. Vitagliano,Catastrogenesis: DM, GWs, and PBHs from ALP string-wall networks,JCAP02(2023) 031, [2207.07126]

    2023

  17. [25]

    D. I. Dunsky and M. Kongsore,Primordial black holes from axion domain wall collapse,JHEP06 (2024) 198, [2402.03426]

    2024

  18. [26]

    Gouttenoire, S

    Y. Gouttenoire, S. F. King, R. Roshan, X. Wang, G. White and M. Yamazaki,Cosmological consequences of domain walls biased by quantum gravity,Phys. Rev. D112(2025) 075007, [2501.16414]. [27]ETcollaboration, A. Abac et al.,The Science of the Einstein Telescope,JCAP03(2026) 081, ...

    2025

  19. [27]

    D. J. Reardon et al.,Search for an Isotropic Gravitational-wave Background with the Parkes Pulsar Timing Array,Astrophys. J. Lett.951(2023) L6, [2306.16215]. [30]EPTA, InPTA:collaboration, J. Antoniadis et al.,The second data release from the European Pulsar Timing Array - III...

    2023 arXiv

  20. [28]

    Xu et al.,Searching for the Nano-Hertz Stochastic Gravitational Wave Background with the Chinese Pulsar Timing Array Data Release I,Res

    H. Xu et al.,Searching for the Nano-Hertz Stochastic Gravitational Wave Background with the Chinese Pulsar Timing Array Data Release I,Res. Astron. Astrophys.23(2023) 075024, [2306.16216]. – 29 – [32]NANOGravcollaboration, A. Afzal et al.,The NANOGrav 15 yr Data Set: Search fo...

    2023 arXiv

  21. [29]

    Kitajima, J

    N. Kitajima, J. Lee, K. Murai, F. Takahashi and W. Yin,Gravitational waves from domain wall collapse, and application to nanohertz signals with QCD-coupled axions,Phys. Lett. B851(2024) 138586, [2306.17146]

    2024

  22. [30]

    Babichev, I

    E. Babichev, I. Dankovsky, D. Gorbunov, S. Ramazanov and A. Vikman,Biased domain walls: faster annihilation, weaker gravitational waves,JCAP10(2025) 103, [2504.07902]

    2025

  23. [31]

    Dankovsky, S

    I. Dankovsky, S. Ramazanov, E. Babichev, D. Gorbunov and A. Vikman,Cosmic domain walls on a lattice: Illusive effects of initial conditions,Phys. Rev. D112(2025) 123521, [2509.25367]

    2025

  24. [32]

    Turok,Grand Unified Strings and Galaxy Formation,Nucl

    N. Turok,Grand Unified Strings and Galaxy Formation,Nucl. Phys. B242(1984) 520–541

    1984

  25. [33]

    Vachaspati and A

    T. Vachaspati and A. Vilenkin,Gravitational Radiation from Cosmic Strings,Phys. Rev. D31(1985) 3052

    1985

  26. [34]

    R. H. Brandenberger,On the Decay of Cosmic String Loops,Nucl. Phys. B293(1987) 812–828

    1987

  27. [35]

    K. D. Olum and J. J. Blanco-Pillado,Field theory simulation of Abelian Higgs cosmic string cusps, Phys. Rev. D60(1999) 023503, [gr-qc/9812040]

    1999 arXiv

  28. [36]

    L. M. Widrow,The Collapse of Nearly Spherical Domain Walls,Phys. Rev. D39(1989) 3576

    1989

  29. [37]

    Hoppe,Conservation laws and formation of singularities in relativistic theories of extended objects, hep-th/9503069

    J. Hoppe,Conservation laws and formation of singularities in relativistic theories of extended objects, hep-th/9503069

    arXiv

  30. [38]

    Eggers and J

    J. Eggers and J. Hoppe,Singularity formation for time-like extremal hypersurfaces,Phys. Lett. B680 (2009) 274–278

    2009

  31. [39]

    Eggers, J

    J. Eggers, J. Hoppe, M. Hynek and N. Suramlishvili,Singularities of relativistic membranes,Geometric Flows1(2015)

    2015

  32. [40]

    J. J. Blanco-Pillado and K. D. Olum,Form of cosmic string cusps,Phys. Rev. D59(1999) 063508, [gr-qc/9810005]

    1999

  33. [41]

    J. J. Blanco-Pillado, D. Jim´ enez-Aguilar and O. Pujol` as,From cusps to swallowtails: Domain wall singularities in 2+1 dimensions,Phys. Rev. D112(2025) 123536, [2509.19457]

    2025

  34. [42]

    Babichev, B

    E. Babichev, B. Gafarov, S. Ramazanov and M. Valencia-Villegas,Hard to shock DBI: wave propagation on planar domain walls,2604.08142

    arXiv

  35. [43]

    Drew and I

    A. Drew and I. Rybak,New Massless Spectra from Cosmic String Cusps,2512.12258

  36. [44]

    V. I. Arnold,Critical Points of Smooth Functions and Their Normal Forms,Russian Mathematical Surveys30(Oct., 1975) 1–75

    1975

  37. [45]

    V. I. Arnold, A. A. Davydov, V. A. Vassiliev and V. M. Zakalyukin,Mathematical models of catastrophes. control of catastrophic processes, iiasa research report (reprint), IIASA, Laxenburg, Austria, June, 2006

    2006

  38. [46]

    Poston and I

    T. Poston and I. Stewart,Catastrophe Theory and Its Applications. Dover books on mathematics. Dover Publications, 1996

    1996

  39. [47]

    Geometry of umbilics on surfaces in the three-space

    G. Gangopadhyay, “Geometry of umbilics on surfaces in the three-space.”https: //gautamgangopadhyay.wordpress.com/wp-content/uploads/2019/02/geometryofumbilics2018.pdf

    2019

  40. [48]

    Berry and C

    M. Berry and C. Upstill,Iv catastrophe optics: Morphologies of caustics and their diffraction patterns, vol. 18 ofProgress in Optics, pp. 257–346. Elsevier, 1980. DOI

    1980

  41. [49]

    Guilfoyle and W

    B. Guilfoyle and W. Klingenberg,Proof of the carath´ eodory conjecture,arXiv:0808.0851 [math.DG] (2008)

    2008

  42. [50]

    Poelaert, J

    D. Poelaert, J. Schniewind and F. Janssens,Surface Area and Curvature of the general Ellipsoid, 1104.5145

    arXiv

  43. [51]

    M. V. Berry and J. H. Hannay,Umbilic points on gaussian random surfaces,Journal of Physics A: Mathematical and General10(1977) 1809. – 30 –

    1977

  44. [52]

    V. I. Arnold, S. M. Gusein-Zade and A. N. Varchenko,Singularities of Differentiable Maps, Volume I: The Classification of Critical Points, Caustics and Wave Fronts, vol. 82 ofMonographs in Mathematics. Birkh¨ auser Boston, Boston, MA, 1985, 10.1007/978-1-4612-5154-5

    1985 doi

  45. [53]

    Garfinkle and T

    D. Garfinkle and T. Vachaspati,Radiation From Kinky, Cuspless Cosmic Loops,Phys. Rev. D36 (1987) 2229

    1987

  46. [54]

    Smale,A classification of immersions of the two-sphere,Transactions of the American Mathematical Society90(1959) 281–290

    S. Smale,A classification of immersions of the two-sphere,Transactions of the American Mathematical Society90(1959) 281–290

    1959

  47. [55]

    G. K. Francis and J. M. Sullivan,Visualizing a sphere eversion,IEEE Transactions on Visualization and Computer Graphics4(1998) 309–323

    1998

  48. [56]

    Bednorz and W

    A. Bednorz and W. Bednorz,Analytic sphere eversion using ruled surfaces,Differential Geometry and its Applications64(2019) 59–79

    2019

  49. [57]

    Max and T

    N. Max and T. Banchoff,Every sphere eversion has a quadruple point, inContributions to Analysis and Geometry(D. N. Clark, G. Pecelli and R. Sachsteder, eds.), pp. 191–209. Johns Hopkins University Press, Baltimore, 1981

    1981

  50. [58]

    J. F. Hughes,Another proof that every eversion of the sphere has a quadruple point,American Journal of Mathematics107(1985) 501–505

    1985

  51. [59]

    The minimax sphere eversion

    G. K. Francis, J. M. Sullivan, R. B. Kusner, K. A. Brakke, C. Hartman and G. Chappell, “The minimax sphere eversion.” Visualization and Mathematics, Ed. Hege, Hans-Christian and Polthier, Konrad, Springer Berlin, Heidelberg, 1997, 3–20,”https://www.researchgate.net/publication...

    1997

  52. [60]

    Andrade et al.,GRChombo: An adaptable numerical relativity code for fundamental physics,J

    T. Andrade et al.,GRChombo: An adaptable numerical relativity code for fundamental physics,J. Open Source Softw.6(2021) 3703, [2201.03458]

    2021

  53. [61]

    J. C. Aurrekoetxea, J. Bamber, S. E. Brady, K. Clough, T. Helfer, J. Marsden et al.,GRDzhadzha: A code for evolving relativistic matter on analytic metric backgrounds,J. Open Source Softw.9(2024) 5956, [2308.08299]

    2024

  54. [62]

    Alcubierre, B

    M. Alcubierre, B. Bruegmann, P. Diener, M. Koppitz, D. Pollney, E. Seidel et al.,Gauge conditions for long term numerical black hole evolutions without excision,Phys. Rev. D67(2003) 084023, [gr-qc/0206072]

    2003 arXiv

  55. [63]

    W. G. Cook, P. Figueras, M. Kunesch, U. Sperhake and S. Tunyasuvunakool,Dimensional reduction in numerical relativity: Modified cartoon formalism and regularization,Int. J. Mod. Phys. D25(2016) 1641013, [1603.00362]

    2016 arXiv

  56. [64]

    J. L. Jaramillo and B. Krishnan,Airy-function approach to binary black hole merger waveforms: The fold-caustic diffraction model,2206.02117

  57. [65]

    J. J. Blanco-Pillado, A. Garc´ ıa Mart´ ın-Caro, D. Jim´ enez-Aguilar and J. M. Queiruga,Effective actions for domain wall dynamics,Phys. Rev. D111(2025) 056007, [2411.13521]

    2025

  58. [66]

    J. C. Aurrekoetxea, J. J. Blanco-Pillado, A. Garc´ ıa Mart´ ın-Caro and J. M. Queiruga,On curvature corrections for field theory cosmic strings,2603.05243. – 31 – A Convergence test As a check of the validity of our field theory simulations, we perform a local convergence test...

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