Mean surfaces in half-pipe space and infinitesimal Teichmüller theory

Farid Diaf

doi:10.5802/jep.311

Mean surfaces in half-pipe space and infinitesimal Teichmüller theory
[Surfaces moyennes dans l’espace half-pipe et théorie de Teichmüller infinitésimale]

Farid Diaf ¹

¹ Univ. Grenoble Alpes, CNRS, IF, 38000 Grenoble, France

Journal de l’École polytechnique — Mathématiques, Tome 12 (2025), pp. 1289-1343.

Abstract (VO)
Résumé

We study a correspondence between smooth spacelike surfaces in the half-pipe space $\mathbb{HP}^3$ and divergence-free vector fields on the hyperbolic plane $\mathbb{H}^2$. We show that a particular case involves harmonic Lagrangian vector fields on $\mathbb{H}^2$, which are related to mean surfaces in $\mathbb{HP}^3$. Consequently, we prove that the infinitesimal Douady-Earle extension is a harmonic Lagrangian vector field that corresponds to a mean surface in $\mathbb{HP}^3$ with prescribed boundary data at infinity.

We establish both existence and, under certain assumptions, uniqueness results for harmonic Lagrangian extension of a vector field on the circle. Finally, we characterize the Zygmund and little Zygmund conditions and provide quantitative bounds in terms of the half-pipe width.

Nous étudions une correspondance entre les surfaces lisses de type espace dans l’espace half-pipe $\mathbb{HP}^3$ et les champs de vecteurs à divergence nulle sur le plan hyperbolique $\mathbb{H}^2$. Nous montrons qu’un cas particulier de cette correspondance fait intervenir des champs de vecteurs harmoniques lagrangiens sur $\mathbb{H}^2$, qui sont liés aux surfaces moyennes dans $\mathbb{HP}^3$. Par conséquent, nous prouvons que l’extension infinitésimale de Douady–Earle est un champ de vecteurs harmonique lagrangien correspondant à une surface moyenne dans $\mathbb{HP}^3$, avec un bord prescrit à l’infini.

Nous établissons à la fois des résultats d’existence et, sous certaines hypothèses, d’unicité d’extension harmonique lagrangienne d’un champ de vecteurs défini sur le cercle. Enfin, nous caractérisons les conditions de Zygmund et de petit Zygmund, et nous fournissons des estimations quantitatives en termes de largeur de l’espace half-pipe.

Reçu le : 2024-09-06
Accepté le : 2025-08-28
Publié le : 2025-09-01

DOI : 10.5802/jep.311

Classification : 53C50, 30F60, 53C42
Keywords: Half-pipe space, Minkowski space, hyperbolic plane, mean surfaces, harmonic vector fields
Mots-clés : Espace half-pipe, espace de Minkowski, plan hyperbolique, surfaces moyennes, champs de vecteurs harmoniques

Affiliations des auteurs :

Farid Diaf ¹

¹ Univ. Grenoble Alpes, CNRS, IF, 38000 Grenoble, France

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

@article{JEP_2025__12__1289_0,
     author = {Farid Diaf},
     title = {Mean surfaces in half-pipe space and infinitesimal {Teichm\"uller} theory},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
     pages = {1289--1343},
     publisher = {\'Ecole polytechnique},
     volume = {12},
     year = {2025},
     doi = {10.5802/jep.311},
     language = {en},
     url = {https://jep.centre-mersenne.org/articles/10.5802/jep.311/}
}

TY  - JOUR
AU  - Farid Diaf
TI  - Mean surfaces in half-pipe space and infinitesimal Teichmüller theory
JO  - Journal de l’École polytechnique — Mathématiques
PY  - 2025
SP  - 1289
EP  - 1343
VL  - 12
PB  - École polytechnique
UR  - https://jep.centre-mersenne.org/articles/10.5802/jep.311/
DO  - 10.5802/jep.311
LA  - en
ID  - JEP_2025__12__1289_0
ER  -

%0 Journal Article
%A Farid Diaf
%T Mean surfaces in half-pipe space and infinitesimal Teichmüller theory
%J Journal de l’École polytechnique — Mathématiques
%D 2025
%P 1289-1343
%V 12
%I École polytechnique
%U https://jep.centre-mersenne.org/articles/10.5802/jep.311/
%R 10.5802/jep.311
%G en
%F JEP_2025__12__1289_0

Farid Diaf. Mean surfaces in half-pipe space and infinitesimal Teichmüller theory. Journal de l’École polytechnique — Mathématiques, Tome 12 (2025), pp. 1289-1343. doi : 10.5802/jep.311. https://jep.centre-mersenne.org/articles/10.5802/jep.311/

Bibliographie
Cité par

[ABB + 07] L. Andersson, T. Barbot, R. Benedetti, F. Bonsante, W. M. Goldman, F. Labourie, K. P. Scannell & J.-M. Schlenker - “Notes on a paper of Mess”, Geom. Dedicata 126 (2007), p. 47-70 | DOI | MR | Zbl

[AGR11] H. Anciaux, B. Guilfoyle & P. Romon - “Minimal Lagrangian surfaces in the tangent bundle of a Riemannian surface”, J. Geom. Phys. 61 (2011) no. 1, p. 237-247 | DOI | MR | Zbl

[Anc14] H. Anciaux - “Spaces of geodesics of pseudo-Riemannian space forms and normal congruences of hypersurfaces”, Trans. Amer. Math. Soc. 366 (2014) no. 5, p. 2699-2718 | DOI | MR | Zbl

[AR14] H. Anciaux & P. Romon - “A canonical structure on the tangent bundle of a pseudo- or para-Kähler manifold”, Monatsh. Math. 174 (2014) no. 3, p. 329-355 | DOI | MR | Zbl

[Bar05] T. Barbot - “Globally hyperbolic flat space–times”, J. Geom. Phys. 53 (2005) no. 2, p. 123-165 | DOI | MR | Zbl

[BB09] R. Benedetti & F. Bonsante - Canonical Wick rotations in $3$ -dimensional gravity, Mem. Amer. Math. Soc., vol. 926, American Mathematical Society, Providence, RI, 2009 | DOI | Zbl

[BEE22] F. Bonsante & C. El Emam - “On immersions of surfaces into $SL (2, ℂ)$ and geometric consequences”, Internat. Math. Res. Notices 2022 (2022) no. 12, p. 8803-8864 | DOI | MR | Zbl

[BF20] T. Barbot & F. Fillastre - “Quasi-Fuchsian co-Minkowski manifolds”, in In the tradition of Thurston. Geometry and topology, Springer, Cham, 2020, p. 645-703 | DOI | Zbl

[Bis25] C. J. Bishop - “Weil-Petersson curves, $β$ -numbers, and minimal surfaces”, Ann. of Math. (2) 202 (2025) no. 1, p. 111-188 | DOI | MR | Zbl

[Bon05] F. Bonsante - “Flat spacetimes with compact hyperbolic Cauchy surfaces”, J. Differential Geom. 69 (2005) no. 3, p. 441-521 | DOI | MR | Zbl

[BS10] F. Bonsante & J.-M. Schlenker - “Maximal surfaces and the universal Teichmüller space”, Invent. Math. 182 (2010) no. 2, p. 279-333 | DOI | MR | Zbl

[BS16] F. Bonsante & A. Seppi - “On Codazzi tensors on a hyperbolic surface and flat Lorentzian geometry”, Internat. Math. Res. Notices 2016 (2016) no. 2, p. 343-417 | DOI | MR | Zbl

[BS17] F. Bonsante & A. Seppi - “Spacelike convex surfaces with prescribed curvature in $(2 + 1)$ -Minkowski space”, Adv. Math. 304 (2017), p. 434-493 | DOI | MR | Zbl

[BS18] F. Bonsante & A. Seppi - “Area-preserving diffeomorphisms of the hyperbolic plane and $K$ -surfaces in anti-de Sitter space”, J. Topology 11 (2018) no. 2, p. 420-468 | DOI | MR | Zbl

[Dan11] J. Danciger - Geometric transitions: from hyperbolic to Ads geometry, Ph. D. Thesis, Stanford University, 2011

[DE86] A. Douady & C. J. Earle - “Conformally natural extension of homeomorphisms of the circle”, Acta Math. 157 (1986), p. 23-48 | DOI | MR | Zbl

[DGK16] J. Danciger, F. Guéritaud & F. Kassel - “Geometry and topology of complete Lorentz spacetimes of constant curvature”, Ann. Sci. École Norm. Sup. (4) 49 (2016) no. 1, p. 1-56 | DOI | Numdam | MR | Zbl

[Dia24] F. Diaf - “The infinitesimal earthquake theorem for vector fields on the circle”, Trans. Amer. Math. Soc. 377 (2024) no. 12, p. 8721-8767 | DOI | MR | Zbl

[Ear88] C. J. Earle - “Conformally natural extension of vector fields from $S^{n - 1}$ to $B^{n}$ ”, Proc. Amer. Math. Soc. 102 (1988) no. 1, p. 145-149 | DOI | MR | Zbl

[EES22] C. El Emam & A. Seppi - “On the Gauss map of equivariant immersions in hyperbolic space”, J. Topology 15 (2022) no. 1, p. 238-301 | DOI | MR | Zbl

[EM07] D. Epstein & V. Markovic - “Extending homeomorphisms of the circle to quasiconformal homeomorphisms of the disk”, Geom. Topol. 11 (2007), p. 517-595 | DOI | MR | Zbl

[FH14] J. Fan & J. Hu - “Characterization of the asymptotic Teichmüller space of the open unit disk through shears”, Pure Appl. Anal. 10 (2014) no. 3, p. 513-546 | DOI | Zbl

[FH22] J. Fan & J. Hu - “Conformally natural extensions of vector fields and applications”, Pure Appl. Anal. 18 (2022) no. 3, p. 1147-1186 | DOI | Zbl

[FS19] F. Fillastre & A. Seppi - “Spherical, hyperbolic, and other projective geometries: convexity, duality, transitions”, in Eighteen essays in non-Euclidean geometry, European Mathematical Society (EMS), Zürich, 2019, p. 321-409 | DOI | Zbl

[FV16] F. Fillastre & G. Veronelli - “Lorentzian area measures and the Christoffel problem”, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) 16 (2016) no. 2, p. 383-467 | DOI | MR | Zbl

[GK05] B. Guilfoyle & W. Klingenberg - “An indefinite Kähler metric on the space of oriented lines”, J. London Math. Soc. (2) 72 (2005) no. 2, p. 497-509 | DOI | MR | Zbl

[GL00] F. P. Gardiner & N. Lakic - Quasiconformal Teichmüller theory, Math. Surv. Monogr., vol. 76, American Mathematical Society, Providence, RI, 2000 | MR | Zbl

[GS92] F. P. Gardiner & D. P. Sullivan - “Symmetric structures on a closed curve”, Amer. J. Math. 114 (1992) no. 4, p. 683-736 | DOI | MR | Zbl

[GS15] Y. Godoy & M. Salvai - “Global smooth geodesic foliations of the hyperbolic space”, Math. Z. 281 (2015) no. 1-2, p. 43-54 | DOI | MR | Zbl

[GT83] D. Gilbarg & N. S. Trudinger - Elliptic partial differential equations of second order, Grundlehren Math. Wissen., vol. 224, Springer, Cham, 1983 | Zbl

[Hit82] N. J. Hitchin - “Monopoles and geodesics”, Comm. Math. Phys. 83 (1982), p. 579-602 | DOI | MR | Zbl

[HSWS13] Y. Hu, J. Song, H. Wei & Y. Shen - “An integral operator induced by a Zygmund function”, J. Math. Anal. Appl. 401 (2013) no. 2, p. 560-567 | DOI | Zbl

[Hu22] J. Hu - “Characterizations of circle homeomorphisms of different regularities in the universal Teichmüller space”, EMS Surv. Math. Sci. 9 (2022) no. 2, p. 321-353 | DOI | MR | Zbl

[Kon92] J. J. Konderak - “On harmonic vector fields”, Publ. Mat. 36 (1992) no. 1, p. 217-228 | DOI | MR | Zbl

[LT93a] P. Li & L.-F. Tam - “Uniqueness and regularity of proper harmonic maps”, Ann. of Math. (2) 137 (1993) no. 1, p. 167-201 | DOI | MR | Zbl

[LT93b] P. Li & L.-F. Tam - “Uniqueness and regularity of proper harmonic maps. II”, Indiana Univ. Math. J. 42 (1993) no. 2, p. 591-635 | DOI | MR | Zbl

[LY86] P. Li & S. T. Yau - “On the parabolic kernel of the Schrödinger operator”, Acta Math. 156 (1986), p. 154-201 | DOI | MR | Zbl

[Mar17] V. Markovic - “Harmonic maps and the Schoen conjecture”, J. Amer. Math. Soc. 30 (2017) no. 3, p. 799-817 | DOI | MR | Zbl

[Mar22] B. Martelli - “An Introduction to geometric topology”, 2022 | arXiv | Zbl

[McM96] C. T. McMullen - Renormalization and $3$ -manifolds which fiber over the circle, Annals of Math. Studies, vol. 142, Princeton Univ. Press, Princeton, NJ, 1996 | DOI | MR | Zbl

[Mes07] G. Mess - “Lorentz spacetimes of constant curvature”, Geom. Dedicata 126 (2007), p. 3-45 | DOI | MR | Zbl

[NS94] K. Nomizu & T. Sasaki - Affine differential geometry, Cambridge Tracts Math., vol. 111, Cambridge University Press, Cambridge, 1994 | MR | Zbl

[NS22] X. Nie & A. Seppi - “Regular domains and surfaces of constant Gaussian curvature in $3$ -dimensional affine space”, Anal. PDE 15 (2022) no. 3, p. 643-697 | DOI | MR | Zbl

[NS23] X. Nie & A. Seppi - “Affine deformations of quasi-divisible convex cones”, Proc. London Math. Soc. (3) 127 (2023) no. 1, p. 35-83 | DOI | MR | Zbl

[PW67] M. H. Protter & H. F. Weinberger - Maximum principles in differential equations, Prentice-Hall, 1967 | MR | Zbl

[RC91] E. Reich & J. Chen - “Extensions with bounded $\bar{\partial}$ -derivative”, Ann. Acad. Sci. Fenn. Ser. A I Math. 16 (1991) no. 2, p. 377-389 | DOI | MR | Zbl

[Roc70] R. T. Rockafellar - Convex analysis, Princeton Math. Series, vol. 28, Princeton University Press, Princeton, NJ, 1970 | DOI | MR | Zbl

[RS22] S. Riolo & A. Seppi - “Geometric transition from hyperbolic to anti-de Sitter structures in dimension four”, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) 23 (2022) no. 1, p. 115-176 | DOI | MR | Zbl

[Sep15] A. Seppi - Surfaces in constant curvature three-manifolds and the infinitesimal Teichmüller theory, Ph. D. Thesis, University of Pavia, 2015

[Sep19] A. Seppi - “Maximal surfaces in Anti-de Sitter space, width of convex hulls and quasiconformal extensions of quasisymmetric homeomorphisms”, J. Eur. Math. Soc. (JEMS) 21 (2019) no. 6, p. 1855-1913 | DOI | MR | Zbl

[Thu86] W. P. Thurston - “Earthquakes in two-dimensional hyperbolic geometry”, in Low-dimensional topology and Kleinian groups (Coventry/Durham, 1984), London Math. Soc. Lecture Note Ser., vol. 112, Cambridge Univ. Press, Cambridge, 1986, p. 91-112 | MR | Zbl

Cité par Sources :