The extremal lengths of conformal Riemannian metrics on Riemann surfaces
HTML articles powered by AMS MathViewer
- by Peijia Liu;
- Proc. Amer. Math. Soc. 152 (2024), 631-638
- DOI: https://doi.org/10.1090/proc/16648
- Published electronically: November 21, 2023
- HTML | PDF | Request permission
Abstract:
We give the definition of extremal lengths of conformal Riemannian metrics on Riemann surfaces. And we obtain the extremal lengths of conformal negatively curved Riemannian metrics on Riemann surfaces.References
- Javier Aramayona and Christopher J. Leininger, Hyperbolic structures on surfaces and geodesic currents, Algorithmic and geometric topics around free groups and automorphisms, Adv. Courses Math. CRM Barcelona, Birkhäuser/Springer, Cham, 2017, pp. 111–149. MR 3752040
- Francis Bonahon, Bouts des variétés hyperboliques de dimension $3$, Ann. of Math. (2) 124 (1986), no. 1, 71–158 (French). MR 847953, DOI 10.2307/1971388
- Francis Bonahon, The geometry of Teichmüller space via geodesic currents, Invent. Math. 92 (1988), no. 1, 139–162. MR 931208, DOI 10.1007/BF01393996
- C. Croke and A. Fathi, An inequality between energy and intersection, Bull. London Math. Soc. 22 (1990), no. 5, 489–494. MR 1082022, DOI 10.1112/blms/22.5.489
- C. Croke, A. Fathi, and J. Feldman, The marked length-spectrum of a surface of nonpositive curvature, Topology 31 (1992), no. 4, 847–855. MR 1191384, DOI 10.1016/0040-9383(92)90013-8
- Moon Duchin, Christopher J. Leininger, and Kasra Rafi, Length spectra and degeneration of flat metrics, Invent. Math. 182 (2010), no. 2, 231–277. MR 2729268, DOI 10.1007/s00222-010-0262-y
- Sa’ar Hersonsky and Frédéric Paulin, On the rigidity of discrete isometry groups of negatively curved spaces, Comment. Math. Helv. 72 (1997), no. 3, 349–388. MR 1476054, DOI 10.1007/s000140050022
- Jürgen Jost, Riemannian geometry and geometric analysis, 7th ed., Universitext, Springer, Cham, 2017. MR 3726907, DOI 10.1007/978-3-319-61860-9
- A. Katok, Entropy and closed geodesics, Ergodic Theory Dynam. Systems 2 (1982), no. 3-4, 339–365 (1983). MR 721728, DOI 10.1017/S0143385700001656
- Steven P. Kerckhoff, The asymptotic geometry of Teichmüller space, Topology 19 (1980), no. 1, 23–41. MR 559474, DOI 10.1016/0040-9383(80)90029-4
- François Labourie, Cross ratios, Anosov representations and the energy functional on Teichmüller space, Ann. Sci. Éc. Norm. Supér. (4) 41 (2008), no. 3, 437–469 (English, with English and French summaries). MR 2482204, DOI 10.24033/asens.2072
- Dídac Martínez-Granado and Dylan P. Thurston, From curves to currents, Forum Math. Sigma 9 (2021), Paper No. e77, 52. MR 4345013, DOI 10.1017/fms.2021.68
- Jean-Pierre Otal, Le spectre marqué des longueurs des surfaces à courbure négative, Ann. of Math. (2) 131 (1990), no. 1, 151–162 (French). MR 1038361, DOI 10.2307/1971511
- Michael Wolf, The Teichmüller theory of harmonic maps, J. Differential Geom. 29 (1989), no. 2, 449–479. MR 982185
- Scott A. Wolpert, Thurston’s Riemannian metric for Teichmüller space, J. Differential Geom. 23 (1986), no. 2, 143–174. MR 845703
Bibliographic Information
- Peijia Liu
- Affiliation: School of Mathematics and Big Data, Foshan University, Foshan, Guangdong 528000, People’s Republic of China
- MR Author ID: 1278307
- ORCID: 0000-0001-8328-4800
- Email: liu_peijia@outlook.com
- Received by editor(s): November 14, 2021
- Received by editor(s) in revised form: June 30, 2022, and February 13, 2023
- Published electronically: November 21, 2023
- Additional Notes: The work was partially supported by NSFC, No: 12271533
- Communicated by: Shelly Harvey
- © Copyright 2023 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 152 (2024), 631-638
- MSC (2020): Primary 32G15; Secondary 57M99
- DOI: https://doi.org/10.1090/proc/16648
- MathSciNet review: 4683845