The harmonic map compactification of Teichmüller spaces for punctured Riemann surfaces

Sakai, Kento

doi:10.1090/ecgd/388

The harmonic map compactification of Teichmüller spaces for punctured Riemann surfaces
HTML articles powered by AMS MathViewer

by Kento Sakai

Conform. Geom. Dyn. 27 (2023), 322-343

DOI: https://doi.org/10.1090/ecgd/388

Published electronically: December 21, 2023

PDF | Request permission

Abstract:

In the paper [The Teichmüller theory of harmonic maps, J. Differential Geom. 29 (1989), no. 2, 449–479], Wolf provided a global coordinate system of the Teichmüller space of a closed oriented surface $S$ with the vector space of holomorphic quadratic differentials on a Riemann surface $X$ homeomorphic to $S$. This coordinate system is via harmonic maps from the Riemann surface $X$ to hyperbolic surfaces. Moreover, he gave a compactification of the Teichmüller space by adding a point at infinity to each endpoint of harmonic map rays starting from $X$ in the space. Wolf also showed this compactification coincides with the Thurston compactification.

In this paper, we extend the harmonic map ray compactification to the case of punctured Riemann surfaces and show that it still coincides with the Thurston compactification.

References

James Eells Jr. and J. H. Sampson, Harmonic mappings of Riemannian manifolds, Amer. J. Math. 86 (1964), 109–160. MR 164306, DOI 10.2307/2373037
Albert Fathi, François Laudenbach, and Valentin Poénaru, Thurston’s work on surfaces, Mathematical Notes, vol. 48, Princeton University Press, Princeton, NJ, 2012. Translated from the 1979 French original by Djun M. Kim and Dan Margalit. MR 3053012, DOI 10.1515/9781400839032
Frederick P. Gardiner, Teichmüller theory and quadratic differentials, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1987. A Wiley-Interscience Publication. MR 903027
Philip Hartman, On homotopic harmonic maps, Canadian J. Math. 19 (1967), 673–687. MR 214004, DOI 10.4153/CJM-1967-062-6
John Hubbard and Howard Masur, Quadratic differentials and foliations, Acta Math. 142 (1979), no. 3-4, 221–274. MR 523212, DOI 10.1007/BF02395062
Jürgen Jost, Compact Riemann surfaces, 3rd ed., Universitext, Springer-Verlag, Berlin, 2006. An introduction to contemporary mathematics. MR 2247485, DOI 10.1007/978-3-540-33067-7
Jochen Lohkamp, Harmonic diffeomorphisms and Teichmüller theory, Manuscripta Math. 71 (1991), no. 4, 339–360. MR 1104989, DOI 10.1007/BF02568411
Yair N. Minsky, Harmonic maps, length, and energy in Teichmüller space, J. Differential Geom. 35 (1992), no. 1, 151–217. MR 1152229
Huiping Pan and Michael Wolf, Ray structures on Teichmüller space, arXiv preprint arXiv:2206.01371, 2022.
J. H. Sampson, Some properties and applications of harmonic mappings, Ann. Sci. École Norm. Sup. (4) 11 (1978), no. 2, 211–228. MR 510549, DOI 10.24033/asens.1344
Richard Schoen and Shing Tung Yau, On univalent harmonic maps between surfaces, Invent. Math. 44 (1978), no. 3, 265–278. MR 478219, DOI 10.1007/BF01403164
Yu-Ru Syau, On the existence of closed geodesics w.r.t. an admissible quadratic differential, Chinese J. Math. 24 (1996), no. 1, 69–79. MR 1399187
Michael Wolf, The Teichmüller theory of harmonic maps, J. Differential Geom. 29 (1989), no. 2, 449–479. MR 982185
Michael Wolf, High energy degeneration of harmonic maps between surfaces and rays in Teichmüller space, Topology 30 (1991), no. 4, 517–540. MR 1133870, DOI 10.1016/0040-9383(91)90037-5