The Douady-Earle extensions are not always harmonic

Jiang, Manman; Liu, Lixin; Yao, Hongyu

doi:10.1090/proc/13047

The Douady-Earle extensions are not always harmonic
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by Manman Jiang, Lixin Liu and Hongyu Yao

Proc. Amer. Math. Soc. 146 (2018), 2853-2865

DOI: https://doi.org/10.1090/proc/13047

Published electronically: March 19, 2018

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Abstract:

In this paper we give examples to show that the Douady-Earle extensions are not always harmonic. Furthermore, we discuss the criterion for the Douady-Earle extensions to be harmonic.

References

S. L. Al’ber, On n-dimensional problems in the calculus of variations in the large, Sov. Math. Dokl 5 (1964), 700–704.
Kazuo Akutagawa, Harmonic diffeomorphisms of the hyperbolic plane, Trans. Amer. Math. Soc. 342 (1994), no. 1, 325–342. MR 1147398, DOI 10.1090/S0002-9947-1994-1147398-9
A. Beurling and L. Ahlfors, The boundary correspondence under quasiconformal mappings, Acta Math. 96 (1956), 125–142. MR 86869, DOI 10.1007/BF02392360
Adrien Douady and Clifford J. Earle, Conformally natural extension of homeomorphisms of the circle, Acta Math. 157 (1986), no. 1-2, 23–48. MR 857678, DOI 10.1007/BF02392590
Clifford J. Earle and James Eells, A fibre bundle description of Teichmüller theory, J. Differential Geometry 3 (1969), 19–43. MR 276999
Clifford J. Earle, Conformally natural extension of vector fields from $S^{n-1}$ to $B^n$, Proc. Amer. Math. Soc. 102 (1988), no. 1, 145–149. MR 915733, DOI 10.1090/S0002-9939-1988-0915733-2
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
Robert Hardt and Michael Wolf, Harmonic extensions of quasiconformal maps to hyperbolic space, Indiana Univ. Math. J. 46 (1997), no. 1, 155–163. MR 1462800, DOI 10.1512/iumj.1997.46.1351
Philip Hartman, On homotopic harmonic maps, Canadian J. Math. 19 (1967), 673–687. MR 214004, DOI 10.4153/CJM-1967-062-6
Jürgen Jost, Two-dimensional geometric variational problems, Pure and Applied Mathematics (New York), John Wiley & Sons, Ltd., Chichester, 1991. A Wiley-Interscience Publication. MR 1100926
Peter Li and Luen-Fai Tam, Uniqueness and regularity of proper harmonic maps, Ann. of Math. (2) 137 (1993), no. 1, 167–201. MR 1200080, DOI 10.2307/2946622
Peter Li and Luen-Fai Tam, Uniqueness and regularity of proper harmonic maps. II, Indiana Univ. Math. J. 42 (1993), no. 2, 591–635. MR 1237061, DOI 10.1512/iumj.1993.42.42027
Peter Li and Luen-Fai Tam, The heat equation and harmonic maps of complete manifolds, Invent. Math. 105 (1991), no. 1, 1–46. MR 1109619, DOI 10.1007/BF01232256
Curtis T. McMullen, Renormalization and 3-manifolds which fiber over the circle, Annals of Mathematics Studies, vol. 142, Princeton University Press, Princeton, NJ, 1996. MR 1401347, DOI 10.1515/9781400865178
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 M. Schoen, The role of harmonic mappings in rigidity and deformation problems, Complex geometry (Osaka, 1990) Lecture Notes in Pure and Appl. Math., vol. 143, Dekker, New York, 1993, pp. 179–200. MR 1201611
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
Yuguang Shi, Luen-Fai Tam, and Tom Y.-H. Wan, Harmonic maps on hyperbolic spaces with singular boundary value, J. Differential Geom. 51 (1999), no. 3, 551–600. MR 1726739
Luen-Fai Tam and Tom Y. H. Wan, Quasi-conformal harmonic diffeomorphism and the universal Teichmüller space, J. Differential Geom. 42 (1995), no. 2, 368–410. MR 1366549
A. J. Tromba, Global analysis and Teichmüller theory, Seminar on new results in nonlinear partial differential equations (Bonn, 1984) Aspects Math., E10, Friedr. Vieweg, Braunschweig, 1987, pp. 167–198. MR 896283
Pekka Tukia, Quasiconformal extension of quasisymmetric mappings compatible with a Möbius group, Acta Math. 154 (1985), no. 3-4, 153–193. MR 781586, DOI 10.1007/BF02392471
P. Tukia and J. Väisälä, Quasiconformal extension from dimension $n$ to $n+1$, Ann. of Math. (2) 115 (1982), no. 2, 331–348. MR 647809, DOI 10.2307/1971394
Tom Yau-Heng Wan, Constant mean curvature surface, harmonic maps, and universal Teichmüller space, J. Differential Geom. 35 (1992), no. 3, 643–657. MR 1163452
Michael Wolf, The Teichmüller theory of harmonic maps, J. Differential Geom. 29 (1989), no. 2, 449–479. MR 982185
Michael Wolf, Infinite energy harmonic maps and degeneration of hyperbolic surfaces in moduli space, J. Differential Geom. 33 (1991), no. 2, 487–539. MR 1094467