×

Harmonic diffeomorphisms of the hyperbolic plane. (English) Zbl 0796.58014

Starting with an immersion (resp., a diffeomorphism) \(\varphi:D(\infty) \to D (\infty)\) of the boundary at \(\infty\) of the Poincaré model \(D\) of the hyperbolic plane, the author finds a harmonic extension (resp., a harmonic diffeomorphism) \(u:D \to D\). Moreover, \(u\) is \(\pm\) holomorphic (resp., conformal) iff \(\varphi\) is conformal. The proof proceeds by construction of a suitable barrier map at \(D(\infty)\) associated to each \(\varphi\) – and to obtain from it an a priori growth estimate. He also constructs entire spacelike constant mean curvature surfaces \(M\) in Minkowski 3-space whose Gauss maps are harmonic diffeomorphisms \(M \to H^ 2\).
Related results for \(D^ m \to D^ n\) have been obtained by P. Li and L.-F. Tam [Invent. Math. 105, No. 1, 1-46 (1991; Zbl 0748.58006)].

MSC:

58E20 Harmonic maps, etc.
30C20 Conformal mappings of special domains
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)

Citations:

Zbl 0748.58006
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Lars V. Ahlfors, Complex analysis, 3rd ed., McGraw-Hill Book Co., New York, 1978. An introduction to the theory of analytic functions of one complex variable; International Series in Pure and Applied Mathematics. · Zbl 0395.30001
[2] Kazuo Akutagawa, The Dirichlet problem at infinity for harmonic mappings between Hadamard manifolds, Geometry of manifolds (Matsumoto, 1988) Perspect. Math., vol. 8, Academic Press, Boston, MA, 1989, pp. 59 – 70.
[3] Kazuo Akutagawa and Seiki Nishikawa, The Gauss map and spacelike surfaces with prescribed mean curvature in Minkowski 3-space, Tohoku Math. J. (2) 42 (1990), no. 1, 67 – 82. · Zbl 0679.53002 · doi:10.2748/tmj/1178227694
[4] Kazuo Akutagawa and Atsushi Tachikawa, Nonexistence results for harmonic maps between noncompact complete Riemannian manifolds, Tokyo J. Math. 16 (1993), no. 1, 131 – 145. · Zbl 0791.53044 · doi:10.3836/tjm/1270128986
[5] Michael T. Anderson and Richard Schoen, Positive harmonic functions on complete manifolds of negative curvature, Ann. of Math. (2) 121 (1985), no. 3, 429 – 461. · Zbl 0587.53045 · doi:10.2307/1971181
[6] P. Avilés, H.I. Choi, and M. J. Micallef, Boundary behavior of harmonic maps of non-smooth domains and complete negatively curved manifolds, Preprint. · Zbl 0805.53037
[7] Hyeong In Choi and Andrejs Treibergs, New examples of harmonic diffeomorphisms of the hyperbolic plane onto itself, Manuscripta Math. 62 (1988), no. 2, 249 – 256. · Zbl 0669.58012 · doi:10.1007/BF01278983
[8] Hyeong In Choi and Andrejs Treibergs, Gauss maps of spacelike constant mean curvature hypersurfaces of Minkowski space, J. Differential Geom. 32 (1990), no. 3, 775 – 817. · Zbl 0717.53038
[9] P. Eberlein and B. O’Neill, Visibility manifolds, Pacific J. Math. 46 (1973), 45 – 109. · Zbl 0264.53026
[10] Richard S. Hamilton, Harmonic maps of manifolds with boundary, Lecture Notes in Mathematics, Vol. 471, Springer-Verlag, Berlin-New York, 1975. · Zbl 0308.35003
[11] Stéfan Hildebrandt, Helmut Kaul, and Kjell-Ove Widman, An existence theorem for harmonic mappings of Riemannian manifolds, Acta Math. 138 (1977), no. 1-2, 1 – 16. · Zbl 0356.53015 · doi:10.1007/BF02392311
[12] Willi Jäger and Helmut Kaul, Uniqueness and stability of harmonic maps and their Jacobi fields, Manuscripta Math. 28 (1979), no. 1-3, 269 – 291. · Zbl 0413.31006 · doi:10.1007/BF01647975
[13] Jürgen Jost, Harmonic maps between surfaces, Lecture Notes in Mathematics, vol. 1062, Springer-Verlag, Berlin, 1984. · Zbl 0542.58002
[14] Jürgen Jost and Hermann Karcher, Geometrische Methoden zur Gewinnung von a-priori-Schranken für harmonische Abbildungen, Manuscripta Math. 40 (1982), no. 1, 27 – 77 (German, with English summary). · Zbl 0502.53036 · doi:10.1007/BF01168235
[15] Peter Li and Luen-Fai Tam, The heat equation and harmonic maps of complete manifolds, Invent. Math. 105 (1991), no. 1, 1 – 46. · Zbl 0748.58006 · doi:10.1007/BF01232256
[16] Peter Li and Luen-Fai Tam, Uniqueness and regularity of proper harmonic maps, Ann. of Math. (2) 137 (1993), no. 1, 167 – 201. · Zbl 0776.58010 · doi:10.2307/2946622
[17] Tilla Klotz Milnor, Harmonic maps and classical surface theory in Minkowski 3-space, Trans. Amer. Math. Soc. 280 (1983), no. 1, 161 – 185. · Zbl 0532.53047
[18] Richard Schoen and Shing Tung Yau, On univalent harmonic maps between surfaces, Invent. Math. 44 (1978), no. 3, 265 – 278. · Zbl 0388.58005 · doi:10.1007/BF01403164
[19] Atsushi Tachikawa, Harmonic mappings from \?^{\?} into an Hadamard manifold, J. Math. Soc. Japan 42 (1990), no. 1, 147 – 153. · Zbl 0701.58019 · doi:10.2969/jmsj/04210147
[20] Andrejs E. Treibergs, Entire spacelike hypersurfaces of constant mean curvature in Minkowski space, Invent. Math. 66 (1982), no. 1, 39 – 56. · Zbl 0483.53055 · doi:10.1007/BF01404755
[21] Peter Li and Luen-Fai Tam, Uniqueness and regularity of proper harmonic maps. II, Indiana Univ. Math. J. 42 (1993), no. 2, 591 – 635. · Zbl 0790.58011 · doi:10.1512/iumj.1993.42.42027
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.