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Harmonic diffeomorphisms and Teichmüller theory. (English) Zbl 0734.30038
The author gets a global parametrization of Teichmüller spaces of noncompact Riemann surfaces of finite type. If a closed geodesic on a surface shrinks to a point then the canonical Poincaré metric on the surface converges outside the geodesic to the Poincaré metric of the limit surface. A homeomorphism between puntured surfaces is homotopic to a unique harmonic diffeomorphism of finite energy. Then the author builds up a Teichmüller theory and gives a new proof of Teichmüller’s theorem for Riemann surfaces of finite type.
Reviewer: V.Oproiu (Iaşi)
MSC:
30F60Teichmüller theory
30F45Conformal metrics (hyperbolic, Poincaré, distance functions)
53C21Methods of Riemannian geometry, including PDE methods; curvature restrictions (global)
58J60Relations of PDE with special manifold structures
32G15Moduli of Riemann surfaces, Teichmüller theory
References:
[1][A1] Abikoff, W.: Degenerating families of Riemann Surfaces, Ann. of Math. 105, 29–44, 1977 · Zbl 0347.32010 · doi:10.2307/1971024
[2][A2] Abikoff, W.: The Real Analytic Theory of Teichmüller Space, LNM 820, Springer, 1980
[3][E] Eberlein, P.: Surfaces of Nonpositive Curvature, Memoirs of AMS 218 (1979)
[4][FK] Farkas, H. and I. Kra: Riemann Surfaces, Graduate Texts in Math. 71, Springer, 1980
[5][H] Heinz, E.: Über das Nichtverschwinden der Funktionaldeterminante bei einer Klasse eineindeutiger Abbildungen, Math. Z. 105, (1968), 87–89 · Zbl 0159.40203 · doi:10.1007/BF01110433
[6][J1] Jost, J.: Two-dimensional variational problems, Vorlesungsreihe no. 6, SFB 256, Bonn
[7][J2] Jost, J.: Harmonic Maps between Surfaces, Lecture Notes in Math. 1062, Springer, 1984
[8][N] Nug, S.: The Complex Analytic Theory of Teichmüller Spaces, Canadian Math. Soc., 1988
[9][S] Sampson, J.H.: Some properties and applications of harmonic mappings, Ann. Ec. Norm. Sup. XI (1978), 211–228
[10][SY1] Schoen, R. and S.T. Yau: On Univalent Harmonic Maps between Surfaces, Invent. Math. 44 (1978), 265–278 · Zbl 0388.58005 · doi:10.1007/BF01403164
[11][SY2] Schoen, R. and S.T. Yau: Compact Group Actions and the Topology of Manifolds with Non-positive Curvature, Topology 18, (1979), 361–380 · Zbl 0424.58012 · doi:10.1016/0040-9383(79)90026-0
[12][SY3] Schoen, R. and S.T. Yau: Harmonic Maps and the Topology of Stable Hypersurfaces and Manifolds of Non-negative Ricci curvature, Comm. Math. Helv., 51 (1976), 333–341 · Zbl 0361.53040 · doi:10.1007/BF02568161
[13][W1] Wolf, M.: The Teichmüller theory of harmonic maps, Journal of Diff. Geom. 29 (1989), 449–479
[14][W2] Wolf, M.: Infinite Energy Harmonic Maps and Degenration of the hyperbolic Surfaces in Moduli Space, preprint
[15][Y] Yau, S.T.: A general Schwarz Lemma for Kähler Manifolds, Amer. J. Math. 100, 1, (1978), 197–203 · Zbl 0424.53040 · doi:10.2307/2373880