Minimal immersions of closed Riemann surfaces

Authors:
J. Sacks and K. Uhlenbeck

Journal:
Trans. Amer. Math. Soc. **271** (1982), 639-652

MSC:
Primary 58E12; Secondary 53A10, 53C42, 58E20

MathSciNet review:
654854

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let be a closed orientable surface of genus larger than zero and a compact Riemannian manifold. If is a continuous map, such that the map induced by it between the fundamental groups of and contains no nontrivial element represented by a simple closed curve in its kernel, then there exists a conformal branched minimal immersion having least area among all branched immersions with the same action on as . Uniqueness within the homotopy class of fails in general: It is shown that for certain -manifolds which fiber over the circle there are at least two geometrically distinct conformal branched minimal immersions within the homotopy class of any inclusion map of the fiber. There is also a topological discussion of those -manifolds for which uniqueness fails.

**[1]**William Abikoff,*Degenerating families of Riemann surfaces*, Ann. of Math. (2)**105**(1977), no. 1, 29–44. MR**0442293****[2]**William Abikoff,*On boundaries of Teichmüller spaces and on Kleinian groups. III*, Acta Math.**134**(1975), 211–237. MR**0435452****[3]**-,*Topics in the real analytic theory of Teichmüller space*, mimeographed notes, Univ. of Illinois at Urbana-Champaign.**[4]**Lipman Bers,*Uniformization, moduli, and Kleinian groups*, Bull. London Math. Soc.**4**(1972), 257–300. MR**0348097****[5]**Lipman Bers,*On boundaries of Teichmüller spaces and on Kleinian groups. I*, Ann. of Math. (2)**91**(1970), 570–600. MR**0297992****[6]**Lipman Bers,*Spaces of degenerating Riemann surfaces*, Discontinuous groups and Riemann surfaces (Proc. Conf., Univ. Maryland, College Park, Md., 1973) Princeton Univ. Press, Princeton, N.J., 1974, pp. 43–55. Ann. of Math. Studies, No. 79. MR**0361051****[7]**Lipman Bers,*An extremal problem for quasiconformal mappings and a theorem by Thurston*, Acta Math.**141**(1978), no. 1-2, 73–98. MR**0477161****[8]**James Eells Jr. and J. H. Sampson,*Harmonic mappings of Riemannian manifolds*, Amer. J. Math.**86**(1964), 109–160. MR**0164306****[9]**Robert Gulliver,*Branched immersions of surfaces and reduction of topological type. II*, Math. Ann.**230**(1977), no. 1, 25–48. MR**0464078****[10]**R. D. Gulliver II, R. Osserman, and H. L. Royden,*A theory of branched immersions of surfaces*, Amer. J. Math.**95**(1973), 750–812. MR**0362153****[11]**Philip Hartman,*On homotopic harmonic maps*, Canad. J. Math.**19**(1967), 673–687. MR**0214004****[12]**Philip Hartman and Aurel Wintner,*On the local behavior of solutions of non-parabolic partial differential equations*, Amer. J. Math.**75**(1953), 449–476. MR**0058082****[13]**John Hempel,*3-Manifolds*, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1976. Ann. of Math. Studies, No. 86. MR**0415619****[14]**Troels Jørgensen,*Compact 3-manifolds of constant negative curvature fibering over the circle*, Ann. of Math. (2)**106**(1977), no. 1, 61–72. MR**0450546****[15]**Luc Lemaire,*Applications harmoniques de surfaces riemanniennes*, J. Differential Geom.**13**(1978), no. 1, 51–78 (French). MR**520601****[16]**L. A. Ljusternik,*The topology of the calculus of variations in the large*, Translated from the Russian by J. M. Danskin. Translations of Mathematical Monographs, Vol. 16, American Mathematical Society, Providence, R.I., 1966. MR**0217817****[17]**R. Schoen and Shing Tung Yau,*Existence of incompressible minimal surfaces and the topology of three-dimensional manifolds with nonnegative scalar curvature*, Ann. of Math. (2)**110**(1979), no. 1, 127–142. MR**541332**, 10.2307/1971247**[18]**J. Sacks and K. Uhlenbeck,*The existence of minimal immersions of 2-spheres*, Ann. of Math. (2)**113**(1981), no. 1, 1–24. MR**604040**, 10.2307/1971131**[19]**D. Sullivan,*Travaux de Thurston sur les groupes quasi-Fuchsiens et les variétés hyperboliques de dimension**fibrées sur*, Séminaire Bourbaki, 32e année (1979/80), no. 554, pp. 1-17.**[20]**W. Thurston,*The geometry and topology of*-*manifolds*, mimeographed notes, Princeton Univ., Princeton, N. J.**[21]**K. Uhlenbeck,*Morse theory by perturbation methods with applications to harmonic maps*, Trans. Amer. Math. Soc.**267**(1981), no. 2, 569–583. MR**626490**, 10.1090/S0002-9947-1981-0626490-X

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
58E12,
53A10,
53C42,
58E20

Retrieve articles in all journals with MSC: 58E12, 53A10, 53C42, 58E20

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1982-0654854-8

Article copyright:
© Copyright 1982
American Mathematical Society