Minimal surfaces in manifolds with actions and the simple loop conjecture for Seifert fibered spaces

Author:
Joel Hass

Journal:
Proc. Amer. Math. Soc. **99** (1987), 383-388

MSC:
Primary 57N10; Secondary 53A10

MathSciNet review:
870806

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Abstract: The Simple Loop Conjecture for -manifolds states that if a -sided map from a surface to a -manifold fails to inject on the fundamental group, then there is an essential simple loop in the kernel. This conjecture is solved in the case where the -manifold is Seifert fibered. The techniques are geometric and involve studying least area surfaces and circle actions on Seifert Fibered Spaces.

**[FHS]**Michael Freedman, Joel Hass, and Peter Scott,*Least area incompressible surfaces in 3-manifolds*, Invent. Math.**71**(1983), no. 3, 609–642. MR**695910**, 10.1007/BF02095997**[G]**L. Z. Gao,*Applications of minimal surfaces theory to topology and Riemannian geometry constructions of negative Ricci curved manifolds*, Thesis, S.U.N.Y. at Stony Brook, 1985.**[Ga]**David Gabai,*The simple loop conjecture*, J. Differential Geom.**21**(1985), no. 1, 143–149. MR**806708****[Gu]**Robert D. Gulliver II,*Regularity of minimizing surfaces of prescribed mean curvature*, Ann. of Math. (2)**97**(1973), 275–305. MR**0317188****[H]**Joel Hass,*Minimal surfaces in Seifert fiber spaces*, Topology Appl.**18**(1984), no. 2-3, 145–151. MR**769287**, 10.1016/0166-8641(84)90006-3**[HK]**J. Hass and L. Karp,*Local properties of minimal submanifolds*(in preparation).**[HS]**J. Hass and P. Scott,*Existence theorems for minimal surfaces in**-manifolds*. I, preprint.**[L]**H. B. Lawson, Jr.,*Minimal varieties in real and complex geometry*, Univ. of Montreal Press, 1973.**[MY]**William H. Meeks III and Shing Tung Yau,*Topology of three-dimensional manifolds and the embedding problems in minimal surface theory*, Ann. of Math. (2)**112**(1980), no. 3, 441–484. MR**595203**, 10.2307/1971088**[O]**Robert Osserman,*A proof of the regularity everywhere of the classical solution to Plateau’s problem*, Ann. of Math. (2)**91**(1970), 550–569. MR**0266070****[S]**Peter Scott,*The geometries of 3-manifolds*, Bull. London Math. Soc.**15**(1983), no. 5, 401–487. MR**705527**, 10.1112/blms/15.5.401**[SU]**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**[SY]**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**[Se]**J. Serrin,*The problem of Dirichlet for quasilinear elliptic differential equations with many independent variables*, Philos. Trans. Roy. Soc. London Ser. A**264**(1969), 413–496. MR**0282058**

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DOI:
http://dx.doi.org/10.1090/S0002-9939-1987-0870806-7

Article copyright:
© Copyright 1987
American Mathematical Society