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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

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
DOI: https://doi.org/10.1090/S0002-9947-1982-0654854-8
MathSciNet review: 654854
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Abstract: Let $ M$ be a closed orientable surface of genus larger than zero and $ N$ a compact Riemannian manifold. If $ u:M \to N$ is a continuous map, such that the map induced by it between the fundamental groups of $ M$ and $ N$ contains no nontrivial element represented by a simple closed curve in its kernel, then there exists a conformal branched minimal immersion $ s:M \to N$ having least area among all branched immersions with the same action on $ {\pi _1}(M)$ as $ u$. Uniqueness within the homotopy class of $ u$ fails in general: It is shown that for certain $ 3$-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 $ 3$-manifolds for which uniqueness fails.


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DOI: https://doi.org/10.1090/S0002-9947-1982-0654854-8
Article copyright: © Copyright 1982 American Mathematical Society