Jost J.  Minimal surfaces and teichmueller theory

Название: 
Minimal surfaces and teichmueller theory 
Автор: 
Jost J. 
Категория: 
Математика

Тип: 
Книга 
Дата: 
31.12.2008 00:04:06 
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35 
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1 Introduction These are the notes from a set of lectures I delivered at National TsingHua University in Hsinchu, Taiwan, in the spring of 1992. In these notes, first the Plateau problem for orientable and nonorientable minimal surfaces of arbitrary genus is solved in Euclidean space and in Riemannian manifolds. We also present as applications some new examples of Plateau problems with infinitely many solutions. In the second chapter, Teichmueller theory for orientable Riemann surfaces is sketched. In particular, we introduce some kind of enlarged Teichmueller space consisting of both orientable and nonorientable surfaces. This space is stratifed by genus. We hope that the constructions presented there will have further applications. In the third chapter, the Dirichlet energy functional is considered as a functional on this enlarged Teichmueller space. Estimates of the behaviour of this functional near lower dimensional strata are desired. These are the basis for a H. MorseConley theory for minimal surfaces of varying genus and orientability character in Riemannian manifolds, at least in those of nonpositive curvature. This will constitute an extension of my work with Struwe [JS]. In my lectures, however, I did not manage to cover this topic because of lack of time, and therefore, it is also not presented in the present lecture notes. I intend to write up the complete theory elsewhere. I thank Professor ShingTung Yau for arranging this stimulating visit at Hsinchu, and the department's chairman, Professor HengLung Lai and his colleagues for making my stay in Taiwan most pleasant, professionally and personally.
1.1 Plateau's problem in Euclidean space. Let 7 be a closed Jordan curve in Euclidean space Ed of dimension d > 2. We want to find a minimal surface С bounded by 7. A promising approach is to minimize area among all surfaces bounded by 7. If this approach is to be made rigorous, one first needs to define what a surface is, what a minimal one is, what is meant by the statement that a surface is bounded by 7, and what the area of a surface is. The difficulty involved here is that it is a priori not clear that the minimum of area is achieved by a smooth surface and the class of smooth surfaces is not closed under any type of convergence that naturally arises in the context of the area problem. Thus if one takes an area minimizing sequence in the class of smooth surfaces bounded by 7 and uses some compactness theorem in order to pass to some kind of weak limit, one may obtain a rather irregular object to which it may not even be easy to assign an area in a meaningful manner. Mathematicians have found various methods to resolve these difficulties. We present here the one that was first successfully applied to Plateau's problem (to find a minimal surface bounded by 7) and that will lead us to the theory with the richest mathematical content 
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