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- W2316062341 abstract "In this chapter we shall present an introduction to the general problem of Plateau that, justifiedly, is often called the Douglas problem. This is the question whether a configuration of several closed curves Γ j may bound multiply connected minimal surfaces that could be of higher genus and even nonorientable. After a brief introduction to the problem we shall exhibit in Section 11.1 several examples of boundary configurations and of minimal surfaces illustrating the general Plateau problem. In Section 11.2 we shall outline an approach to Teichmuller theory of compact oriented Riemann surfaces which, in Section 11.3, is extended to oriented Riemann surfaces with boundary. These results are basic for the variational treatment of the general Plateau problem as we have to deal with variations of the complex structure of two-manifolds. The second basic tool is a compactness theorem of Mumford which will be combined with Courant’s condition of cohesion to obtain nondegenerate solutions of the variational problem connected with the general Plateau problem. This variational problem will be discussed in Section 11.5. We shall outline the crucial ideas for solving this problem and state some of the principal existence results due to Douglas, Courant and Shiffman. Finally, in Section 11.6, we shall tackle the general Plateau problem in combination with an obstacle problem in order to obtain a sufficient condition for the solvability of the general Douglas-Plateau problem that can be formulated in geometric-topological terms." @default.
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- W2316062341 date "1992-01-01" @default.
- W2316062341 modified "2023-09-23" @default.
- W2316062341 title "The General Problem of Plateau" @default.
- W2316062341 doi "https://doi.org/10.1007/978-3-662-08776-3_6" @default.
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