On the number of simply connected minimal surfaces spanning a curve / [electronic resource] A. J. Tromba.

Bibliography: p. 118-121.

I. A review of the Euler characteristic of a Palais-Smale vector field II. Analytical preliminaries - the Sobelev spaces III. The global formulation of the problem of Plateau IV. The existence of a vector field associated to the Dirichlet functional $E_\alpha $ V. A proof that the vector field $X^\alpha $, associated to $E_\alpha $, is Palais-Smale VI. The weak Riemannian structure on $\mathcal {N}_\alpha $ VII. The equivariance of $X^\alpha $ under the action of the conformal group VIII. The regularity results for minimal surfaces IX. The Fr�echet derivative of the minimal surface vector field $X$ and the surface fibre bundle X. The minimal surface vector field $X$ is proper on bounded sets XI. Non-degenerate critical submanifolds of $\mathcal {N}_\alpha $ and a uniqueness theorem for minimal surfaces XII. The spray of the weak metric XIII. The transversality theorem XIV. The Morse number of minimal surfaces spanning a simple closed curve and its invariance under isotopy

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Electronic reproduction. Providence, Rhode Island : American Mathematical Society. 2012

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