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On the number of simply connected minimal surfaces spanning a curve

About this Title

A. J. Tromba

Publication: Memoirs of the American Mathematical Society
Publication Year: 1977; Volume 12, Number 194
ISBNs: 978-0-8218-2194-7 (print); 978-1-4704-0155-9 (online)
DOI: https://doi.org/10.1090/memo/0194
MathSciNet review: 0649458

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Table of Contents

Chapters

  • 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 Fréchet 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