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On immersions of constant mean curvature: Compactness results and finiteness theorems for Plateau's problem

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Abstract

Assuming stability and integral conditions we show that a sequence of immersed surfaces of constant mean curvatureH converges to an immersedH-surface. The latter theorem depends on an oscillation estimate forH-surfaces based on an isoperimetric inequality. These compactness results are utilized to prove that certain Jordan curvesΓ only bound finitely many stable and unstable, immersed, smallH-surfaces.

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Communicated by J. C. C.Nitsche

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Sauvigny, F. On immersions of constant mean curvature: Compactness results and finiteness theorems for Plateau's problem. Arch. Rational Mech. Anal. 110, 125–140 (1990). https://doi.org/10.1007/BF00873495

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Keywords

  • Neural Network
  • Complex System
  • Nonlinear Dynamics
  • Electromagnetism
  • Integral Condition