## Dirichlet’s Principle, Conformal Mapping, and Minimal SurfacesIt has always been a temptation for mathematicians to present the crystallized product of their thoughts as a deductive general theory and to relegate the individual mathematical phenomenon into the role of an example. The reader who submits to the dogmatic form will be easily indoctrinated. Enlightenment, however, must come from an understanding of motives; live mathematical development springs from specific natural problems which can be easily understood, but whose solutions are difficult and demand new methods of more general significance. The present book deals with subjects of this category. It is written in a style which, as the author hopes, expresses adequately the balance and tension between the individuality of mathematical objects and the generality of mathematical methods. The author has been interested in Dirichlet's Principle and its various applications since his days as a student under David Hilbert. Plans for writing a book on these topics were revived when Jesse Douglas' work suggested to him a close connection between Dirichlet's Principle and basic problems concerning minimal sur faces. But war work and other duties intervened; even now, after much delay, the book appears in a much less polished and complete form than the author would have liked." |

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### Contents

1 | |

Semicontinuity of Dirichlets integral Dirichlets Principle for cir | 11 |

Proof of Dirichlets Principle for general domains | 23 |

Alternative proof of Dirichlets Principle | 31 |

Conformal mapping of simply and doubly connected domains | 38 |

Plateaus Problem | 95 |

The General Problem of Douglas | 141 |

W Conformal Mapping of Multiply Connected Domains | 167 |

Minimal surfaces with partly free boundaries | 206 |

Minimal surfaces spanning closed manifolds | 213 |

Unstable minimal surfaces with prescribed polygonal boundaries | 223 |

Unstable minimal surfaces in rectifiable contours | 236 |

Bibliography Chapters I to WI | 245 |

Dirichlet integrals for harmonic functions | 266 |

Variation of the Greens function | 292 |

319 | |

W Conformal Mapping of Multiply Connected DomainsContinued | 184 |

WI Minimal Surfaces with Free Boundaries and Unstable Minimal Sur | 199 |

Supplementary Notes 1977 | 331 |

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### Common terms and phrases

admissible function admissible vectors analytic function arbitrarily small arbitrary assume boundary curve boundary points boundary slit boundary value problem branch points catenoid cells Chapter circular closed curve closed manifold closed subdomain condition conformal mapping consider constant continuous contour converge coordinates corresponding defined degeneration denote derivatives differential Dirichlet's integral Dirichlet's Principle disk domain G doubly connected equation existence Figure finite number formula free boundary function f(z genus zero greatest lower bound Green’s function harmonic function harmonic vectors Hence inequality infinity interior Jordan arc Jordan curves kernel least area lemma lim inf mapping theorem minimizing sequence neighborhood obtain parameter domain piecewise smooth plane domain Plateau's problem point of G polygon proof proved radius Riemann domain Riemann surface semicontinuity simply connected single-valued solution stationary streamlines sufficiently small surface G surface of least tends to zero tion transformation uniformly unit circle univalent unstable minimal surfaces vanishes variational problem w-plane