## Two-Dimensional Geometric Variational ProblemsThis monograph treats variational problems for mappings from a surface equipped with a conformal structure into Euclidean space or a Riemannian manifold. Presents a general theory of such variational problems, proving existence and regularity theorems with particular conceptual emphasis on the geometric aspects of the theory and thorough investigation of the connections with complex analysis. Among the topics covered are: Plateau's problem, the regularity theory of solutions, a variational approach for obtaining various conformal representation theorems, a general existence theorem for harmonic mappings, and a new approach to Teichmuller theory via harmonic maps. |

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### Contents

Regularity and uniqueness results | 32 |

Conformal representation | 85 |

Existence results | 106 |

Copyright | |

3 other sections not shown

### Common terms and phrases

apply argument assume assumption ball boundary values bounded centred choose closed compact complete compute condition conformal conformal map consequence consider constant construction contained continuous converges uniformly convex coordinates Corollary corresponding covering curvature curve define definition denote depends derivatives diffeomorphism disk domain energy Equation equicontinuous equivalent estimate everywhere example exists fixed follows free boundary function fundamental genus geodesic geometry given harmonic map hence holds holomorphic quadratic differential homeomorphic homotopic implies independent induced integral interior Lemma limit metric minimal surface monotonically Moreover namely non-trivial normal obtain oriented parametrized particular positive possible present problem projection proof of Theorem prove radius regularity Remark replacement result Riemannian manifold satisfies sequence side smooth solution space structure subsequence sufficiently Suppose uniqueness unit variational vector weak weakly zero