Two-dimensional geometric variational problems
Two-dimensional geometric variational problems
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发表时间：
1991
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通讯作者：
J. Jost
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作者：
J. Jost
第一部分例子、定义和基本结果：Plateau问题二维共形不变变分问题调和映射、共形映射和全纯二次微分全纯二次微分曲面在R ~ 3中的一些应用高斯映射。第二部分正则性与唯一性结果：调和坐标调和映射的唯一性弱解的连续性孤立奇点的可移性高正则性Hartmann-Wintner引理及其某些结果分支点的渐近展开式单变调和映射的泛函行列式的下推估计.第三部分保角表示：与S2和圆域同胚的曲面和高亏格的闭曲面的保角表示。第四部分存在结果：局部存在定理-能量极小化映射存在性的简单证明一般存在定理一般存在定理的推论和后果边界条件不存在结果-具有全纯二次微分的映射的存在性不稳定极小曲面存在性的另一个证明黎曼流形中的Plateau-Douglas问题。第五部分曲面间的调和映射：调和同态的存在性，局部计算，非正曲像度量的推论，调和同态，Kneser定理，调和映射和分支覆盖。第六部分调和映射与Teichmuller空间：基本定义，Teichmuller定理的拓扑结构和可微结构，复结构，能量作为区域度量的函数，度量结构，Weil-Petersson度量，Kahler性质。
Part 1 Examples, definitions, and elementary results: Plateau's problem two dimensional conformally invariant variational problems harmonic maps, conformal maps and holomorphic quadratic differentials some applications of holomorphic quadratic differentials surfaces in R3 the Gauss map. Part 2 Regularity and uniqueness results: harmonic co-ordinates uniqueness of harmonic maps continuity of weak solutions removability of isolated singularities higher regularity the Hartmann-Wintner lemma and some of its consequences asymptotic expansions at branch points estimates from below for the functional determinant of univarlent harmonic mappings. Part 3 Conformal representation: conformal representation of surfaces homeomorphic to S2 and circular domains and closed surfaces of higher genus. Part 4 Existence results: the local existence theorem - an easy proof of the existence of energy minimizing maps the general existence theorem corollaries and consequences of the general existence theorem - boundary conditions nonexistence results - existence of maps with holomorphic quadratic differentials another proof of the existence of unstable minimal surfaces the Plateau-Douglas problem in Riemannian manifolds. Part 5 Harmonic maps between surfaces: the existence of harmonic diffeomorphisms local computations consequences for nonpositively curved image metrics harmonic diffeomorphisms Kneser's Theorem harmonic maps and branched coverings. Part 6 Harmonic maps and Teichmuller spaces: the basic definitions the topological and differentiable structure of Teichmuller's Theorem the complex structure the energy as a function of the domain metric the metric structure the Weil-Petersson metric Kahler property.