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Boundary and holder regularities of douady-earle extensions and eigenvalues of laplace operators acting on riemann surfaces.

机译：作用于riemann曲面上的douady-eare扩展的边界和持有者规则以及拉普拉斯算子的特征值。

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摘要

Douady-Earle extensions of homeomorphisms of the unit circle are of particular interest in understanding contractibility and complex structures of Teichmueller and assymptotic Teichmueller spaces. Motivated by questions in analysis and partial differential equations, one can ask how regular the Douady-Earle extensions can be on the closed unit disk if one puts sufficient regularity on the circle homeomorphisms to start with. In first part of this thesis which consists of the first four chapters, we prove that Douady-Earle extensions of Holder continuous circle homeomorphisms are Holder continuous with the same Holder exponent, and Douady-Earle extensions of circle diffeomorphisms are diffeomorphisms of the closed unit disk. Eigenvalues of Laplace operators on Riemannian manifolds are widely studied by differential geometers. But when the manifold is a hyperbolic Riemann surface, the problem becomes more special, because the collar lemma and the minimax principles allow us to construct functions which produce lower and upper bounds on eigenvalues on that Riemann surface. In the second part of this thesis consisting of chapters 5 and 6, we show, using the minimax principles, given any small positive number epsilon and given any big natural number k, we can construct a Riemann surface whose k-th eigenvalue is less than epsilon. The result was first proved by Burton randol, here we provide a much simpler and geometric proof.
机译：单位圆的同胚性的杜亚迪-厄尔扩展对理解Teichmueller和渐近Teichmueller空间的可收缩性和复杂结构特别感兴趣。受分析和偏微分方程问题的启发，人们可以问，如果人们对圆同胚性设定足够的正则性，那么在封闭的单位圆盘上，杜亚迪-厄尔扩展可以有多规则。在由前四章组成的本论文的第一部分中，我们证明了Holder连续圆同胚性的Douady-Earle扩展是具有相同Holder指数的Holder连续，并且圆微分形的Douady-Earle扩展是闭合单位圆盘的微分。 。黎曼流形上的拉普拉斯算子的特征值已被微分几何广泛研究。但是，当流形是双曲Riemann曲面时，问题变得更加特殊，因为项圈引理和极小极大原理使我们能够构造函数，从而在该Riemann曲面上生成特征值的上下界。在本论文的第二部分（由第5章和第6章组成）中，我们展示了使用极小极大原理，给定任何小的正数ε和给定的自然数k，我们可以构造第k个特征值小于epsilon。结果首先由Burton randol证明，在这里我们提供了更简单和几何的证明。

著录项

  • 作者

    Pal, Susovan.;

  • 作者单位

    Rutgers The State University of New Jersey - New Brunswick.;

  • 授予单位 Rutgers The State University of New Jersey - New Brunswick.;
  • 学科 Mathematics.;Applied Mathematics.
  • 学位 Ph.D.
  • 年度 2013
  • 页码 43 p.
  • 总页数 43
  • 原文格式 PDF
  • 正文语种 eng
  • 中图分类
  • 关键词

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