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Boundary and Holder regularities of Douady-Earle extensions and eigenvalues of Laplace operators acting on Riemann surfaces
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Pal, Susovan. Boundary and Holder regularities of Douady-Earle extensions and eigenvalues of Laplace operators acting on Riemann surfaces. Retrieved from https://doi.org/doi:10.7282/T3S46Q0K

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TitleBoundary and Holder regularities of Douady-Earle extensions and eigenvalues of Laplace operators acting on Riemann surfaces
NamePal, Susovan (author); Luo, Feng (chair); Gilman, Jane (internal member); Li, YanYan (internal member); Hu, Jun (outside member); Rutgers University; Graduate School - New Brunswick
Date Created2013
Other Date2013-10 (degree)
SubjectMathematics, Riemann surfaces, Teichmüller spaces, Homeomorphisms
Extentvii, 36 p. : ill.
DescriptionDouady-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
NotePh.D.
NoteIncludes bibliographical references
NoteIncludes vita
Noteby Susovan Pal
Genretheses, ETD doctoral
Persistent URLhttps://doi.org/doi:10.7282/T3S46Q0K
Languageeng
CollectionGraduate School - New Brunswick Electronic Theses and Dissertations
Organization NameRutgers, The State University of New Jersey
RightsThe author owns the copyright to this work.
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