Douady-Earle extensions of homeomorphisms of the unit circle are of particular interest in understanding contractibility and complex structures of Teichmueller and assymp-totic 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

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... , 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 and given any ii big natural number k, we can construct a Riemann surface whose k-th eigenvalue is less than. The result was first proved by Burton randol, here we provide a much simpler and geometric proof iii