Date of Degree

5-2015

Document Type

Dissertation

Degree Name

Ph.D.

Program

Mathematics

Advisor

Jeremy Kahn

Subject Categories

Mathematics

Keywords

conformal embeddings; extremal length; h-principle; quadratic differentials; Teichmuller's theorem

Abstract

We prove that if two conformal embeddings between Riemann surfaces with finite topology are homotopic, then they are isotopic through conformal embeddings. Furthermore, we show that the space of all conformal embeddings in a given homotopy class deformation retracts into a point, a circle, a torus, or the unit tangent bundle of the codomain, depending on the induced homomorphism on fundamental groups. Quadratic differentials play a central role in the proof.

Included in

Mathematics Commons

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