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Harmonic surfaces in 3-manifolds and the simple loop theorem

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Vladimir Markovich
Fri, 2019-07-12 12:00 - 13:00
MPIM Lecture Hall

Denote by ${\eufm M}(\Sigma)$ the space of hyperbolic metrics on a closed, orientable surface $\Sigma$ and by ${\eufm M}(M)$ the space of negatively curved Riemannian metrics on a closed, orientable 3-manifold $M$. We show that the set of metrics for which the corresponding harmonic map is in Whitney's general position is an open, dense, and connected subset of ${\eufm M}(\Sigma)\times {\eufm M}(M)$. The main application of this result is the proof of the Simple Loop Theorem for hyperbolic 3-manifolds. Consequences regarding minimal surfaces will be mentioned.

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