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The geometry of locally symmetric manifolds via natural maps

Harmonic maps from surfaces into locally symmetric manifolds of non-positive curvature can be used to characterize such manifolds up to isometry. Namely, if the fundamental group of such a manifold is isomorphic to a surface group, then for every conformal structure on the corresponding surface,  there exists a unique harmonic map whose energy defines a function on Teichmueller space. This energy profile is intimately related to the geometry of the locally symmetric manifold. The main goal of this project is to explore this energy profile to study the geometry of the following classes of such locally symmetric manifolds.

1) Doubly degenerate hyperbolic 3-manifolds and random hyperbolic three-manifolds.

2) Locally symmetric manifolds in the Hitchin component of character variety obtained from a surface in the Fuchsian locus by grafting.

The approach builds on an energy profile for surfaces obtained from hyperbolic surfaces by grafting. Furthermore, other natural maps from surfaces will also be studied to achieve the main goal.

Publications

Extending earlier work of Tian, we show that if a manifold admits a metric that is almost hyperbolic in a suitable sense, then there exists an Einstein metric that is close to the given metric in the $$C^{2,\alpha}$$-topology. In dimension 3 the original manifold only needs to have finite volume, and the volume can be arbitrarily large. Applications include a new proof of the hyperbolization of 3-manifolds of large Hempel distance yielding some new geometric control on the hyperbolic metric, and an analytic proof of Dehn filling and drilling that allows the filling and drilling of arbitrary many cusps and tubes.

Related project(s):
51The geometry of locally symmetric manifolds via natural maps

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Team Members

Prof. Dr. Ursula Hamenstädt