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
We show that for every \(n \geq 2\) and \(D > 0\) there exist a convex domain \(\Omega \subseteq \mathbb{H}^n\) with diameter \(D\) and a convex potential \(V\) on \(\Omega\) such that the fundamental gap of the operator \(-\Delta + V\) is strictly smaller than the fundamental gap of \(-\Delta\). In comparison to previous work, this result requires more refined control of the eigenfunctions.
Related project(s):
51The geometry of locally symmetric manifolds via natural maps
We show that if a closed manifold of dimension at least four admits a negatively curved metric that is almost Einstein in a suitable sense, then it admits a genuine Einstein metric of negative sectional curvature. Importantly, the pinching constant measuring the almost-Einstein condition neither depends on an upper bound for the diameter or volume, nor on a lower bound for the injectivity radius.
Related project(s):
51The geometry of locally symmetric manifolds via natural maps
For every \(n \geq 4\) we construct infinitely many mutually not homotopic closed manifolds of dimension \(n\) which admit a negatively curved Einstein metric but no locally symmetric metric.
| Journal | To apper in Journal of the European Mathematical Society (JEMS). |
| Link to preprint version |
Related project(s):
51The geometry of locally symmetric manifolds via natural maps
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
Team Members
Prof. Dr. Ursula Hamenstädt
Project leader
Rheinische Friedrich-Wilhelms-Universität Bonn
M.Sc. Frieder Jäckel
Doctoral student
Rheinische Friedrich-Wilhelms-Universität Bonn