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.