The geometry of locally symmetric spaces via natural maps

• Applicant: Professorin Dr. Ursula Hamenstädt
• Subject Area: Mathematics
• Term: since 2020
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 441720292

Harmonic maps of closed surfaces into locally symmetric manifolds M of non-positive curvature without Euclidean factors are an important tool for the geometric understanding of such manifolds. The energy of such a map in a fixed (sufficiently complicated) homotopy class is a smooth function on Teichmüller space of the surface. Goal of the project is a geometric characterization of the manifold via this energy profile in the following situations. 1) The manifold is of dimension 3, non-compact and quasi-Fuchsian or doubly degenerate.2) The manifold is closed, of dimension 3, hyperbolic and given by a minimal Heegaard decomposition. 3) Grafting is an operation which associates to a hyperbolic surface a new metric, which is not hyperbolic any more. 4) Via a grafting construction one can construct explicit deformations of Fuchsian points in the Hitchin component of surface group representations into PSL(n,R). The energy profile gives information about minimal representatives and degenerations in the character variety.

• DFG Programme: Priority Programmes
• Subproject of: SPP 2026: Geometry at Infinity