Geometric structures via circle patterns

• Applicant: Dr. Carl Lutz
• Subject Area: Mathematics
• Term: since 2025
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 554919336

The primary objective of this project is to investigate the structure of the space of circle patterns and its relationship with hyperbolic 3-manifolds, the classical (smooth) notion of conformality, and mathematical physics. A defining characteristic of conformal maps is their ability to preserve infinitesimal circles. By considering patterns of circles of positive radius, one obtains a discrete model of conformality that shares numerous properties and structural characteristics with its smooth counterpart. This approach has proven to be highly effective for surfaces that are locally modeled on a metric geometry, such as Euclidean geometry, yielding strong existence and uniqueness results for the associated discretely conformal maps. However, the concept of a circle, and thus a circle pattern, is sensible in the broader context of complex projective structures. This offers the appealing possibility to investigate complex projective structures and their underlying (smooth) conformal structures via circle patterns. In particular, it is conjectured that the space of complex projective structures on which a given circle pattern may be realized is a section of the fibration of the space of complex projective structures over Teichmüller space. This project aims to investigate these phenomenons. A fundamental property of circle patterns is their intimate relationship to polyhedra in hyperbolic 3-space. As a consequence of this relationship, realization problems for circle patterns are naturally associated with Cauchy and Alexandrov type realization problems for polyhedra. In the former case, the objective is to identify a polyhedron with prescribed dihedral angles, whereas in the latter, the metric on the boundary is prescribed. One of the most influential classical results in complex analysis is the Riemann mapping theorem. This theorem states that any simply connected proper subdomain of the complex plane can be conformally mapped to the open unit disk. Koebe extended this result to multiply connected domains. Discrete versions of the Riemann mapping theorem are known. The objective of this investigation is to examine the extension of these results to multiply connected discrete domains. This question is equivalent to an Alexandrov problem, where the boundary metric possesses certain reflectional symmetries. In recent years, circle patterns have emerged in statistical mechanics as a powerful tool for relating the parameters of a statistical model to the geometry of the underlying graph. For this reason, the proposed project offers a compelling opportunity to investigate the relationships between statistical mechanics and the geometry of hyperbolic space through the lens of circular patterns. It is an intriguing prospect to examine the ways in which both fields can mutually benefit from each other.

• DFG Programme: WBP Fellowship
• International Connection: Luxembourg
• Host: Professor Dr. Jean-Marc Schlenker