Extremal Problems in Conformal Geometry: Moduli and Dynamics
Fuente:
Zenodo
Guardado en:
| Autor principal: | |
|---|---|
| Formato: | Recurso digital |
| Publicado: |
Zenodo
2025
|
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866901180752330752 |
|---|---|
| author | SÉRGIO DE ANDRADE, PAULO |
| author_facet | SÉRGIO DE ANDRADE, PAULO |
| contents | This paper explores the deep connections between extremal problems, the geometry of moduli spaces, and the behavior of dynamical systems within the framework of conformal geometry. We investigate the problem of finding optimal maps between Riemann surfaces, specifically those that minimize conformal distortion. These extremal quasiconformal mappings are central to the study of Teichmüller theory. The methodology relies on variational principles and the analysis of the Beltrami equation, which characterizes quasiconformal maps. We demonstrate that the unique solutions to these extremal problems are Teichmüller mappings, which are intrinsically linked to holomorphic quadratic differentials. This connection allows for a detailed description of the geometry of Teichmüller space, which is the moduli space of marked conformal structures. Furthermore, we analyze the dynamical properties of the Teichmüller geodesic flow on this space. The results establish a bridge between the static nature of extremal problems and the long-term, ergodic behavior of geometric structures under this flow. The interplay reveals how solutions to optimization problems in geometry encode profound information about the dynamics on the corresponding moduli spaces, with implications for complex dynamics and low-dimensional topology. |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_17690000 |
| institution | Zenodo |
| language | |
| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Extremal Problems in Conformal Geometry: Moduli and Dynamics SÉRGIO DE ANDRADE, PAULO This paper explores the deep connections between extremal problems, the geometry of moduli spaces, and the behavior of dynamical systems within the framework of conformal geometry. We investigate the problem of finding optimal maps between Riemann surfaces, specifically those that minimize conformal distortion. These extremal quasiconformal mappings are central to the study of Teichmüller theory. The methodology relies on variational principles and the analysis of the Beltrami equation, which characterizes quasiconformal maps. We demonstrate that the unique solutions to these extremal problems are Teichmüller mappings, which are intrinsically linked to holomorphic quadratic differentials. This connection allows for a detailed description of the geometry of Teichmüller space, which is the moduli space of marked conformal structures. Furthermore, we analyze the dynamical properties of the Teichmüller geodesic flow on this space. The results establish a bridge between the static nature of extremal problems and the long-term, ergodic behavior of geometric structures under this flow. The interplay reveals how solutions to optimization problems in geometry encode profound information about the dynamics on the corresponding moduli spaces, with implications for complex dynamics and low-dimensional topology. |
| title | Extremal Problems in Conformal Geometry: Moduli and Dynamics |
| url | https://doi.org/10.5281/zenodo.17690000 |