PRIMES: Recurrent Flows and Harmonic Functions on Infinite Riemann Surfaces

Riemann surfaces are fundamental objects underlying various fields of mathematics and mathematical physics. Riemann surfaces appear as geometric objects on which functions are naturally defined; they describe two-dimensional geometric shapes and break higher-dimensional objects into simple components, even parametrizing strings in physics. A Riemann surface can be visualized as the surface of a three-dimensional object, such as a donut, a coffee mug, a cooking pot with two handles, or a (very long) picket fence. Finite Riemann surfaces have tame topology, and they have been studied for over a century, with many fundamental questions being successfully answered. A Riemann surface is infinite if its topology is not tame; in other words, if it has infinite complexity. An example is the surface of an infinitely long picket fence. Infinite Riemann surfaces have gained considerable attention in the last fifteen years, and most of this research is from the perspective of topology and extensions of the theory on finite surfaces. In contrast, the primary focus of this project is to investigate the analytic and geometric properties of infinite Riemann surfaces. An interplay of analytic and geometric methods will be applied. The award will also involve training graduate students in research and exposing undergraduate students to research practices, thereby providing training and increasing the number of STEM professionals in the workforce. This project will focus on the ergodic theory of the geodesic flow on infinite Riemann surfaces, its relationship to the broader question of classifying infinite Riemann surfaces according to various classes of harmonic maps that they do or do not support, and its connection to the Teichmüller theory of infinite Riemann surfaces. By constructing partial foliations with finite Dirichlet integrals and using hyperbolic geometry methods, various sufficient conditions for the Fenchel-Nielsen coordinates of infinite Riemann surfaces to exhibit ergodic geodesic flows are provided. This approach has partially solved a conjecture of Kahn-Marković, which states that any Riemann surface can be mapped to one with ergodic geodesic flow by twisting along a pants decomposition. An extension of these ideas will completely resolve the conjecture, while also providing sufficient conditions for Riemann surfaces to belong to various classes, as well as linking these methods to the questions in potential theory for infinite graphs. Finally, the project proposes to study the differences between the Teichmüller spaces of two surfaces: one with an ergodic geodesic flow and another with a non-ergodic geodesic flow. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.