Probabilistic and Analytic Aspects of the Loewner Energy

Massachusetts Institute Of Technology, Cambridge MA

Investigators

Abstract

This project concerns research in the areas of probability and complex analysis. The Loewner energy is a quantity measuring the roundness of a simple planar loop. It arises from the asymptotic behaviors of the Schramm-Loewner evolution (SLE), a model of random fractal curves. SLE plays a central role in random conformal geometry and two-dimensional statistical mechanics that study the macroscopic geometry of systems with given information on the microscopic level. Surprisingly, this probabilistically motivated Loewner energy can be described using fundamental concepts from seemingly disparate branches of mathematics and mathematical physics, including geometric function theory, Teichmüller theory, conformal field theory, and string theory. These links suggest deep connections between random conformal geometry and those branches. This research project aims at revealing these connections and exploring how the variety of perspectives around the Loewner energy can bring new insights to probability theory and other fields. The results are expected also to reveal new facets of the mathematical architecture underlying theoretical physics. The Loewner energy of a Jordan curve is defined as the Dirichlet energy of its driving function via the Loewner differential equation. Finite energy curves can, therefore, be viewed as the Cameron-Martin space of SLE, which has a multiple of Brownian motion as driving function. This definition of both Loewner energy and SLE depends strongly on the parametrization of the curves. However, an equivalent and intrinsic description of the Loewner energy was discovered using determinants of Laplacians and is known to be the Kähler potential of the Weil-Petersson metric on the universal Teichmüller space. This research project first aims to provide similar intrinsic descriptions of SLE loops via the canonical measures on the welding homeomorphisms, then studies generalizations of the Loewner energy to other scenarios involving multi-chords or higher genus surfaces, analytic identities inspired by results from random conformal geometry, and the relation to minimal surfaces in the hyperbolic 3-space. 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.