Definition Loewner energy is a mathematical functional defined on curves, particularly in the context of complex analysis and probability theory, quantifying the "cost" or "complexity" associated with a curve generated via the Loewner differential equation. It is often associated with the Schramm-Loewner evolution (SLE) and has connections to conformal field theory, Teichmüller theory, and the integrability of random curves.
Overview The Loewner energy arises in the study of planar curves that can be encoded through the Loewner equation, a differential equation that describes the evolution of conformal maps associated with growing curves in the complex plane. In deterministic settings, the Loewner energy measures the regularity of the driving function of the Loewner equation. Curves with finite Loewner energy are known to be rectifiable and belong to the Weil-Petersson class of quasicircles. The energy functional is invariant under Möbius transformations and plays a role in the large deviation principles of SLE as the parameter tends to zero.
In probabilistic contexts, the Loewner energy serves as the rate function for such large deviations, linking deterministic variational problems with the asymptotic behavior of stochastic processes. It has also been related to the Dirichlet energy of the logarithmic derivative of conformal maps and appears in mathematical physics, particularly in connection with the Kähler geometry of universal Teichmüller space.
Etymology/Origin The term "Loewner energy" originates from the name of the mathematician Karl Loewner (later Charles Loewner), who introduced the Loewner differential equation in 1923 as a tool in geometric function theory, notably to make progress on the Bieberbach conjecture. The specific concept of "Loewner energy" was formally defined and studied in the 2010s by researchers including Yilin Wang, who explored its connections to SLE and Teichmüller theory.
Characteristics
- The Loewner energy is defined for rectifiable Jordan curves in the Riemann sphere.
- It is conformally invariant; specifically, it is invariant under Möbius transformations.
- It is closely tied to the $H^{1/2}$ norm of the driving function in the Loewner equation.
- Curves with finite Loewner energy form a subgroup of quasisymmetric homeomorphisms known as the Weil-Petersson Teichmüller space.
- It is a deterministic quantity that emerges as the large deviation rate function for Schramm-Loewner evolution (SLE$_\kappa$) as $\kappa \to 0$.
- The energy can be expressed in terms of the Dirichlet energy of the uniformizing conformal map.
Related Topics
- Schramm-Loewner evolution (SLE)
- Conformal invariance
- Loewner differential equation
- Teichmüller theory
- Weil-Petersson geometry
- Quasiconformal mappings
- Large deviations theory
- Deterministic Loewner chains
- Universal Teichmüller space