Interplay Between Loewner and Dirichlet Energies via Conformal Welding and Flow-Lines

Fredrik Viklund,Yilin Wang

doi:10.1007/s00039-020-00521-9

Abstract

The Loewner energy of a Jordan curve is the Dirichlet energy of its Loewner driving term. It is finite if and only if the curve is a Weil–Petersson quasicircle. In this paper, we describe cutting and welding operations on finite Dirichlet energy functions defined in the plane, allowing expression of the Loewner energy in terms of Dirichlet energy dissipation. We show that the Loewner energy of a unit vector field flow-line is equal to the Dirichlet energy of the harmonically extended winding. We also give an identity involving a complex-valued function of finite Dirichlet energy that expresses the welding and flow-line identities simultaneously. As applications, we prove that arclength isometric welding of two domains is sub-additive in the energy, and that the energy of equipotentials in a simply connected domain is monotone. Our main identities can be viewed as action functional analogs of both the welding and flow-line couplings of Schramm–Loewner evolution curves with the Gaussian free field.

Highlights

Let η be a Jordan curve in C = C ∪ {∞}
We describe cutting and welding operations on finite Dirichlet energy functions defined in the plane, allowing expression of the Loewner energy in terms of Dirichlet energy dissipation
We show that the Loewner energy of a unit vector field flow-line is equal to the Dirichlet energy of the harmonically extended winding

Summary

Introduction

There is another coupling known as the forward SLE/GFF coupling, of critical importance, e.g., in the imaginary geometry framework of Miller-Sheffield [Dub[09], MS16]: very loosely speaking, an SLEκ curve may be coupled with a GFF Φ and thought of as a (measurable) flow-line of the vector field eiΦ/χ, where χ = 2/γ − γ/2 Given these and similar observations, it is possible to guess our identities via heuristic large deviation arguments analogous to (1.5) in the small γ limit. Let us remark that the complex identity, Corollary 3.13, which expresses both welding and flow-line identities simultaneously, is the finite energy analog of the mating of trees theorem of Duplantier, Miller, and Sheffield [DMS14]. This analogy is not as apparent as in the other cases and details will appear elsewhere [VW19]. We choose the orientation for bounded curves so that Ω is the bounded component

Preliminaries

Proofs of Main Results

A Trace Operators on Chord-Arc Curves