The Loewner Energy of Loops and Regularity of Driving Functions

Abstract

Loewner driving functions encode simple curves in 2-dimensional simply connected domains by real-valued functions. We prove that the Loewner driving function of a C 1,β curve (differentiable parametrization with β-Hölder continuous derivative) is in the class C 1,β−1/2 if 1/2 < β ≤ 1, and in the class C 0,β+1/2 if 0 ≤ β ≤ 1/2. This is the converse of a result of Carto Wong [27] and is optimal. We also introduce the Loewner energy of a rooted planar loop and use our regularity result to show the independence of this energy from the basepoint. Recently, in [6] and [23] the chordal Loewner energy ∞ 0Ẇ (t) 2 /2 dt of η was introduced independently, and basic properties (such as rectifiability) of curves with finite energy were obtained. The chordal Loewner energy a priori depends on the orientation of η, namely viewed as a curve from 0 to ∞ or from ∞ to 0. However, the second author [23] proved the direction-independence (or reversibility).In this paper, we generalize the definition of Loewner energy to simple loops on the Riemann sphere γ : R →Ĉ where γ is continuous, 1-periodic and injective on [0, 1) : We just observe that the limit when ε → 0 of the chordal energy of γ [ε, 1] in the simply connected domainĈ \ γ[0, ε] exists in [0, ∞] (Proposition 2.8), and define it as the loop Loewner energy of γ rooted at γ(0). Note that circles have loop energy 0. Intuitively, the loop energy measures how much the Jordan curve γ[0, 1] differs from a circle seen from the root γ(0), in a Möbius invariant fashion. The loop Loewner energy generalizes * the chordal Loewner energy: Indeed, if we apply z → z 2 to a chord η from 0 to ∞ in H, the positive real line together with the image of η forms a loop γ through ∞. It is clear that its loop energy rooted at ∞ (i.e. we parametrize the loop such that γ(0) = ∞) equals the chordal energy of η.Note also that the loop energy neither depends on any increasing reparametrization of γ fixing the root, nor on the direction of parametrization. The latter fact basically comes from the chordal reversibility, which can be used to show thatγ(t) = γ(1 − t) has the same energy as γ (see Section 2.2 for details). But it depends a priori on the root γ(0) where the limit is taken, not only on the Jordan curve γ[0, 1]. However, our first main result states:Theorem 1.1. The loop Loewner energy is root-invariant.In particular, this result shows that the loop Loewner energy is a conformal invariant on the set of unrooted loops on the Riemann sphere, which attains its minimum 0 only on circles.In our proof of the root-invariance, we approximate the curve by well-chosen regular curves and are led to the following question: What can we say about the relation between the regularity of the driving function and the regularity of the curve? Prior to this work, only one direction was well understood. Slightly imprecisely, the following results state that C α driving functions generate C α+1/2 curves for α > 1/2, where C α is understood with the usual convention as C n,β , where n is the integer part of α and β =...

Equivalent descriptions of the loewner energy

“…The Loewner loop energy is therefore a non-negative and Möbius invariant quantity on the set of free loops, which vanishes only on circles. The proof in [31] relies on the reversibility of chordal Loewner energy and a certain type of surgeries on the loop to displace the root. This root-invariance suggests that the framework of loops provides even more symmetries and invariance properties than the chordal case when one studies Loewner energy.…”

“…It is clear that the loop energy depends a priori on the root γ(0) and the orientation of the parametrization, since the change of root/orientation induces non-trivial changes on the driving function that is in general hard to track. However, the main result of [31] shows that the Loewner loop energy of γ only depends on the image of γ. In this section we prove the identity (Theorem 6.1) that will give other approaches to the parametrization independence of the loop energy in Section 7 and 8.…”

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

“…Although finite energy curves are not C 1 and may exhibit slow spirals [21] (and so are not Lipschitz), there is still enough regularity to take traces using disk averages of elements in E(C) (see Appendix A) giving rise to H 1/2 spaces along the curves and to employ BMO-space estimates. Conformal invariance properties and the good interplay between arclength and harmonic measure on finite energy curves are also important for the analysis.…”

“…Finite energy curves are not C 1 however, and may, e.g., exhibit slow spirals, and C 3/2−ε does not imply finite energy, see [21]. Since η is rectifiable, f ∈ H 1 (the Hardy space) and has non-tangential limits a.e.…”

“…The Sobolev space trace operator is usually defined for domains with Lipschitz boundary. We are interested in domains bounded by finite energy curves (see Section 2.3), which are chord-arc but not necessarily Lipschitz [21]. This appendix recalls and develops the facts needed for this paper.…”