Minkowski content and natural parameterization for the Schramm–Loewner evolution

Abstract

We prove the existence and nontriviality of the d-dimensional 4 Minkowski content for the Schramm-Loewner evolution (SLEκ) with κ < 8 and d = 1 + κ 8. We show that this is a multiple of the natural parameterization.1. Introduction. A number of measures on paths or clusters on twodimensional lattices arising from critical statistical mechanical models are believed to exhibit some kind of conformal invariance in the scaling limit. Schramm [13] introduced a one-parameter family of such processes, now called the (chordal) Schramm-Loewner evolution with parameter κ (SLE κ ) and showed that these give the only possible limits for conformally invariant processes in simply connected domains satisfying a certain "domain Markov property." He defined the process as a probability measure on curves from 0 to ∞ in H and then used conformal invariance to define the process in other simply connected domains.The definition of the process in H uses the half-plane Loewner equation. Suppose γ : (0, t] → H is a curve with γ(0) = 0, and let γ t = γ(0, t]. Let H t denote the unbounded component of H \ γ t . We assume that γ is noncrossing in the sense that for all s < t, γ[s, ∞) ⊂ H s , and γ[s, t] ∩ H s is nonempty. Let g t : H t → H be the unique conformal transformation with g t (z) − z = o(1) as z → ∞. Then for every a > 0, there exists a reparameterization of the curve such that the following holds:

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“…To prove the claim, we need to show that there is a constant C ∈ (1, ∞) depending on κ, ε, R such that for any simply connected domain D that contains the disc {|z| < R}, for any distinct prime ends a, b of D, and for any w ∈ C with |w| ≤ ε, we have 1/C ≤ G D;a,b (w)/G D;a,b (0) ≤ C. For this purpose, we use the explicit expression of the chordal SLE κ Green's function (cf. [9,Formula (8)]):…”