Boundary Green's functions and Minkowski content measure of multi-force-point SLE$_\kappa(\underline\rho)$

TL;DR

This paper establishes the existence of a Minkowski content measure for multi-force-point SLE curves, linking geometric measure theory with conformal invariance properties in complex analysis.

Contribution

It extends the concept of Minkowski content measure to multi-force-point SLE curves and proves its conformal covariance and existence using Green's function estimates.

Findings

Existence of a Minkowski content measure for multi-force-point SLE curves.

The measure coincides with known measures in special cases.

The measure satisfies conformal covariance properties.

Abstract

We consider a transient chordal SLE$_\kappa(\rho_1,\dots,\rho_m)$ curve $\eta$ in $\mathbb{H}$ from $w$ to $\infty$ with force points $ v_1> \cdots >v_m$ in $(-\infty,w^-]$, which intersects and is not boundary-filling on $(-\infty,v_m)$. The main result is that there is an atomless locally finite Borel measure $\mu_\eta$ on $\eta\cap (-\infty,v_m]$ such that for any $v<v_m$, the $d$-dimensional Minkowski content of $\eta\cap [v,v_m]$ exists and equals $\mu_\eta [v,v_m]$, where $d=\frac{(\sum \rho_j+4)(\kappa-4-2\sum \rho_j)}{2\kappa}$ is the Hausdorff dimension of $\eta\cap [v,v_m]$. In the case that all $\rho_j=0$, this measure agrees with the covariant measure derived in [Alberts-Sheffield, 2011] for chordal SLE$_\kappa$ up to a multiplicative constant. %Such measure, called Minkowski content measure, satisfies conformal covariance properties. We call such measure a Minkowski content measure, extend it to a class of subsets of ${\mathbb{R}}^n$, and prove that they satisfy conformal covariance. To construct the Minkowski content measure on $\eta\cap [v,v_m]$, we follow the standard approach to derive the existence and estimates of the one- and two-point boundary Green's functions of $\eta$ on $(-\infty,v_m)$, which are the limits of the rescaled probability that $\eta$ passes through small discs or open real intervals centered at points on $(-\infty,v_m)$.