Boundary Green’s functions and Minkowski content measure of multi-force-point SLE[formula omitted

Abstract

We consider a transient chordal SLEκ(ρ1,…,ρm) curve η in H from w to ∞ with force points v1>⋯>vm in (−∞,w−], which intersects and is not boundary-filling on (−∞,vm). The main result is that there is an atomless locally finite Borel measure μη on η∩(−∞,vm] such that for any v<vm, the d-dimensional Minkowski content of η∩[v,vm] exists and equals μη[v,vm], where d=(∑ρj+4)(κ−4−2∑ρj)2κ is the Hausdorff dimension of η∩[v,vm]. In the case that all ρj=0, this measure agrees with the covariant measure derived in Alberts and Sheffield (2011) for chordal SLEκ up to a multiplicative constant. We call such measure a Minkowski content measure, extend it to a class of subsets of Rn, and prove that they satisfy conformal covariance. To construct the Minkowski content measure on η∩[v,vm], we follow the standard approach to derive the existence and estimates of the one- and two-point boundary Green’s functions of η on (−∞,vm), which are the limits of the rescaled probability that η passes through small disks or open real intervals centered at points on (−∞,vm).