Minkowski content and natural parameterization for the Schramm-Loewner evolution

TL;DR

This paper establishes the existence of the Minkowski content for SLE curves with parameter 5<8, linking it to the natural parameterization and providing a geometric measure of these fractal curves.

Contribution

It proves the existence and nontriviality of the Minkowski content for 5<8 SLE curves and relates it to their natural parameterization.

Findings

Minkowski content exists and is nontrivial for 5<8 SLE with dimension d=1+5/8

The Minkowski content is proportional to the natural parameterization of SLE curves

Provides a geometric measure for fractal SLE curves

Abstract

We prove the existence and nontriviality of the $d$-dimensional 4 Minkowski content for the Schramm-Loewner evolution ($\mathrm {SLE}_{\kappa}$) with $\kappa<8$ and $d=1+\frac{\kappa}{8}$. We show that this is a multiple of the natural parameterization.