Basic properties of the natural parametrization for the Schramm-Loewner evolution

TL;DR

This paper advances the understanding of the natural parametrization of SLE{ extkappa}, establishing its domain independence, H"older continuity, and convergence properties, thereby strengthening its theoretical foundation.

Contribution

It improves the proof of the natural parametrization's existence and proves new properties like domain invariance and H"older continuity, with bounds on the Green's function.

Findings

Natural parametrization is domain-independent.

Natural parametrization is H"older continuous.

Convergence of expectations for the natural length.

Abstract

The natural paramterization or length for the Schramm-Loewner evolution (SLE{\kappa}) is the candidate for the scaling limit of the length of discrete curves for \kappa < 8. We improve the proof of the existence of the parametrization and use this to establish some new results. In particular, we show that the natural parametrization is independent of domain and it is H\"older continuous with respect to the capacity parametrization. We also give up-to-constants bounds for the two-point Green's function. Although we do not prove the conjecture that the natural length is given by the appropriate Minkowski content, we do prove that the corresponding expectations converge.