Spherical Minkowski content and natural parametrization on the sphere for Schramm-Loewner evolution

TL;DR

This paper establishes the existence of a natural parametrization for Schramm-Loewner evolution (SLE) on the sphere by proving the finiteness of its spherical Minkowski content, offering new insights into conformal invariance in 2D models.

Contribution

It introduces the spherical Minkowski content for SLE on the sphere and proves its finiteness, providing a new natural parametrization framework.

Findings

Spherical Minkowski content of SLE$_{ppa}$ is finite.

The content serves as a natural parametrization of SLE on the sphere.

Provides foundational properties of SLE on $S^2$.

Abstract

Studying SLE$_{\kappa}$ on $S^2$ provides a new and interesting perspective for the conformality of some 2-dimensional physical models. We prove the existence and some basic properties of the spherical Minkowski content of SLE$_{\kappa}$, which is finite and can be viewed as the natural parametrization of SLE$_{\kappa}$ on $S^2$.