A M\"obius invariant discretization of O'Hara's M\"obius energy

TL;DR

This paper presents a new M"obius invariant discretization of O'Hara's M"obius energy, ensuring invariance under M"obius transformations and demonstrating convergence to the continuous energy.

Contribution

It introduces a novel discretization based on Doyle and Schramm's cosine formula that is invariant under M"obius transformations and proves its $\

Findings

Discretization is invariant under M"obius transformations.

Discretized energies $\

Discretization converges to the continuous M"obius energy under natural assumptions.

Abstract

We introduce a new discretization of O'Hara's M\"obius energy. In contrast to the known discretizations of Simon and Kim and Kusner it is invariant under M\"obius transformations of the surrounding space. The starting point for this new discretization is the cosine formula of Doyle and Schramm. We then show $\Gamma$-convergence of our discretized energies to the M\"obius energy under very natural assumptions.