Discrete Lorentz surfaces and s-embeddings I: isothermic surfaces

TL;DR

This paper establishes a geometric framework linking s-embeddings of isoradial graphs to Lorentzian geometry, demonstrating that certain s-embeddings lift to discrete S-isothermic surfaces, advancing the understanding of discrete maximal surfaces.

Contribution

It introduces a geometric interpretation of s-embeddings via Lorentz sphere congruences and identifies isothermic s-embeddings that lift to discrete S-isothermic surfaces, answering a key open question.

Findings

S-embeddings correspond to Lorentz sphere congruences.

Isothermic s-embeddings lift to discrete S-isothermic surfaces.

Ising weights of isothermic s-embeddings lie in a specific subvariety.

Abstract

S-embeddings were introduced by Chelkak as a tool to study the conformal invariance of the thermodynamic limit of the Ising model. Moreover, Chelkak, Laslier and Russkikh introduced a lift of s-embeddings to Lorentz space, and showed that in the limit the lift converges to a maximal surface. They posed the question whether there are s-embeddings that lift to maximal surfaces already at the discrete level, before taking the limit. This paper is the first in a two paper series, in which we answer that question in the positive. In this paper we introduce a correspondence between s-embeddings (incircular nets) and congruences of touching Lorentz spheres. This geometric interpretation of s-embeddings enables us to apply the tools of discrete differential geometry. We identify a subclass of s-embeddings -- isothermic s-embeddings -- that lift to (discrete) S-isothermic surfaces, which were introduced by Bobenko and Pinkall. S-isothermic surfaces are the key component that will allow us to obtain discrete maximal surfaces in the follow-up paper. Moreover, we show here that the Ising weights of an isothermic s-embedding are in a subvariety.