Minimal surfaces from infinitesimal deformations of circle packings

TL;DR

This paper establishes a connection between infinitesimal deformations of circle packings and discrete minimal surfaces, providing a unified framework that extends Koebe type surfaces to general discrete minimal surfaces.

Contribution

It introduces a Weierstrass representation linking circle packing deformations to discrete minimal surfaces and extends Koebe type surfaces to a broader class.

Findings

Infinitesimal deformations correspond to discrete minimal surfaces.

Every Koebe type minimal surface extends to a general discrete minimal surface.

Unified framework for discrete minimal surfaces via Steiner's formula.

Abstract

We study circle packings with the combinatorics of a triangulated disk in the plane and parametrize deformations of circle packings in terms of vertex rotation and cross ratios. We show that there is a Weierstrass representation formula relating infinitesimal deformations of circle packings to discrete minimal surfaces of Koebe type. Furthermore, every minimal surface of Koebe type can be extended naturally to a discrete minimal surface of general type. In this way, discrete minimal surfaces via Steiner's formula are unified.