Minimal horizontal triods: Analysis and computation

TL;DR

This paper studies minimal length networks connecting three points in the Heisenberg group, characterizes their properties, and introduces a flow to find critical points, supported by numerical experiments in sub-Riemannian geometry.

Contribution

It proves existence and characterization of minimal horizontal triods in the Heisenberg group and develops a curve shortening flow with numerical validation.

Findings

Existence of minimal horizontal triods established.

Characterization of these triods provided.

Numerical experiments illustrate the geometric landscape.

Abstract

In this article we investigate the question of finding a network configuration of minimal length connecting three given points in the Heisenberg group. After proving existence of (possibly degenerate) minimal horizontal triods, we investigate their characterization. We then formulate a horizontal curve shortening flow that deforms any given suitable initial triod into a critical point for the length functional. Numerical experiments based on a stable fully discrete finite element scheme provide useful insights into the rich landscape of this sub-Riemannian geometry.