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This article analyzes minimal length connections among three points in a geometric space, suggesting improved understanding of sub-Riemannian geometry.
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.
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Nürnberg et al. (2026) studied this question.
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