Periodic networks of fixed degree minimizing length

TL;DR

This paper investigates the optimal periodic networks in Euclidean space with fixed degree, minimizing length under volume constraints, and identifies specific minimizers for various dimensions and degrees.

Contribution

It characterizes minimal length periodic networks with prescribed degree in Euclidean space, providing explicit solutions for certain degrees and dimensions.

Findings

Minimizers identified for 3D networks with degrees 3 to 6.

Length estimates provided for degrees greater than 6.

Unique minimizers found for degrees n+1 and 2n in general dimensions.

Abstract

We study networks in $\R^n$ which are periodic under a lattice of rank~$n$ and have vertices of prescribed degree $d\ge 3$. We minimize the length of the quotient networks, subject to the constraint that the fundamental domain has $n$-dimensional volume~$1$. For $n=3$ and degree $3\leq d\leq 6$ we determine the minimizing networks with the least number of vertices in the quotient, while for $d\ge 7$ we state a length estimate. For general $n$, we determine the unique minimizers with $d=n+1$ and $d=2n$.