Calibrations for minimal networks in a covering space setting

TL;DR

This paper introduces a calibration method for minimal networks in a covering space framework, providing explicit examples and a new approach to identify minimizers in geometric covering problems.

Contribution

It develops a calibration concept in a covering space setting and introduces calibration in families to determine minimal Steiner networks.

Findings

Proves minimality of Steiner configurations for regular hexagon and pentagon

Provides explicit calibration examples in a covering space context

Introduces a novel family-based calibration approach

Abstract

In this paper we define a notion of calibration for an equivalent approach to the classical Steiner problem in a covering space setting and we give some explicit examples. Moreover we introduce the notion of calibration in families: the idea is to divide the set of competitors in a suitable way, defining an appropriate (and weaker) notion of calibration. Then, calibrating the candidate minimizers in each family and comparing their perimeter, it is possible to find the minimizers of the minimization problem. Thanks to this procedure we prove the minimality of the Steiner configurations spanning the vertices of a regular hexagon and of a regular pentagon.