Symmetry of uniaxial global Landau-de Gennes minimizers in the theory of nematic liquid crystals

TL;DR

This paper proves that in the Landau-de Gennes model for nematic liquid crystals on a sphere, uniaxial energy minimizers at low temperatures must resemble the radial-hedgehog solution, implying they cannot be purely uniaxial at sufficiently low temperatures.

Contribution

It establishes the structure of uniaxial minimizers in the low-temperature regime and shows they must be radial-hedgehog solutions, revealing limitations on uniaxiality.

Findings

Uniaxial minimizers are necessarily radial-hedgehog solutions at low temperatures.

Purely uniaxial minimizers do not exist for sufficiently low temperatures.

The result applies to nematic liquid crystals on spherical droplets with homeotropic boundary conditions.

Abstract

We study uniaxial energy-minimizers within the Landau-de Gennes theory for nematic liquid crystals on a three-dimensional spherical droplet subject to homeotropic boundary conditions. We work in the low-temperature regime and show that uniaxial energy-minimizers necessarily have the structure of the well-studied radial-hedgehog solution in the low-temperature limit. An immediate consequence of this result is that Landau-de Gennes energy minimizers cannot be purely uniaxial for sufficiently low temperatures.