On symmetry of energy minimizing harmonic-type maps on cylindrical surfaces

TL;DR

This paper investigates the symmetry properties of energy-minimizing harmonic maps on cylindrical surfaces, revealing conditions for symmetry and symmetry-breaking, with implications for models in liquid crystals and micromagnetics.

Contribution

It proves that global minimizers are cylindrically symmetric and introduces sharp inequalities to analyze the energy landscape, advancing understanding of symmetry in these models.

Findings

Minimizers are z-invariant and axially symmetric under certain conditions

Established sharp Poincaré-type inequalities on cylinders

Identified symmetry-breaking phenomena in energy minimizers

Abstract

The paper concerns the analysis of global minimizers of a Dirichlet-type energy functional in the class of $\mathbb{S}^2$-valued maps defined in cylindrical surfaces. The model naturally arises as a curved thin-film limit in the theories of nematic liquid crystals and micromagnetics. We show that minimal configurations are $z$-invariant and that energy minimizers in the class of weakly axially symmetric competitors are, in fact, axially symmetric. Our main result is a family of sharp Poincar\'e-type inequality on the circular cylinder, which allows establishing a nearly complete picture of the energy landscape. The presence of symmetry-breaking phenomena is highlighted and discussed. Finally, we provide a complete characterization of in-plane minimizers, which typically appear in numerical simulations for reasons we explain.