A note on $n$-axially symmetric harmonic maps from $B^3$ to $S^2$ minimizing the relaxed energy

TL;DR

This paper constructs explicit examples of n-axially symmetric harmonic maps from the 3-ball to the 2-sphere that minimize relaxed energy, highlighting differences from the n=1 case.

Contribution

It provides explicit constructions of energy-minimizing n-axially symmetric maps for n>1, contrasting with known results for n=1.

Findings

Explicit examples of energy-minimizing maps for n>1

Demonstrates non-trivial vertical and graph parts in minimizers

Contrasts with previous results for n=1

Abstract

For any n>1 we give an explicit example of an n-axially symmetric Cartesian current in B^3 x S^2 with non-trivial vertical part and non-constant graph part minimizing the relaxed Dirichlet energy among the n-axially symmetric Cartesian currents with the same boundary. This stands in sharp contrast with a results of Hardt, Lin and Poon for the case n=1.