Partial Regularity of Stable Stationary Harmonic Maps into Certain Lie Groups

TL;DR

This paper proves that stable stationary harmonic maps into most compact simple Lie groups have singular sets of Hausdorff codimension at least four, with explicit examples showing this bound is sharp.

Contribution

It establishes a partial regularity result for harmonic maps into certain Lie groups, identifying the minimal possible dimension of their singular sets.

Findings

Singular set of stable stationary harmonic maps has Hausdorff codimension at least four.

Examples show the singular set cannot have lower codimension.

Results exclude certain Lie groups like $ ext{Sp}(n)$ for $n ext{ } ext{large}$, $E_8$, $F_4$, $G_2$.

Abstract

Let $M$ be a compact Riemannian manifold, and let $G$ be a compact simple Lie group with bi-invariant metric that is not $\operatorname{Sp}(n)$ for $n \geq 8$, $E_{8}$, $F_{4}$, or $G_{2}$. We show that the singular set of any stable stationary harmonic map $u : M \to G$ has Hausdorff codimension at least four. We also find examples of maps into these manifolds with codimension four singularities to show that we cannot reduce the dimension of the singular set any further.