A remark on isolated removable singularity of harmonic maps in dimension two

TL;DR

We address removability of isolated singularities for harmonic maps in dimension two under a small self-similar bound on the gradient. The authors establish the existence of a small universal $\varepsilon_0>0$ such that a harmonic map $u:B_R(0)\setminus\{0\}\to N$ with $|\nabla u|\le\frac{\varepsilon_0}{|x|}$ and a vanishing boundary energy limit extends smoothly across the origin; when $R=\infty$, the same hypotheses force $u$ to be constant. The method combines a monotonicity formula, a Sacks–Uhlenbeck removal argument, and a radial-energy comparison using harmonic competitors on annuli to bootstrap regularity. The results provide a 2D Liouville-type rigidity for isolated singularities under a quantified gradient bound and yield a partial answer to a posed question about rigidity in the steady Ericksen–Leslie system. This contributes to the understanding of harmonic map regularity and singularity removability in two dimensions with applications to related geometric-analytic problems.

Abstract

For a ball $B_R(0)\subset\mathbb{R}^2$, we provide sufficient conditions such that a harmonic map $u\in C^\infty(B_R(0)\setminus\{0\}, N)$, with a self-similar bound on its gradient, belongs to $C^\infty(B_R(0))$. Those conditions also guarantee the triviality of such harmonic maps when $R=\infty$.

Theorems & Definitions (5)

• Theorem 1.2
• Corollary 1.3
• Lemma 2.1
• proof
• Remark 2.2