Minimizing Harmonic Maps on the Unit Ball with Tangential Anchoring

TL;DR

The paper analyzes minimizing harmonic maps $u:\Omega\to \mathbb{S}^2$ on the unit ball with tangential boundary conditions $u\cdot\nu=0$. It develops a tangential boundary extension and a tangential inner variation framework to derive a near-monotonicity formula, achieving optimal regularity up to the boundary and isolating singularities. Under a symmetry condition expressed through a nonnegative functional $\mathcal{T}$, the authors prove equivariance about the $e_3$-axis and show that, when $u_\theta=0$, exactly two boundary defects occur at antipodal points with no interior defects. They further provide both upper and lower energy bounds via explicit competitors and level-set analysis, highlighting the energy landscape of these constrained harmonic maps and raising open questions about extensions to weaker anchoring or more general models.

Abstract

Since the seminal work of Schoen-Uhlenbeck, many authors have studied properties of harmonic maps satisfying Dirichlet boundary conditions. In this article, we instead investigate regularity and symmetry of $\mathbb{S}^2-$valued minimizing harmonic maps subject to a tangency constraint in the model case of the unit ball in $\mathbb{R}^{3}$. In particular, we obtain a monotonicity formula respecting tangentiality on a curved boundary in order to show optimal regularity up to the boundary. We introduce novel sufficient conditions under which the minimizer must exhibit symmetries. Under a symmetry assumption, we present a delineation of the singularities of minimizers, namely that a mimimizer has exactly two point singularities, located on the boundary at opposite points.

Figures (5)

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Theorems & Definitions (53)

• Definition 1.1: Equivariance
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Remark 1.5: Ginzburg-Landau
• Remark 1.6: General domains
• Remark 1.7: Energy in 3D
• Proposition 2.1
• proof
• Corollary 2.2