Minimizing Harmonic Maps on the Unit Ball with Tangential Anchoring

TL;DR

This paper studies the regularity and symmetry of harmonic maps into the sphere with tangential boundary conditions on the unit ball, establishing boundary regularity, symmetry conditions, and singularity structure.

Contribution

It introduces a monotonicity formula respecting tangentiality, provides conditions for symmetry, and characterizes boundary singularities of harmonic maps.

Findings

Established boundary regularity up to the boundary.

Identified conditions under which minimizers exhibit symmetry.

Characterized singularities as exactly two boundary points at opposite locations.

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.