Area-minimizing unit vector fields on some spherical annuli

TL;DR

This work establishes a sharp lower bound on the area of a unit vector field defined on spherical annuli in $\mathbb{S}^2$ using the Sasaki metric to measure area. By imposing boundary-tangent conditions along the parallels and a constant-angle constraint, the authors derive $area(V) \ge 2\pi \cos(\alpha_0) + K_{\alpha_0}$ and identify the explicit minimizer with $\theta(\alpha) = \arcsin(\cot(\alpha_0) \tan(\alpha)) + \frac{\pi}{2}$, which attains equality. The results extend prior area-minimization findings on punctured spheres to spherical annuli and connect to the broader framework involving Poincaré indices and Pontryagin fields. The paper also situates the main theorem within the landscape of known area-minimizing unit vector fields on $\mathbb{S}^2$ and discusses how the minimizing field is constructed from the angular data along parallels. Overall, it provides explicit sharp bounds and a concrete minimizer for area on a natural class of curved domains in $\mathbb{S}^2$ with potential implications for tangent-bundle geometry and variational problems on surfaces.

Abstract

We establish in this paper a sharp lower bound for the area of a unit vector field $V$ defined on some spherical annuli in the Euclidean sphere $\mathbb{S}^2$.

Figures (2)

• Figure 1: Visual representation of the area-minimizing unit vector field $V_{k}$ on $\mathbb{S}^2\backslash\{N,S\}$
• Figure 2: Visual representation of the area-minimizing unit vector field $V$ on $A_{\alpha_0}$

Theorems & Definitions (6)

• Theorem 1
• Theorem 2: Brito, Chacón and Johnson, 2
• Theorem 3: Brito, Gomes and Gonçalves, 4
• Theorem 4: Brito et al., 3
• Definition 5
• proof