On Brezis Open Problem 3.1

Technical Summary: Uniqueness of Weak Harmonic Maps with Latitude Circle Trace

Problem Statement
This paper addresses a long-standing open question regarding the uniqueness of weak harmonic maps from the unit disk $B_1 \subset \mathbb{R}^2$ to the unit sphere $S^2$. Specifically, it considers the Dirichlet problem for the harmonic map equation:
$$-\Delta u = |\nabla u|^2 u \quad \text{in } B_1, \quad u|_{\partial B_1} = g_R,$$
where the boundary trace $g_R(e^{i\theta}) = (R \cos \theta, R \sin \theta, \sqrt{1-R^2})$ describes a latitude circle traversed once at constant speed, with $0 < R < 1$.

In 1983, Brézis and Coron established the existence of two explicit solutions, $u_+$ and $u_-$, corresponding to conformal and anti-conformal fillings of the spherical caps defined by the boundary trace. They proved these maps are the unique energy minimizers within their respective relative homotopy classes (degrees $\pm 1$). However, the question remained whether non-minimizing critical points (saddle points) could exist for the same boundary data. This was formally posed as Open Problem 3.1 in Brézis' "Favorite Open Problems" list. The paper aims to determine if the two Brézis–Coron maps exhaust the set of all weak harmonic maps with this specific trace.

Methodology
The authors employ a "boundary rigidity" argument that combines geometric analysis, complex analysis, and variational inequalities. The proof proceeds in two primary stages:

1. Boundary Rigidity via Auxiliary Potentials and Isoperimetric Inequalities:

   • Auxiliary Potential: Associated with the harmonic map $u$, the authors define a global potential $X: B_1 \to \mathbb{R}^3$ satisfying $X_x = -u \times u_y$ and $X_y = u \times u_x$. This construction is rooted in the Gauss map/constant mean curvature (CMC) correspondence.
   • Projected Curve and Algebraic Area: They analyze the boundary curve $\Gamma = X|_{\partial B_1}$ and its orthogonal projection $\gamma$ onto the horizontal plane $x_3=0$. Using a projection-Jacobian identity, they prove that the signed (algebraic) area enclosed by $\gamma$ is exactly $\pi R^2$.
   • Pohozaev Identity and Length Bounds: The Pohozaev identity for the Hopf differential provides an exact integral bound on the length of $\Gamma$. Specifically, $\int_0^{2\pi} |u_r(1, \theta)|^2 d\theta = 2\pi R^2$.
   • Isoperimetric Saturation: By applying the planar isoperimetric inequality to the projected curve $\gamma$ (relating its length to its algebraic area) and comparing it with the length of $\Gamma$, the authors establish a chain of inequalities:
     $$4\pi^2 R^2 \leq L_h^2 \leq L^2 \leq 4\pi^2 R^2.$$
     Since the lower bound (from the exact algebraic area) and the upper bound (from the Pohozaev identity) coincide, all inequalities must be equalities. This "saturation" forces pointwise identities on the boundary: $|u_r| \equiv R$ and $u_r \cdot u_\theta \equiv 0$.

2. Conformal Classification:

   • Vanishing Hopf Differential: The boundary identities derived above imply that the Hopf differential $q(z) = u_z \cdot u_z$ vanishes on $\partial B_1$. Since $q$ is holomorphic in $B_1$, the maximum modulus principle forces $q \equiv 0$ throughout the disk. Thus, the map $u$ is weakly conformal.
   • Stereographic Projection and Laurent Series: Using the south-pole stereographic projection, the map is represented by a complex-valued function $w$ on a boundary collar. The vanishing of the Hopf differential implies $w$ is either holomorphic or anti-holomorphic.
   • Uniqueness on the Collar: Given the non-vanishing boundary derivative and the specific trace $w(e^{i\theta}) = \rho e^{i\theta}$, a Laurent series expansion shows that $w$ must be either $\rho z$ (holomorphic) or $\rho/\bar{z}$ (anti-holomorphic).
   • Global Extension: By the real analyticity of solutions to the elliptic system, the identification on the collar extends to the entire disk, yielding exactly the two Brézis–Coron solutions.

Key Contributions and Results

• Resolution of Open Problem 3.1: The paper proves Theorem 1.1, stating that for $0 < R < 1$, the only weak harmonic maps in $H^1(B_1; S^2)$ with trace $g_R$ are the two explicit Brézis–Coron maps $u_+$ and $u_-$. This definitively answers the question by ruling out the existence of any non-minimizing critical points.
• Boundary Rigidity Mechanism: The authors introduce a novel rigidity argument that converts the exact algebraic area of a projected potential curve into the vanishing of the Hopf differential. This mechanism isolates a geometric feature of the "once-around" latitude trace that is invisible to standard variational classification methods.
• Application to H-Systems: The paper derives Corollary 1.2, applying the uniqueness result to two-dimensional $H$-systems (conformal immersions with constant mean curvature). It classifies conformal immersions $Y: B_1 \to \mathbb{R}^3$ with prescribed unit normal traces, showing they must be rigid translations of the model disks $Z_+$ or $Z_-$.
• Endpoint Analysis: The authors analyze the limits $R \to 1$ (equator) and $R \to 0$ (pole). They show the result holds for $R=1$ (non-degenerate), while $R=0$ represents a degenerate case where the boundary collapses to a point, yielding only the constant map.

Significance
The paper provides a definitive affirmative answer to Brézis' Open Problem 3.1. Its significance lies in demonstrating that for the specific case of a once-around latitude circle, the "multiplicity" phenomenon observed in higher-winding or more general non-convex boundary data does not occur. The proof establishes that the two natural conformal and anti-conformal fillings exhaust all critical points, not just the minimizers.

The work highlights a special geometric rigidity inherent to the once-around trace, driven by the interplay between the Pohozaev identity, the Hopf differential, and the sharp planar isoperimetric inequality. This approach offers a new perspective on uniqueness problems in harmonic map theory, distinguishing cases where variational methods alone are insufficient from those where geometric rigidity forces uniqueness. The authors note that this mechanism collapses for higher winding numbers ($m \geq 2$), where multiple critical points are expected, thereby clarifying the topological and geometric boundaries of uniqueness in this field.