Hélein's bounded Coulomb-frame conjecture for maps into the sphere

Let $D_1$ be the unit disc in $\mathbb R^2$, let $\mathbb S^2$ be the unit sphere in $\mathbb R^3$, and let $\mathbf n:D_1\to\mathbb S^2$ satisfy

$$\int_{D_1}|\nabla\mathbf n|^2\,dX\leq 8\pi-\delta,\qquad \delta>0.$$

An orthonormal moving frame $(\mathbf e_1,\mathbf e_2)$ along $\mathbf n$ consists of vectors orthogonal to $\mathbf n$ such that $(\mathbf e_1(X),\mathbf e_2(X),\mathbf n(X))$ has positive orientation and $\mathbf n\cdot(\mathbf e_1\times\mathbf e_2)>0$. The conjecture also concerns a function $f$ on $D_1$ satisfying the displayed first-order and Dirichlet equations below.

Hélein's conjecture. There exist an orthonormal moving frame $(\mathbf e_1,\mathbf e_2)$ and a function $f$ such that

$$\|\mathbf e_i\|_{W^{1,2}(D_1)}\leq c(\delta),\qquad i=1,2,$$ $$\partial_1f=-\mathbf e_1\cdot\partial_2\mathbf e_2,\qquad \partial_2f=\mathbf e_1\cdot\partial_1\mathbf e_2\quad\text{in }D_1,$$ $$-\Delta f=\partial_1\mathbf e_1\cdot\partial_2\mathbf e_2-\partial_2\mathbf e_1\cdot\partial_1\mathbf e_2\quad\text{in }D_1,\qquad f=0\quad\text{on }\partial D_1,$$

and

$$\|\nabla f\|_{L^2(D_1)}\leq c(\delta),\qquad \|f\|_{L^\infty(D_1)}\leq c(\delta).$$

This is the $n=3$ formulation of Hélein's conjecture on Coulomb frames for maps into the oriented Grassmannian. It removes the assumption that $\mathbf n$ is the Gauss map of a surface and seeks bounded conformal-factor-type control under the sharp energy threshold $8\pi$; the supplied source presents the assertion as the conjecture proved in the paper.