= Finite sigma-model energy need not give a limit at infinity
For arbitrary maps $\phi:\mathbb R^2\to S^2$, integrability of the Dirichlet <energy> does not imply continuous extension to the <one-point compactification>. For large $r$, take
$$
\phi(r,\vartheta)=(\sin a(r),0,\cos a(r)),\qquad a(r)=\log\log r,
$$
and interpolate smoothly to a constant on a bounded disk. Its exterior <energy> is
$$
\frac12\int_{|x|\geq R}|\nabla\phi|^2\,d^2x
=\pi\int_R^\infty\frac{dr}{r\log^2r}=\frac{\pi}{\log R}<\infty.
$$
The field has no limit because $a(r)$ keeps winding. A continuous extension at the point at infinity instead requires $\phi(x)\to\phi_\infty$ uniformly outside sufficiently large disks. This fixed-limit boundary condition is standard for a <sigma-model lump>, but is extra information beyond finite <energy> for arbitrary fields.
Back to article page
