Surveys in Differential Geometry

Volume 13 (2008)

Existence of Faddeev knots

Pages: 149 – 222



Fengbo Hang

Fanghua Lin

Yisong Yang


In this paper, we present an existence theory for absolute minimizers of theFaddeev knot energies in the general Hopf dimensions. These minimizers aretopologically classified by the Hopf-Whitehead invariant, $Q$, represented asan integral of the Chern-Simons type. Our method involves an energydecomposition relation and a fractionally powered universal topological growthlaw. We prove that there is an infinite subset $\mathbb{S}$ of the set of allintegers such that for each $N\in{\mathbb{S}}$ there exists an energyminimizer in the topological sector $Q=N$. In the compact setting, we showthat there exists an absolute energy minimizer in the topological sector $Q=N$for any given integer $N$ that may be realized as a Hopf-Whitehead number. Wealso obtain a precise energy-splitting relation and an existence result forthe Skyrme model.

Published 1 January 2008