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# Surveys in Differential Geometry

## Volume 13 (2008)

### Existence of Faddeev knots

Pages: 149 – 222

DOI: https://dx.doi.org/10.4310/SDG.2008.v13.n1.a6

#### Authors

#### Abstract

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