  Existence of a complex structure on the $6$ dimensional sphere is proved in
this paper. The proof is based on re-interpreting a hypothetical complex
structure as a classical ground state of a Yang--Mills--Higgs-like theory on
$S^6$. This classical vacuum solution is then constructed by Fourier expansion
(dimensional reduction) from the obvious one of a similar theory on the $14$
dimensional exceptional compact Lie group ${\rm G}_2$.
