Complex structure on the six dimensional sphere from a spontaneous   symmetry breaking
  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 ${\rm G}_2$. 
Gabor Etesi