  The moduli space of Higgs bundles has two stratifications. The
Bialynicki-Birula stratification comes from the action of the non-zero complex
numbers by multiplication on the Higgs field, and the Shatz stratification
arises from the Harder-Narasimhan type of the vector bundle underlying a Higgs
bundle. While these two stratification coincide in the case of rank two Higgs
bundles, this is not the case in higher rank. In this paper we analyze the
relation between the two stratifications for the moduli space of rank three
Higgs bundles.
