The Bogomolov theorem says if $V$ is a rank 2 vector bundle on an algebraic surfaces $S$ is $H$-stable (in the sense of Mumford-Takemoto) for some ample divisor $H$, then $c_1^2(V) \leq 4c_2(V)$ holds.
My question is for which bundle $V$ and surface $S$, this inequality becomes an equality? Is there any classification of such pairs?
Thanks!