We concerns here with the continuity on the geometry of the second
Riemannian $L^p$-Sobolev best constant $B_0(p,g)$ associated to the
AB program. Precisely, for $1 \leq p \leq 2$, we prove that
$B_0(p,g)$ depends continuously on $g$ in the $C^2$-topology.
Moreover, this topology is sharp for $p = 2$. From this discussion,
we deduce some existence and $C^0$-compactness results on extremal