  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 <= p <= 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 functions.
