Blow-up analysis of conformal metrics of the disk with prescribed Gaussian and geodesic curvatures

"Bubbling" of the prescribed curvature flow on the torus

“…In the latter case, moreover, similar to our analysis of the prescribed curvature flow on S 2 in [19], in (5.30) we exhibit a 2-dimensional "shadow flow" of ordinary differential equations showing that the motion of the center of mass of the evolving metrics is essentially driven by a combination of the gradients of the functions f and j, where we extend j as a harmonic function on the disc. Quite miraculously, we are able to relate this combination to the gradient of the function (1.24) J = j + j 2 + f introduced by Jevnikar et al in [12]. On the other hand, our results show that in contrast to their Theorem 1.1 it seems that not only critical points z 0 ∈ ∂B of J may arise as concentration points of the flow.…”

“…Main results. Even though condition (1.23) is somewhat weaker than the conditions required by Jevnikar et al in [12], in Corollary 4.4 below we will show that the conclusion of their Theorem 1.1 still holds and that either a subsequence of the conformal metrics g l = e 2u(t l ) g 0 converges in H 3/2 (B) ∩ H 1 (∂B) and uniformly to a metric g ∞ = e 2u∞ g 0 inducing a solution of (1.3), (1.4), or the flow subsequentially concentrates at a boundary point z 0 ∈ ∂B in the sense of measures, exhibiting blow-up in a spherical cap.…”