A compactness result for scalar-flat metrics on low dimensional manifolds with umbilic boundary

Blow-up phenomena for linearly perturbed Yamabe problem on manifolds with umbilic boundary

“…In both classical and boundary Yamabe problems, as a corollary of the compactness of solutions, people get that the problem is also stable for perturbation from below of the critical exponent. So stability is proved for scalar flat manifolds in [22], for manifolds with non umbilic boundary in [3,32], and for umbilic boundary manifolds whose Weyl tensor never vanishes on ∂M in [23,24]. On the other hand, in the present paper (and in [26] for umbilic boundary manifolds) we prove that Yamabe boundary problem is unstable with respect of perturbation form above of the critical exponent.…”

“…, n, obtaining that Wλ √ ε,q + λ√ εV λ √ ε,q + φ λ √ ε,q solves also (26), and so the proof is complete Proof of Theorem 1. By our assumption on the second fundamental form and by (24), we have that the function ϕ(q) defined in Proposition 7 is strictly negative on ∂M . We recall as well, that the number C defined in the same proposition is positive.…”

“…When the trace-free part of the second fundamental form is non-zero everywhere on the boundary the boundary of the manifold is said non umbilic and Almaraz exploited this condition to bypass the Positive Mass Theorem. This strategy has been recently adapted to manifold with umbilic boundary, that is when the tensor of the second funtamental form vanishes on the boundary, by the authors, provided that the Weyl tensor is always different from zero on the boundary and n ≥ 6 in [23,24]. Dimension n = 24 appears to be relevant also in boundary Yamabe problems, in fact Almaraz in proved that for n ≥ 25 there exists manifold with umbilic boundary for which compactness of solutions fails.…”

Blowing up solutions for supercritical Yamabe problems on manifolds with umbilic boundary

“…Very recently, compactness has been proved by the authors in [15] for manifold with umbilic boundary when n > 8 and the Weyl tensor of M is everywhere non zero on the boundary ∂M . In [17] the authors extend the compactness result to manifold of dimension n = 6, 7, 8, when the boundary is umbilic and the Weyl tensor of M is everywhere non zero on ∂M .…”

“…An improvement of the estimate ( 20) is performed in [17], where the authors give a more precise description of the function v q as a sum of an harmonic function with explicit rational functions. This description, for n = 8, leads to the inequality, proved in [17, Lemma 19],…”