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

Abstract

Let (M, g) a compact Riemannian n-dimensional manifold. It is well know that, under certain hypothesis, in the conformal class of g there are scalar-flat metrics that have ∂M as a constant mean curvature hypersurface. Also, under certain hypothesis, it is known that these metrics are a compact set. In this paper we prove that, both in the case of umbilic and non-umbilic boundary, if we linearly perturb the mean curvature term hg with a negative smooth function α, the set of solutions of Yamabe problem is still a compact set.2000 Mathematics Subject Classification. 35J65, 53C21.

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

“…The proof of this result is similar to prove of [19,Lemma 6] and will be postponed in the Appendix At this point we can prove the main result of this section.…”

“…As a final remark, we notice that in [19] we ask that the manifold is umbilic, the Weyl tensor is never vanishing and that n ≥ 11. The assumption on the dimension in this paper is technical, since we ask some integrability condition when performing the Ljapounov Schmidt procedure.…”

“…In fact, we have that the set of solutions is compact -and hence (1) is stable-perturbing the mean curvature from below while we construct a blowing up sequence when the perturbation is everywhere positive on the boundary, and for a class of perturbation which are positive in at least one point on the boundary. The result of compactness is dealt in [16] both in umbilic and non umbilic case while for the construction of blowing up sequences for umbilic boundary manifold we refer to [19].…”

“…In deed, in the present paper we perform more effective computations in Lemma 6. This method could be applied verbatim in paper [19], so we can reformulate the main result in dimension n ≥ 8.…”

Compactness results for linearly perturbed Yamabe problem on manifolds with boundary

“…where ε is a small positive parameter and α : M → R is a smooth function. We can prove that the sign of the function α on ∂M has an effect on compactness and non compactness of solutions of (1.2): in [13] we proved the existence of blowing up solution of (1.2) when α > 0 in the case of ∂M non umbilic and n ≥ 7 and in [12] we proved an analogous result in the case of n ≥ 11 and the Weyl tensor not vanishing on ∂M .…”