Compactness of scalar-flat conformal metrics on low-dimensional manifolds with constant mean curvature on boundary

Abstract

We concern C 2 -compactness of the solution set of the boundary Yamabe problem on smooth compact Riemannian manifolds with boundary provided that their dimensions are 4, 5 or 6. By conducting a quantitative analysis of a linear equation associated with the problem, we prove that the trace-free second fundamental form must vanish at possible blow-up points of a sequence of blowing-up solutions. Together with the positive mass theorem, we conclude the C 2 -compactness holds for all 4-manifolds (which may be non-umbilic). For 5-and 6-manifolds, we also prove that the C 2 -compactness is true if the trace-free second fundamental form on the boundary never vanishes.

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

“…In this section we give a an accurate description of a solution γ of (2.17), similarly to [18]. First we split γ = Φ + E where Φ = Φ1 + Φ2 is a polynomial function and Φ1 , Φ2 solve, respectively…”

“…Proof. The first claim is proved in[18, Lemma A.1] (in particular in formula (A.2)). The second claim is proved again in [18, Lemma A.1], while the last claim corresponds to[18, Lemma A.2].…”

“…If the dimension of the manifold is n ≥ 7 and the trace-free second fundamental form in non zero everywhere on ∂M , Almaraz in [1] proved compactness. Very recently, Kim Musso and Wei [18] showed that compactness continues to hold when n = 4 and when n = 6, 7 and the trace-free second fundamental form in non zero everywhere on ∂M . Compactness was proved also by the authors in [11] for manifold with umbilic boundary when n = 8 and the Weyl tensor of the boundary is always different from zero, or if n > 8 and the Weyl tensor of M is always different from zero on the boundary.…”

“…In the case of low dimensional manifolds this requires a very accurate pointwise estimate of the correction term which seems not to have an explicit form in the case of boundary Yamabe problem. This process is somewhat inspired to the strategy used by Kim Musso and Wei [18] to estimate the correction term on low dimensional manifold with non umbilic boundary.…”

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

“…Concerning problem (1), Felli and Ould Ahmedou in [10] have proved compactness when M is locally conformally flat and the boundary is umbilic and Almaraz in [3] has proved compactness when n ≥ 7 and the trace free second fundamental form is non zero everywhere on ∂M , that is any point of the boundary is non umbilic. In [12] Kim Musso and Wei showed that compactness continues to hold when n = 4 and when n = 6, 7 and the trace free second fundamental form is non zero everywhere on the boundary.…”