We concern -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. Applying this result and the positive mass theorem, we deduce the -compactness for all 4-manifolds (which may be non-umbilic). For the 5-dimensional case, we also establish that a sum of the second-order derivatives of the trace-free second fundamental form is non-negative at possible blow-up points. We essentially use this fact to obtain the -compactness for all 5-manifolds. Finally, we show that the -compactness on 6-manifolds is true if the trace-free second fundamental form on the boundary never vanishes.
Cite this article
Monica Musso, Juncheng Wei, Seunghyeok Kim, Compactness of scalar-flat conformal metrics on low-dimensional manifolds with constant mean curvature on boundary. Ann. Inst. H. Poincaré Anal. Non Linéaire 38 (2021), no. 6, pp. 1763–1793DOI 10.1016/J.ANIHPC.2021.01.005