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

We concern C²-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 C²-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 C²-compactness for all 5-manifolds. Finally, we show that the C²-compactness on 6-manifolds is true if the trace-free second fundamental form on the boundary never vanishes.