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. Applying this result and the positive mass theorem, we deduce the $C^2$-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 $C^2$-compactness for all 5-manifolds. Finally, we show that the $C^2$-compactness on 6-manifolds is true if the trace-free second fundamental form on the boundary never vanishes.

我们关心 $C^2$- 具有边界的光滑紧凑黎曼流形上的边界 Yamabe 问题的解集的紧凑性，前提是它们的维数为 4、5 或 6。通过对与问题相关的线性方程进行定量分析，我们证明了无迹第二个基本形式必须在一系列爆破解的可能爆破点消失。应用这个结果和正质量定理，我们推导出$C^2$-所有 4 个歧管（可能是非脐带）的紧凑性。对于 5 维情况，我们还确定无迹第二基本形式的二阶导数之和在可能的爆发点处是非负的。我们基本上使用这个事实来获得$C^2$-所有 5 个歧管的紧凑性。最后，我们证明$C^2$如果边界上的无迹第二基本形式永远不会消失，则 6 流形上的紧凑性为真。