A Remark on the Concentration Compactness Principle in Critical Dimension

We prove some refinements of the concentration compactness principle for Sobolev space W^{1, n} on a smooth compact Riemannian manifold of dimension n. As an application, we extend Aubin's theorem for functions on (Formula presented.) with zero first-order moments of the area element to the higher-order moments case. Our arguments are very flexible and can be easily modified for functions satisfying various boundary conditions or belonging to higher-order Sobolev spaces.