  We prove some refinements of 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 $\mathbb{S}^{n}$ with
zero first order moments of the area element to 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.
