Moser-Trudinger inequalities of vector bundle over a compact Riemannian   manifold of dimension 2
  Let $(M,g)$ be a 2-dimensional compact Riemannian manifold. In this paper, we use the method of blowing up analysis to prove several Moser-Trdinger type inequalities for vector bundle over $(M,g)$. We also derive an upper bound of such inequalities under the assumption that blowing up occur. 
Yuxiang Li, Pan Liu, Yunyan Yang