  Motivated by a recent work of X. Chen and M. Zhu (Commun. Math. Stat., 1
(2013) 369-385), we establish a Trudinger-Moser inequality on compact
Riemannian surface without boundary. The proof is based on blow-up analysis
together with Carleson-Chang's result (Bull. Sci. Math. 110 (1986) 113-127).
This inequality is different from the classical one, which is due to L. Fontana
(Comment. Math. Helv., 68 (1993) 415-454), since the Gaussian curvature is
involved. As an application, we improve Chen-Zhu's result as follows: A
modified Liouville energy of conformal Riemannian metric has a uniform lower
bound, provided that the Euler characteristic is nonzero and the volume of the
conformal surface has a uniform positive lower bound.
