A Trudinger-Moser inequality on compact Riemannian surface involving Gaussian curvature

TL;DR

This paper establishes a new Trudinger-Moser inequality on compact Riemannian surfaces that incorporates Gaussian curvature, extending classical results and providing bounds for conformal metrics under certain topological and geometric conditions.

Contribution

It introduces a curvature-involved inequality on compact surfaces and improves existing bounds on Liouville energy for conformal metrics.

Findings

Established a Trudinger-Moser inequality involving Gaussian curvature.

Proved a uniform lower bound for a modified Liouville energy under specific conditions.

Utilized blow-up analysis and Carleson-Chang's result in the proof.

Abstract

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.