Extremal Sections for a Trudinger–Moser Functional on Vector Bundle over a Closed Riemann Surface.

Trudinger–Moser inequalities are important tools in partial differential equations and geometric analysis. Although there have been many results in this regard, there are few studies on vector valued function spaces. Years ago, joined with Y. Li and P. Liu, the second named author established a Trudinger–Moser inequality on vector bundle over a closed Riemann surface. In that article (Calc Var 28:59–87, 2007), we do not settle down the existence of extremal bundle sections. In the present paper, we have two purposes, one is to prove the existence of extremal bundle sections, and the other is to generalize such an inequality to a stronger one with remainder terms. The proof is based on a much more delicate analysis and a very careful construction of a sequence of bundle sections. [ABSTRACT FROM AUTHOR]