On the Moser-Trudinger inequality for surfaces

This paper is devoted to the Moser-Trudinger inequality on smooth riemanniansurfaces. We establish that the constants involved can be chosen to depend on only 3parameters, which are the systole, isoperimetric constant and curvature of the surface. Wehave two analogous statements, considering respectively infinite-volume surfaces and closedsurfaces. As an application, we show that there are sequences of coverings of a hyperbolicclosed surface which admit a uniform Moser-Trudinger inequality.