  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.
