  Using the method of blow-up analysis, we obtain two sharp Trudinger-Moser
inequalities on a compact Riemann surface with smooth boundary, as well as the
existence of the corresponding extremals. This generalizes early results of
Chang-Yang [7] and the first named author [32], and complements Fontana's
inequality of two dimensions [15]. The blow-up analysis in the current paper is
far more elaborate than that of [32], and particularly clarifies several
ambiguous points there. In precise, we prove the existence of isothermal
coordinate systems near the boundary, the existence and uniform estimates of
the Green function with the Neumann boundary condition. Also our analysis can
be applied to the Kazdan-Warner problem and the Chern-Simons Higgs problem on
compact Riemman surfaces with smooth boundaries.
