Blow-up analysis involving isothermal coordinates on the boundary of compact Riemann surface

TL;DR

This paper establishes sharp Trudinger-Moser inequalities on compact Riemann surfaces with boundary using advanced blow-up analysis, and proves the existence of extremals, extending previous results and providing new insights into boundary coordinate systems and Green functions.

Contribution

It introduces a more detailed blow-up analysis for boundary problems, proving the existence of isothermal coordinates and Green functions, and extends inequalities to broader geometric contexts.

Findings

Proved sharp Trudinger-Moser inequalities on surfaces with boundary.

Established existence of extremals for these inequalities.

Developed detailed blow-up analysis and boundary coordinate systems.

Abstract

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.