Blow-up analysis on line bundles over a compact Riemann surface with smooth boundary (Q6577485)

The Trudinger-Moser inequality (TMI) on \((M,g)\), a closed Riemann surface, states: \N\[\N\sup_{u\in E}\int_M\exp(\gamma u^2)dv_g<\infty,\text{ for }\gamma\le 4\pi,\tag{1}\N\]\Nwhere supremum is taken over all \(u\in E:=\left\{u\in W^{1,2}(M),\displaystyle\int_Mudv_g=0,\text{ and }\int_M|\nabla_gu|dv_g\le 1\right\}\) such that \(W^{1,2}(M)\) represents the Sobolev space of order two on \(M\), \(\nabla_g\) (resp \(dv_g\)) is the standard gradient operator (resp. the Riemann volume element w.r.t to \(g\)). The purpose of the author is to provide a trace embedding inequality (a distinct from the classical TMI) for a smooth line bundle over a compact Riemann surface with smooth boundary.