The asymptotic behavior of Chern-Simons Higgs model on a compact Riemann surface with boundary

We study the self-dual Chern-Simons Higgs equation on a compact Riemann surface with Neuman boundary condition. In [23], we show that the Chern-Simons Higgs equation with parameter λ > 0 has at least two solutions (uλ1, uλ2) for λ sufficiently large, which satisfy that uλ1 → -u0 almost everywhere as λ → ∞, and that uλ2 → -∞ almost everywhere as λ → ∞, where u0 is a (negative) Green function on M. In this paper, we study the asymptotic behavior of the solutions as λ → ∞, and prove that uλ2 - uλ2-bar converges to a solution of the Kazdan-Warner equation if the geodesic curvature of the boundary ∂M is negative, or the geodesic curvature is nonpositive and the Gauss curvature is negative where the geodesic curvature is zero. (author)