Blow-up Analysis of Large Conformal Metrics With Prescribed Gaussian And Geodesic Curvatures

Abstract: In this talk, we consider a compact Riemannian surface (M,g) with nonempty boundary and negative Euler characteristic. Given two smooth non-constant functions f in M and h in the boundary of M with max f = max h = 0, under a suitable condition on the maximum points of f and h, we prove that for sufficiently small positive constants λ and  μ, there exist at least two distinct conformal metrics g_{λ,μ}=e^{2u_{μ,λ}}g and g^{λ,μ}=e^{2u^{μ,λ}}g with prescribed sign-changing Gaussian and geodesic curvature equal to f+μ and h+λ, respectively. Additionally, we employ the method Borer et al. (2015) used to study the blowing up behavior of the large solution u^{μ,λ} when μ↓0 and λ↓0. This is joint work with R. Caju (Universidad de Chile) and T. Cruz (UFAL).