Blow-up analysis of Large conformal metrics with prescribed Gaussian and geodesic curvatures

2024/02/04 by Rayssa Caju, Caju, Rayssa, Tiarlos Cruz +3
Mathematics · Physics and Astronomy · #35B44 #35J20 #35J60 #58J32 #Advanced Differential Geometry Research #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.2402.02467

openalex publication_date 2024/02/04 · openalex created_date 2024/02/07 · openalex updated_date 2026/07/28

Abstract

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 ∂ 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,λg and g^λ,μ=e^2u^μ,λg with prescribed sign-changing Gaussian and geodesic curvature equal to f + μ and h + λ, respectively. Additionally, we employ the method used in Borer et al. (2015) to study the blowing up behavior of the large solution u^μ,λ when μ\downarrow 0 and λ\downarrow 0. Finally, we derive a new Liouville-type result for the half-space, eliminating one of the potential blow-up profiles.

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