Liouville type theorems for conformal Gaussian curvature equation

2008/10/29 by Li Ma, Yihong Du, Ma, Li +1
Computer Science · Mathematics · #35J60 #45G10 #53Cxx #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #math.AP #math.DG #msc:35J60 #msc:45G10 #msc:53Cxx

paper · pdf · doi:10.48550/arxiv.0810.5301

14 pages

openalex publication_date 2008/10/29 · arxiv created 2009/08/18 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this note, we study Liouville type theorem for conformal Gaussian curvature equation (also called the mean field equation) -Δu=K(x)e^u, in R² where K(x) is a smooth function on R². When K(x)=K(x₁) is a sign-changing smooth function in the real line R, we have a non-existence result for the finite total curvature solutions. When K is monotone non-decreasing along every ray starting at origin, we can prove a non-existence result too. We use moving plane method and moving sphere method.

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