Solutions of the singular Yamabe problem near singular boundaries

2025/12/29 by Weiming Shen, Shen, Weiming, Zhehui Wang +3
Mathematics · Computer Science · #Nonlinear Partial Differential Equations #Advanced Mathematical Modeling in Engineering #Geometric Analysis and Curvature Flows

paper · doi:10.48550/arxiv.2512.23331

Abstract

In this paper, we investigate the asymptotic behaviors of solutions to the singular Yamabe problem with negative constant scalar curvature near singular boundaries and derive optimal estimates, where the background metrics are not assumed to be conformally flat. Specifically, we demonstrate that for a wide class of Lipschitz domains with asymptotic conical structure, the local positive solutions are well approximated by the positive solutions in the tangent cones at singular boundary points. This extends the results of [10, 12, 26].

Citations

• Scattering on singular Yamabe spaces
• Two flow approaches to the Loewner-Nirenberg problem on manifolds
• The Loewner-Nirenberg Problem in Cones
• Isolated singularities of solutions to the Yamabe equation in dimension 6
• Asymptotic expansions of solutions of the Yamabe equation and the σ_k-Yamabe equation near isolated singular points
• Chern-Gauss-Bonnet formula for singular Yamabe metrics in dimension four
• Boundary expansion for the Loewner-Nirenberg problem in domains with conic singularities
• Escobar-Yamabe compactifications for Poincare-Einstein manifolds and rigidity theorems
• Volume renormalization for singular Yamabe metrics
• The Loewner-Nirenberg problem in singular domains
• Refined asymptotics for constant scalar curvature metrics with isolated singularities
• Constant scalar curvature metrics with isolated singularities
• A construction of singular solutions for a semilinear elliptic equation\n using asymptotic analysis
• Moduli Spaces of Singular Yamabe Metrics
• Asymptotic symmetry and local behavior of semilinear elliptic equations with critical sobolev growth
• The existence of weak solutions with prescribed singular behavior for a conformally invariant scalar equation
• Conformally flat manifolds, Kleinian groups and scalar curvature

Related

• Asymptotic behavior of complete conformal metric near singular boundary
• Isolated Singularities of Solutions to the Yamabe Equation with boundary in Dimension 3 and 4
• Isolated singularities of solutions to the Yamabe equation in dimension 6
• The asymptotically flat scalar-flat Yamabe problem with boundary
• A singular Yamabe problem on manifolds with solid cones