Asymptotic behavior of solutions to the σ k -Yamabe equation near isolated singularities

2009/11/02 by Zheng-Chao Han, YanYan Li, Eduardo V. Teixeira · 75 citations
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Curvature #Geometry #Gravitational singularity #Mathematical analysis #Mathematical physics #Mathematics #Nonlinear Partial Differential Equations #Physics #Scalar curvature #Sectional curvature #Sigma #Singularity #Spectral Theory in Mathematical Physics #Yamabe flow #math.AP #math.DG #msc:35A24 #msc:35B40 #msc:53A30 #msc:58J05

paper · pdf · doi:10.1007/s00222-010-0274-7

published in Inventiones mathematicae 182(3), 635-684 (Springer Science+Business Media) · 55 pages

arxiv created 2009/11/02 · openalex publication_date 2010/08/18 · arxiv updated 2015/05/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

σ_k-Yamabe equations are conformally invariant equations generalizing the classical Yamabe equation. In an earlier work YanYan Li proved that an admissible solution with an isolated singularity at 0∈ \mathbb Rⁿ to the σ_k-Yamabe equation is asymptotically radially symmetric. In this work we prove that an admissible solution with an isolated singularity at 0∈ \mathbb Rⁿ to the σ_k-Yamabe equation is asymptotic to a radial solution to the same equation on \mathbb Rⁿ ∖ \0\. These results generalize earlier pioneering work in this direction on the classical Yamabe equation by Caffarelli, Gidas, and Spruck. In extending the work of Caffarelli et al, we formulate and prove a general asymptotic approximation result for solutions to certain ODEs which include the case for scalar curvature and σ_k curvature cases. An alternative proof is also provided using analysis of the linearized operators at the radial solutions, along the lines of approach in a work by Korevaar, Mazzeo, Pacard, and Schoen.

Citations

• A symmetry problem in potential theory
• Conformally flat manifolds, Kleinian groups and scalar curvature
• Refined asymptotics for constant scalar curvature metrics with isolated singularities

Cited by

• Local Analysis of Solutions of Fractional Semi-Linear Elliptic Equations with Isolated Singularities
• Asymptotic Behavior of Rupture Solutions for the Elliptic MEMS Equation with Hénon-Type and External Pressure Terms
• On the σ₂-Nirenberg problem on \mathbbS²
• Liouville-type theorems, radial symmetry and integral representation of solutions to Hardy-Hénon equations involving higher order fractional Laplacians
• Comparison principles and Lipschitz regularity for some nonlinear degenerate elliptic equations
• Asymptotic expansions of solutions of the Yamabe equation and the σ_k-Yamabe equation near isolated singular points
• Constant σ_k-curvature metrics with Delaunay type ends
• Asymptotic behavior of positive solutions to a nonlinear biharmonic equation near isolated singularities
• On the σ_k-Nirenberg problem
• σ₂ Yamabe problem on conic 4-spheres
• Local estimates for conformal Q-curvature equations
• The Loewner-Nirenberg problem in singular domains
• On local behavior of singular positive solutions to nonlocal elliptic equations
• Asymptotic symmetry and local behavior of solutions of higher order conformally invariant equations with isolated singularities

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• Asymptotic expansions of solutions of the Yamabe equation and the σ_k-Yamabe equation near isolated singular points
• Isolated singularities of solutions to the Yamabe equation in dimension 6
• Classification of singular radial solutions to the σ_k Yamabe equation on annular domains
• Isolated Singularities of Solutions to the Yamabe Equation with boundary in Dimension 3 and 4
• Singular solutions to the Yamabe equation with prescribed asymptotics