Topological rigidity of compact manifolds supporting Sobolev-type inequalities

2019/07/29 by Csaba Farkas, Alexandru Kristály, Farkas, Csaba +3
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #math.AP

paper · pdf · doi:10.48550/arxiv.1907.12197

7 pages

arxiv created 2019/07/29 · openalex publication_date 2019/07/29 · arxiv updated 2019/07/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let (M,g) be an n-dimensional (n≥ 3) compact Riemannian manifold with Ric_(M,g)≥ (n-1)g. If (M,g) supports an AB-type critical Sobolev inequality with Sobolev constants close to the optimal ones corresponding to the standard unit sphere (\mathbb Sⁿ,g₀), we prove that (M,g) is topologically close to (\mathbb Sⁿ,g₀). Moreover, the Sobolev constants on (M,g) are precisely the optimal constants on the sphere (\mathbb Sⁿ,g₀) if and only if (M,g) is isometric to (\mathbb Sⁿ,g₀); in particular, the latter result answers a question of V.H. Nguyen.

Related

• Rigidity and almost rigidity of Sobolev inequalities on compact spaces with lower Ricci curvature bounds
• An alternative proof of a rigidity theorem for the sharp Sobolev constant
• Quantitative stability of Sobolev inequalities on compact Riemannian manifolds
• Compactness for the Hardy-Sobolev equation on manifolds
• Critical Hardy--Sobolev Inequalities