Topological rigidity of compact manifolds supporting Sobolev-type inequalities

• DOI: 10.48550/arxiv.1907.12197
• Publication Date: 2019-01-01
• Authors: Csaba Farkas; Alexandru Kristály; Ágnes Mester
• Keywords: Unit sphere; Rigidity (electromagnetism); Manifold (fluid mechanics); Riemannian manifold; Sobolev inequality; Constant (computer programming)

Let $(M,g)$ be an $n$-dimensional $(n\geq 3)$ compact Riemannian manifold with Ric$_{(M,g)}\geq (n-1)g$. If supports AB-type critical Sobolev inequality constants close to the optimal ones corresponding standard unit sphere $(\mathbb S^n,g_0)$, we prove that is topologically S^n,g_0)$. Moreover, on are precisely S^n,g_0)$ if and only isometric S^n,g_0)$; in particular, latter result answers a question of V.H. Nguyen.