# Topological rigidity of compact manifolds supporting Sobolev-type inequalities

## TL;DR

This paper proves that compact manifolds with Ricci curvature bounds supporting near-optimal Sobolev inequalities are topologically close to the sphere, and characterizes when they are isometric to the sphere based on Sobolev constants.

## Contribution

It establishes topological rigidity results for manifolds supporting Sobolev inequalities close to the sphere's optimal constants, answering a question by V.H. Nguyen.

## Key findings

- Manifolds with Ricci curvature ≥ (n-1)g supporting near-optimal Sobolev inequalities are topologically close to the sphere.
- Exact optimal Sobolev constants characterize the manifold as isometric to the sphere.
- The results provide a rigidity criterion linking Sobolev inequalities to manifold geometry.

## Abstract

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