Second-order Sobolev inequalities on a class of Riemannian manifolds with nonnegative Ricci curvature

Abstract

Let $(M,g)$ be an $n-$dimensional complete open Riemannian manifold with nonnegative Ricci curvature verifying $\rho {\Delta }_g\rho \ge n-5\ge 0$, where ${\Delta }_g$ is the Laplace-Beltrami operator on $(M,g)$ and $\rho$ is the distance function from a given point. If $(M,g)$ supports a second-order Sobolev inequality with a constant $C>0$ close to the optimal constant $K_0$ in the second-order Sobolev inequality in ${\mathbb{R}}^n$, we show that a global volume non-collapsing property holds on $(M,g)$. The latter property together with a Perelman-type construction established by Munn (J. Geom. Anal., 2010) provide several rigidity results in terms of the higher-order homotopy groups of $(M,g)$. Furthermore, it turns out that $(M,g)$ supports the second-order Sobolev inequality with the constant $C=K_0$ if and only if $(M,g)$ is isometric to the Euclidean space ${\mathbb{R}}^n$.

arXiv comment10 pages; to appear in Bulletin of the London Mathematical Society