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

Abstract

Abstract. Let (M, g) be an n−dimensional complete open Riemannian manifold with nonnegative Ricci curvature verifying ρ∆gρ ≥ n − 5 ≥ 0, where ∆g is the Laplace-Beltrami operator on (M, g) and ρ 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 K0 in the second-order Sobolev inequality in 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 = K0 if and only if (M, g) is isometric to the Euclidean space R n .