REAL

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

Barbosa, Ezequiel and Kristály, Alexandru (2017) Second-order Sobolev inequalities on a class of Riemannian manifolds with nonnegative Ricci curvature. BULLETIN OF THE LONDON MATHEMATICAL SOCIETY. ISSN 0024-6093 (In Press)

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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 .

Item Type: Article
Subjects: Q Science / természettudomány > QA Mathematics / matematika > QA73 Geometry / geometria
Q Science / természettudomány > QA Mathematics / matematika > QA74 Analysis / analízis
Depositing User: Dr. Alexandru Kristaly
Date Deposited: 07 Nov 2017 08:46
Last Modified: 07 Nov 2017 08:46
URI: http://real.mtak.hu/id/eprint/67173

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