NDA@SZTAKI

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

Barbosa, Ezequiel

KristĂĄly, Alexandru

Cambridge University Press

2017

QA73 Geometry / geometria

QA74 Analysis / analĂ­zis

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
.

angol

Article

PeerReviewed

info:eu-repo/semantics/article

text

http://real.mtak.hu/67173/1/Barbosa-Kristaly-final.pdf

Barbosa, Ezequiel and KristĂĄly, 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)

http://real.mtak.hu/67173/

http://onlinelibrary.wiley.com/doi/10.1112/blms.12107/full