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

TL;DR

This paper investigates second-order Sobolev inequalities on certain Riemannian manifolds with nonnegative Ricci curvature, establishing volume non-collapsing and rigidity results that characterize Euclidean space.

Contribution

It proves that near-optimal second-order Sobolev inequalities imply volume non-collapsing and rigidity, characterizing Euclidean space among manifolds with nonnegative Ricci curvature.

Findings

Support for second-order Sobolev inequality implies volume non-collapsing.

Equality case of the inequality characterizes Euclidean space.

Rigidity results relate higher-order homotopy groups to geometric structure.

Abstract

Let $(M,g)$ be an $n-$dimensional complete open Riemannian manifold with nonnegative Ricci curvature verifying $\rho\Delta_g \rho \geq n- 5\geq 0$, 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 $\mathbb R^n$.