Asymptotic cones, bi-Lipschitz ultraflats, and the geometric rank of geodesics

TL;DR

This paper investigates the relationship between geodesics in non-positively curved manifolds and their asymptotic cones, revealing new geometric constraints and providing a novel proof of Gromov's rigidity theorem for higher rank spaces.

Contribution

It establishes a link between bi-Lipschitz flats in asymptotic cones and Jacobi fields on original geodesics, leading to new rigidity results and constraints on quasi-isometries.

Findings

Bi-Lipschitz flats in asymptotic cones imply existence of orthogonal, parallel Jacobi fields.

Constraints on quasi-isometries between NPCR manifolds derived from asymptotic cone analysis.

New proof of Gromov's rigidity theorem for higher rank locally symmetric spaces.

Abstract

Given a geodesic inside a simply-connected, complete, non-positively curved Riemannian (NPCR) manifold M, we get an associated geodesic inside the asymptotic cone Cone(M). Under mild hypotheses, we show that if the latter is contained inside a bi-Lipschitz flat, then the original geodesic supports a non-trivial, orthogonal, parallel Jacobi field. As applications we obtain (1) constraints on the behavior of quasi-isometries between complete, simply connected, NPCR manifolds, and (2) constraints on the NPCR metrics supported by certain manifolds, and (3) a correspondence between metric splittings of complete, simply connected NPCR manifolds, and metric splittings of its asymptotic cones. Furthermore, combining our results with the Ballmann-Burns-Spatzier rigidity theorem and the classic Mostow rigidity, we also obtain (4) a new proof of Gromov's rigidity theorem for higher rank locally symmetric spaces.