Quasiisometries between negatively curved Hadamard manifolds

TL;DR

This paper proves that any quasiisometry between universal covers of certain negatively curved compact manifolds is close to a bilipschitz homeomorphism, revealing a strong geometric rigidity in these spaces.

Contribution

It establishes a rigidity result showing quasiisometries are approximated by bilipschitz homeomorphisms in negatively curved Hadamard manifolds, extending understanding of their geometric structure.

Findings

Quasiisometries are within finite distance of bilipschitz homeomorphisms.

Rigidity holds for manifolds with negative sectional curvature and dimension not equal to 4.

Results contribute to the understanding of geometric mappings in negatively curved spaces.

Abstract

Let X, Y be the universal covers of two compact Riemannian manifolds (with dimension not equal to 4) with negative sectional curvature. Then every quasiisometry between them lies at a finite distance from a bilipschitz homeomorphism.