Pincement du plan hyperbolique complexe

TL;DR

This paper investigates the $L^p$-cohomology of rank one symmetric spaces, demonstrating its Hausdorff property under certain conditions and establishing geometric rigidity results for complex hyperbolic planes.

Contribution

It proves the Hausdorff property of $L^p$-cohomology beyond curvature pinching ranges and shows that complex hyperbolic plane cannot be quasiisometric to certain pinched manifolds.

Findings

$L^p$-cohomology is Hausdorff for specific $p$ values.

No quasiisometry between complex hyperbolic plane and strictly -1/4-pinched manifolds.

Method limitations prevent extension to other rank one symmetric spaces.

Abstract

$L^p$-cohomology of rank one symmetric spaces of noncompact type is shown to be Hausdorff for values of $p$ where this does not follow from curvature pinching. Using the multiplicative structure on $L^p$-cohomology, it is shown that no simply connected Riemannian manifold with strictly -1/4-pinched sectional curvature can be quasiisometric to complex hyperbolic plane. Unfortunately, the method does not extend to other rank one symmetric spaces.