Rigidity and $L^2$ cohomology of hyperbolic manifolds

TL;DR

This paper investigates the rigidity phenomena of hyperbolic manifolds at the critical threshold where certain cohomology and harmonic form spaces vanish, extending understanding of their geometric and topological properties.

Contribution

It establishes new rigidity results for hyperbolic manifolds precisely at the critical exponent where previous vanishing theorems hold.

Findings

Rigidity results at the critical exponent for hyperbolic manifolds.

Extension of vanishing theorems to borderline cases.

Insights into the structure of $L^2$ cohomology in hyperbolic geometry.

Abstract

When $X=\Gamma\backslash \H^n$ is a real hyperbolic manifold, it is already known that if the critical exponent is small enough then some cohomology spaces and some spaces of $L^2$ harmonic forms vanish. In this paper, we show rigidity results in the borderline case of these vanishing results.