Exponential gaps in the length spectrum

TL;DR

This paper demonstrates that in certain small neighborhoods of negatively curved metrics, the length spectrum can be exponentially separated from below, contrasting with the generic case where this is false.

Contribution

It establishes the existence of negatively curved metrics with exponentially separated length spectra in small metric neighborhoods, a property previously known to be false generically.

Findings

Existence of negatively curved metrics with exponentially separated length spectrum

Separation property holds in small neighborhoods of the metric space

Contrasts with the generic case where such separation is false

Abstract

We present a separation property for the gaps in the length spectrum of a compact Riemannian manifold with negative curvature. In arbitrary small neighborhoods of the metric for some suitable topology, we show that there are negatively curved metrics with a length spectrum exponentially separated from below. This property was previously known to be false generically.