The Cheeger constant of an asymptotically locally hyperbolic manifold and the Yamabe type of its conformal infinity

TL;DR

This paper investigates the Cheeger constant of asymptotically locally hyperbolic manifolds and establishes a link between its value and the Yamabe invariant of the conformal infinity, providing insights into geometric analysis and conformal geometry.

Contribution

It proves that the Cheeger constant equals n if and only if the Yamabe invariant of the conformal infinity is non-negative under certain curvature conditions.

Findings

Cheeger constant is bounded above by n.

Cheeger constant equals n iff Yamabe invariant is non-negative.

Results answer a question posed by J. Lee.

Abstract

Let (M, g) be an (n + 1)-dimensional asymptotically locally hyperbolic (ALH) manifold with a conformal compactification whose conformal infinity is ($\partial$M, [$\gamma$]). We will first observe that Ch(M, g) $\le$ n, where Ch(M, g) is the Cheeger constant of M. We then prove that, if the Ricci curvature of M is bounded from below by --n and its scalar curvature approaches --n(n+1) fast enough at infinity, then Ch(M, g) = n if and only Y($\partial$M, [$\gamma$]) $\ge$ 0, where Y($\partial$M, [$\gamma$]) denotes the Yamabe invariant of the conformal infinity. This gives an answer to a question raised by J. Lee [L].