The Cheeger Constant of an Asymptotically Locally Hyperbolic Manifold and the Yamabe Type of Its Conformal Infinity

摘要

Let (M, g) be an (n+1)-dimensional asymptotically locally hyperbolic manifold with a conformal compactification whose conformal infinity is (partial derivative M, [gamma]). We will first observe that Ch(M, g) <= 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 derivative M, [gamma]) >= 0, where Y(partial derivative M, [gamma]) denotes the Yamabe invariant of the conformal infinity. This gives an answer to a question raised by Lee (Commun. Anal. Geom. 2:253-271, 1995).