On the Isoperimetric Constant of Symmetric Spaces of Noncompact Type

From this result one can easily deduce I (H) = n 1. For a detailed discussion of Cheeger’s constant and related results, one can consult [Cha, Chapter 6]. In general, it is very di¢ cult to know if the isoperimetric constant is positive or not and it is almost impossible to compute it explicitly if it is known to be positive. In this short note, we prove that the isoperimetric constant is positive for all symmetric spaces of noncompact type and compute it explicitly. Let f M; g be a Cartan-Hadamard manifold (i.e. complete, simply-connected with nonpositive curvature) and Sf M its unit tangent bundle. For any p 2 M and u 2 Spf M we have a nonnegative symmetric operator Ru : Tpf M ! Tpf M de…ned by