Isoperimetric inequalities, isometric actions and the higher Newman numbers

Abstract

M will be a compact connected n-dimensional Riemannian manifold. If M contains a closed connected k-dimensional, 2 ≤ k < n, minimal immersed submanifold of M, we define the kth isoperimetric number of M, Ñ _k (M), as the infimum of the volumes of all such submanifolds. We obtain a number of interesting estimates for Ñ _k (M), for both general and special manifolds, which appear to be new.

Next we turn to isometric actions and a 1931 theorem of M. H. A. Newman involving the size of orbits of group actions on manifolds. We introduce the higher Newman numbers N _k (M), 1 ≤ k ≤ n. Roughly speaking, if M admits isometric actions of compact connected Lie groups with k-dimensional principal orbits, N _k (M) is defined as the infimum over all such actions of the maximum ‘volume’ of all maximal dimensional orbits. We observe that N _k (M) ≥ Ñ _k (M), 2 ≤ k < n, provided N _k (M) is defined; hence our prior estimates for the isoperimetric numbers of M apply directly to the higher Newman numbers.

As a ‘best possible’ candidate we conjecture that N _k (M) ≥ vol S ^k(i(M)/π), 1 ≤ k ≤ n, where i(M) denotes the radius of injectivity of M and S ^k(i(M)/π) denotes the standard k-sphere of radius i(M)/π. We verify the conjecture for various special cases. We conclude the paper by studying Newman's theorem for compact connected Lie groups with invariant metrics and obtaining a lower bound for the size of small subgroups.