Isoperimetric inequalities,isometric actions and the higher Newman numbers

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)/), 1 k n, where i(M) denotes the radius of injectivity of M and S^k(i(M)/) denotes the standard k-sphere of radius i(M)/. 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.