Isoperimetric inequalities, isometric actions and the higher Newman numbers
This paper mainly deals with isoperimetric inequalities for Riemannian submanifolds. Let M be a compact connected n-dimensional Riemannian manifold. If M contains a closed connected k-dimensional, \(2\leq k\leq n\), minimal submanifold, the authors define the k-th isoperimetric number of M, denoted by \(\tilde N_ k(M)\), as the infimum of the volumes of all such submanifolds. Then, a number of interesting estimates of \(\tilde N_ k(M)\) are given for both general Euclidean submanifolds and special minimal submanifolds, which appear to be new. From these the authors propose the following conjecture: \(\tilde N_ k(M)\geq Vol S^ k(i(M)/\pi)\), \(2\leq k\leq n\), where i(M) denotes the radius of injectivity of M and \(S^ k(r)\) denotes the standard k-sphere of radius r. They verify the conjecture for various special cases. The remainder of this paper turns to isometric actions and the higher Newman numbers \(N_ k(M)\), \(1\leq k\leq n\). The latter is introduced by the present authors. At the end of this paper they study Newman's theorem for compact connected Lie groups with invariant metrics.
scientific article; zbMATH DE number 4058351
scientific article