Isoperimetric inequalities, isometric actions and the higher Newman numbers
type: Item
instance of: scholarly article
title: Isoperimetric inequalities, isometric actions and the higher Newman numbers
zbMATH Open document ID: 0648.53032
DOI: 10.1007/BF00183026
author: Hsu-Tung Ku
author: L. N. Mann
author: Mei-chin Ku
published in: Geometriae Dedicata
publication date: 1988-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 53C40
zbMATH DE Number: 4058351
zbMATH Keywords: isoperimetric inequalities
zbMATH Keywords: minimal submanifold
zbMATH Keywords: isoperimetric number
zbMATH Keywords: volumes
zbMATH Keywords: isometric actions
zbMATH Keywords: Newman numbers
zbMATH Keywords: Lie groups
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf00183026
OpenAlex ID: W2059327612
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description: scientific article; zbMATH DE number 4058351