Comparing the relative volume with a revolution manifold as a model
type: Item
instance of: scholarly article
title: Comparing the relative volume with a revolution manifold as a model
zbMATH Open document ID: 0794.53036
DOI: 10.1007/BF02760949
author: Ana Lluch
author: Vicente Miquel
published in: Israel Journal of Mathematics
publication date: 1994-09-01T00:00:00Z
review text: Given a pair \((P,M)\), where \(M\) is an \(n\)-dimensional connected compact Riemannian manifold and \(P\) is a connected compact hypersurface of \(M\), the relative volume of \((P,M)\) is the quotient \(\text{volume}(P)/ \text{volume} (M)\). In this paper the authors give a comparison theorem for the relative volume of such a pair, with some bounds on the Ricci curvature of \(M\) and the mean curvature of \(P\), with respect to that of a model pair \(({\mathcal P},{\mathcal M})\) where \({\mathcal M}\) is a revolution manifold and \({\mathcal P}\) a ``parallel'' of \({\mathcal M}\). From \textit{P. Berard} and \textit{S. Gallot} [Semin. Gaulaouic-Meyer- Schwartz, Equations Dériv. Partielles 1983-1984, Exp. No. 15, 34 p. (1984; Zbl 0542.53025)] and \textit{S. Gallot} [On the geometry of differentiable manifolds, Workshop, Rome/Italy 1986, Astérisque 163- 164, 31-91 (1988; Zbl 0674.53001)], a compact revolution manifold is a \(C^ \infty\) (compact) Riemannian manifold \((M^ \varphi; \langle\;,\;\rangle)\) of dimension \(n\) for which there are two points \({\mathcal N}\) and \({\mathcal S}\) of \(M^ \varphi\), a real number \(L>0\), a function \(\varphi: [0,2L]\to [0,+\infty)\) and a diffeomorphism \(f: (0,2L)\times S^{n-1}\to M-\{{\mathcal N},{\mathcal S}\}\) such that at each \((s,x)\in (0,2L)\times S^{n- 1}\) we have \(f^*\langle\;,\;\rangle= ds^ 2+ \varphi^ 2(s)\langle\;,\;\rangle_{S^{n-1}}\). A compact revolution manifold \(M^ \varphi\) is symmetric if the curvature \(\varphi(t)\) is symmetric with respect to \(L\) (i.e. \(\rho(2L-t)= \rho(t)\), or, equivalently, \(\rho(L-t)= \rho(L+t)\)). When \(M^ \varphi\) is symmetric it is said that \(M^ \varphi\) is lengthened if \(\rho\) is decreasing on the interval \([0,L]\), and \(M^ \varphi\) is called flattened if \(\rho\) is increasing on \([0,L]\). The main result: Theorem 1.6. Let \(M^ \varphi\) be a compact lengthened convex revolution manifold with sectional curvature \(\rho(t)\). Let \(R\in [0,L]\), and let \(…
Mathematics Subject Classification ID: 53C40
Mathematics Subject Classification ID: 53C20
zbMATH DE Number: 480487
zbMATH Keywords: relative volume
zbMATH Keywords: comparison theorem
zbMATH Keywords: Ricci curvature
zbMATH Keywords: mean curvature
reviewed by: Ioan Pop
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02760949
OpenAlex ID: W2054536568
description: scientific article; zbMATH DE number 480487