The isoperimetric profile of infinite genus surfaces
type: Item
instance of: scholarly article
title: The isoperimetric profile of infinite genus surfaces
zbMATH Open document ID: 1225.53036
author: Procopis Psaltis
published in: Geometriae Dedicata
publication date: 2010-11-23T00:00:00Z
review text: The isoperimetric profile of a surface at \(t\) is the infimum of the lengths of the boundaries of subsets whose area is \(t\). \textit{P. Papasoglou} [Trans. Am. Math. Soc. 361, No.~10, 5139--5162 (2009; Zbl 1183.53030)] showed that for a surface of finite genus, equipped with a Riemannian metric and a simplicial triangulation, the isoperimetric profile satisfies a gap condition with respect to \(t\), i.e., when it grows faster than \(\sqrt t\), it grows as a linear function. The main result of the paper under review is that the result does not hold for surfaces of infinite genus. The construction of such surfaces is based on the existence of infinite sequences of expander graphs with fixed regularity. More precisely, starting with such a sequence, a new sequence is constructed by adding terminal vertices to the graphs. Then, for each graph in the sequence, a surface is constructed by thickening the edges. That is done by replacing the edges by cylinders and gluing, with the appropriate triangulation. The choice of triangulation allows the gluing of the cylinders at the vertices. The construction of the surfaces with the required property starts by making holes in the hyperbolic plane and gluing the surfaces that were constructed above. That produces a surface of infinite genus whose isoperimetric profile does not have a gap with respect to \(t\). It can grow with any rate between \(\sqrt t\) and \(t\).
reviewed by: Stratos Prassidis
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 53C23
Mathematics Subject Classification ID: 20F65
zbMATH DE Number: 5818360
zbMATH Keywords: isoperimetric
zbMATH Keywords: profile
zbMATH Keywords: inequality
zbMATH Keywords: surfaces
zbMATH Keywords: simplicial triangulation
zbMATH Keywords: isoperimetric profile
zbMATH Keywords: expander graphs
zbMATH Keywords: gluing
MaRDI profile type: Publication
full work available at URL: s10711-010-9468-9
OpenAlex ID: W2069450750
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DOI: 10.1007/S10711-010-9468-9
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description: scientific article; zbMATH DE number 5818360