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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)


Uniform bounds for isoperimetric problems

Authors: Jerrold Siegel and Frank Williams
Journal: Proc. Amer. Math. Soc. 107 (1989), 459-464
MSC: Primary 55P99; Secondary 58E05
MathSciNet review: 984815
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Abstract: In this paper we generalize our previous joint work with Allan Calder on the width of homotopies by considering an arbitrary finite polyhedral pair $ \left( {W,V} \right)$ rather than $ \left( {I,\left\{ {0,1} \right\}} \right)$. We show that given appropriate topological conditions on a Riemannian manifold $ M$, with respect to $ \left( {W.V} \right)$, there are bounds, $ {B_q}\left( {a,\left( {W,V} \right),M} \right)$, such that if $ F:K \times W \to M$ is a map with $ {\text{Lip}}\left( {F\left\vert {\left( {K \times V} \right)} \right.} \right) < a$, then $ F$ can be deformed $ {\text{rel}}\left( {K \times V} \right)$ to $ F'$ with $ {\text{Lip}}\left( {F'} \right) < {B_q}\left( {a,\left( {W,V} \right),M} \right) + \varepsilon $, where $ \varepsilon > 0$ is arbitrary and $ \dim \left( K \right) = q$.

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PII: S 0002-9939(1989)0984815-2
Article copyright: © Copyright 1989 American Mathematical Society

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