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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
DOI: https://doi.org/10.1090/S0002-9939-1989-0984815-2
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 ${\text {Lip}}\left ( {F\left | {\left ( {K \times V} \right )} \right .} \right ) < a$, 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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Article copyright: © Copyright 1989 American Mathematical Society