Uniform bounds for isoperimetric problems

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 $\dim \left ( K \right ) = q$.