Isoperimetric inequalities for multivarifolds (Q1819726)

The present paper is to announce an application of the multivarifold theory to isoperimetric inequalities for parametrized films. The main result of this paper is following: Let \(A\subset {\mathbb{R}}^ n\) be a subset contractible in itself, \(C\subset A\) a compact subset of \({\mathbb{R}}^ n\), U a neighbourhood of A in \({\mathbb{R}}^ n\) and \(\alpha: U\to A\) a retraction, satisfying a Lipschitz condition with coefficient \(\xi\) on the subset \(\{x|_{\rho}(x,C)\leq a\}\), where a is some positive number. If \(V\in J{\mathcal C}^ 0_ k({\mathbb{R}}^ n)\), spt \(V\subset C\), \(\partial_ TV=0\), \(2n^{2k}C^ n_ kM_ k(V)\leq a^ k\) and \(V=T(h_ 0)V_ 0\) where \(V_ 0\in {\mathcal R}_ k({\mathfrak H})\), \({\mathfrak H}\) is a Riemannian manifold, which can be the boundary of some Riemannian manifold, and \(h_ 0: {\mathfrak H}\to {\mathbb{R}}^ n\) is a Lipschitzian mapping, then there exists \(W\in J{\mathcal C}^ 0_{k+1}({\mathbb{R}}^ n)\), such that Spt \(W\subset A\), \(\partial_ TW=V\) and \[ [M_{k+1}(W)]^{k/k+1}\leq 2n^{(k+2)/(k+1)}C^ n_ k\xi^ kM_ k(V). \]