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Isoperimetric inequalities of Euclidean type in metric spaces. (English) Zbl 1084.53037

The author aims to answer a question raised in the so-called isoperimetric inequalities of Euclidean type problems for metric spaces. More generally, a topological space \(X\) is said to admit an isoperimetric inequality of Euclidean type in dimension \(k\in{\mathbb N}\), if it is possible to associate to \(X\) a sequence \({\mathbf I}_{k+1}(X)\mathop{\to}\limits^{\partial}{\mathbf I}_{k}(X)\mathop{\to}\limits^{\partial}{\mathbf I}_{k-1}(X)\), and a “volume function” \({\mathbf M}:{\mathbf I}_{s}(X)\to{\mathbb R}\), \(s=k-1,k,k+1\), such that \({\mathbf M}(S)\leq D{\mathbf M}(T)^{(k+1)/K}\), for any \(T\in \ker (\partial)\subset {\mathbf I}_k(X)\), and for some \(S\in{\mathbf I}_{k+1}(X)\), with \(\partial S=T\). Here \(D\) is a constant depending only on \(X\) and \(k\). An example of such a space is given by Federer and Fleming with \(X={\mathbb R}^n\) and \({\mathbf I}_{k}(X)\) the space of \(k\)-dimensional integral currents, \(k=\{1,\dots, n\}\), and other ones by Gromov, where \(X\) is a finite dimensional normed space, or a suitable complete Riemannian manifold, and \({\mathbf I}_{k}(X)\) the space of Lipschitz \(k\)-chains.
The present work by Wenger is nearer to a more recent work by Ambrosio and Kirchheim, that considered \(X\) as a general metric space and \({\mathbf I}_{k}(X)\) the space of \(k\)-dimensional metric integral currents. The main result is that a complete metric space \((X,d)\) such that \({\mathbf I}_{k}(X)\) satisfies a “cone-type inequality”, admits an isoperimetric inequality of Euclidean type in dimension \(k\), if it is so at dimension \(k-1\). As a by-product he solves also some generalized Plateau problems. In particular he proves that a complete metric space \((X,d)\) that admits an isoperimetric inequality of Euclidean type at dimension \(k\), (\({\mathbf I}_{k}(X)\)), for any \(T\in\ker(\partial)\subset{\mathbf I}_{k}(X)\), with compact support, there exists \(S\in {\mathbf I}_{k+1}((X)_\omega)\) such that \(\partial S=i_*(T)\) and \({\mathbf M}(S)\leq\text{ inf}\{{\mathbf M}(S'): S'\in{\mathbf I}_{k+1}(X),\partial S'=T\}\). Here \(\omega\) is a non-principal ultra-filter on \({\mathbb N}\), and \((X)_\omega\) is the ultra-completion of \(X\), i.e., the set of equivalence classes of bounded sequences \((x_n)_{n\in{\mathbb N}}\) in \(X\), together with the metric given by \(d_\omega((x_n),(x'_n))\equiv\text{ lim}_\omega d(x_n,x'_n)\). (The space \(X\) can be isometrically embedded into \((X)_\omega\) by means of the map \(i:X\to(X)_\omega\), given by \(i(x)=(x_n)\), with \(x_n=x\), for any \(n\). Then \(i_*\) denotes the induced mapping \(i_*:{\mathbf I}_{k}(X)\to {\mathbf I}_{k}((X)_\omega)\).)
Remark. The results given in this paper are obtained essentially by an elegant use of functional analysis techniques. Let us emphasize also, that the main issues of this paper are solutions of Plateau problems. With respect to this we can underline that by using a completely different approach, i.e., by working in the framework of the geometric theory of PDE’s, and by using variational techniques besides integral bordism groups for PDE’s, it is possible to solve generalized Plateau problems, constrained by PDE’s. (See some recent papers by the reviewer of this paper.)

MSC:

53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
54E50 Complete metric spaces
53C65 Integral geometry
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