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Löwner’s conjecture, the Besicovitch barrel, and relative systolic geometry. (English. Russian original) Zbl 1039.53044

Sb. Math. 193, No. 4, 473-486 (2002); translation from Mat. Sb. 193, No. 4, 3-16 (2002).
The \(k\)-systole of a compact Riemannian manifold \((M,g)\), \(\text{sys}_k(M;g)\), is the infimum of the volumes of integer Lipschitz \(k\)-cycles of \((M,g)\) not homological to zero. Similarly one defines the stable \(k\)-systole of \((M,g)\), \(\text{stsys}_k (M; g)\), where the mass of \(k\)-cycles is used instead of the volume.
The stable intersystolic inequality says that for every closed Riemannian manifold \((M,g)\) and every \(1\leq k\leq m=\dim M\) one has \[ \text{vol}(M,g)\geq c(M)\cdot \text{stsys}_k (M;g)\cdot \text{stsys}_{m-k}(M;g), \] where the positive constant \(c(M)\) depends only on the topology of \(M\) (many authors contributed in this final result starting with Löwner and Besicovitch). On the other hand, in a previous paper the author proved that given a closed manifold \(M\) then for every \(1\leq k<m/2\) the infimum of \(\text{vol}(M,g)/(\text{sys}_k(M;g)\cdot\text{stsys}_{m-k} (M; g))\) over all Riemannian metrics \(g\) vanishes. This phenomenon is usually referred to as the intersystolic \((k,m- k)\)-freedom or softness, while the opposite case as the intersystolic rigidity.
The paper is devoted to the study of intersystolic rigidity and softness of compact manifolds with boundary. To begin with, the author proves the intersystolic rigidity of the pair \((\text{sys}_1, \text{sys}_m)\) for any cobordism between orientable \(m\)-manifolds with boundary. The proof relies on the Kronrod-Federer coarea formula. As a consequence, one obtains intersystolic rigidity for the pair \((\text{sys}_1(X,\partial X;g),\text{sys}_{m-1}(X;g))\), where \(X\) is an orientable \(m\)-manifold, whose boundary has more than 1 component. Furthermore, intersystolic \((1,m-1)\)-rigidity or softness depends on orientability properties of \(X\), \(\partial X\) and connectedness of the boundary \(\partial X\). A complete picture of this is obtained. As an example of the results it is mentioned that if \(m>2\) and \(X\) is orientable with connected boundary, or neither \(X\) nor \(\partial X\) are orientable, then the intersystolic softness takes place for the pair \((\text{sys}_1(X, \partial X),\text{stsys}_{m-1}(X))\).
The general case \((k,m -k)\) is less understood, nevertheless, for every \(1\leq k<m/2\) the intersystolic softness is proved for the pair \((\text{sys}_k(X), \text{stsys}_{m-k}(X,\partial X))\), where \(X\) is a compact \(m\)-manifold with boundary. The proof is a modification of arguments for closed manifolds.
Several open questions are formulated.

MSC:

53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
53C20 Global Riemannian geometry, including pinching
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