Minimal volume with length bounded below (Q558082)

Let \(M\) be a compact Riemannian manifold, \(\Gamma\) a family of compact curves in \(M\) endowed with the uniform convergence topology, \(\rho:\Gamma\to\mathbb R\) a bounded continuous function. The following problem is addressed in the paper: Minimize the volume of all Riemannian metrics or the volume of metrics in a conformal class such that the length of any curve \(\gamma\in\Gamma\) is not smaller than \(\rho(\gamma)\). A number of well known problems fits in this setup. For example, if \(\Gamma\) consists of all closed and noncontractible curves in a closed manifold, then we have the isosystolic problem studied by many authors. If \(\gamma(0)\), \(\gamma(1)\in\partial M\) for every \(\gamma:[0,1]\to M\in\Gamma\), and \(\rho(\gamma)=\text{ dist}(\gamma(0),\gamma(1))\), then we have the boundary distance problem. The author proves the existence and uniqueness of a weak \(L^p\)-solution in every conformal class and for each \(1<p<\infty\). The proof of the existence is based on strong convexity of the space of conformal factors, while the uniqueness is known earlier for constant \(\rho\). Necessary and sufficient equivalent conditions for minimality within a conformal class are given (theorem~6), which include the Bavard and Beurling conditions. This result generalizes and extends the main result from \textit{C.~Bavard} [Comment. Math. Helv. 67, 146--166 (1992; Zbl 0783.53041)]. It turns out that a local minimum for the volume of Riemannian metrics is the minimum within its conformal class (i.e., within a conformal class, there is no local minimum different from the minimum). Using this, necessary conditions for a local minimum for the Riemannian volume are derived from theorem~6. Finally, another necessary condition for local minimality (theorem~10) is given in terms of density of vectors tangent to curves in \(\Gamma\) with minimal possible length. Applications to the boundary distance problem are presented.