The fractal nature of Riem/Diff. I.
Let \(\text{Met}(M)=\text{Riem}(M)/\text{Diff}(M)\) be the space of Riemannian metrics on a smooth manifold \(M\), up to reparametrization. In this paper, the authors study certain variational problems on the space \(\text{Met}(M)\), especially they expose the richness of this space and consider special aspects of the geometry of the space of isometry classes of Riemannian metrics, with a 2-sided curvature bound, on a fixed compact smooth manifold of dimension at least five. Using a number of tools from logic/computer science, differential geometry, and topology, the authors study the diameter functional \(\text{diam}(M)\) and its critical points, as well as their distribution (density) within the space and the structure of their neighborhoods. For example, it is shown that for any sufficiently large \(x\) the set of Riemannian structures satisfying \(\text{diam}(M) \leq x\) is not connected. Moreover, the number of components grows at least exponentially, and in order to bridge the gap between the components one must pass through structures such that the diameter cannot be bounded by any effectively computable function of \(x\). The set of local minima of the diameter is infinite. Furthermore, the authors show that for every closed smooth manifold \(M\) of dimension \(n> 4\) there are infinitely many local minima of the diameter functional \(\text{diam}(M)\) on the subset \(\text{AI}(M)\) of \(\text{Met}(M)\) consisting of isometry classes of Riemannian metrics with curvature bounded in absolute value by 1.
scientific article; zbMATH DE number 2027203