Roughly isometric minimal immersions into Riemannian manifolds

A given metric (length-) space $X$ (whether compact or not) is roughly isometric to any one of its Kanai graphs $G$, which in turn can be {\em{geometrized}} by considering each edge of $G$ as a 1-dimensional manifold with an associated metric $g$ giving the 'correct' length of the edge. In this talk we will mainly be concerned with {\em{minimal}} isometric immersions of such geometrized approximations $(G, g)$ of $X$ into Riemannian manifolds $N$ with bounded curvature. When such an immersion exists, we will call it an $X$-web in $N$. Such webs admit a natural 'geometric' extension of the intrinsic combinatorial discrete Laplacian, and we will show that they share several analytic and geometric properties with their smooth (minimal submanifold) counterparts in $N$. The intrinsic properties thus obtained may hence serve as roughly invariant descriptors for the original metric space $X$.