Filling invariants of closed curves: isoperimetric and isodiametric functions etc.; area filling in nilpotent and solvable groups; filling length, filling radius and Morse landscape; filling in k–dimensional cycles for k ≥ 2; filling on the large scale and volume distortion of subspaces.
Given a circle S in a Riemannian manifold X (this S may have double points) there is a variety of invariants characterizing the optimal (i.e. minimal) size of surfaces in X filling in S i.e. having S for their boundary. Here are some of them.
I. Filling area, denoted Fill Area S. This is the infimal area of compact surfaces (or better to say of 2–chains D in X, filling in S. (Here one should specify the implied coefficient field. Usually we speak of chains with integral coefficients.)
II. Filling area of genus g, denoted Fillg Area S, where the infimum of area is taken over the surfaces of genus g filling in S (If we insist on Z–coefficient these surfaces must be orient able.) Notice, that here and in future “surface” means a surface which is Lipschitz mapped into X, where the map is by no means required to be one–to–one.
A particularly important special case is that of Fill0 Area S where the implied surface is a disk.
III. Filling diameter Fill0 Diam S. This is the infimal diameter of disks D with some Riemannian metrics for which there are 1–Lipschitz (i.e. contracting) maps D → X sending the boundary onto S with degree 1.