Smallest volume representatives of homology

Question

Given a Riemannian manifold, I have a notion of volume for each of my chains, so it makes sense to ask for a representative of a homology class with the smallest volume. Are there conditions for when such a representative exists?

I know it is not always possible: for example, the generator of $H^1$ of the punctured plane doesn't have a shortest representative. But perhaps if we required our space to be compact?

I'm especially interested in the case where the class has a submanifold representative. In that case, I would like to run a generalized mean curvature flow on the representative to get one with (locally) smallest volume.

Answer 1

Finding volume-minimizing representatives of homology classes is one of the major applications of geometric measure theory.

It's a theorem of Federer and Fleming that any nontrivial integral $k$-homology class of a smooth compact $n$-manifold (with $n>k$) can be represented by an integer-multiplicity rectifiable current of least volume. That's a bit far from a submanifold representation but there is a fair amount of work that establishes regularity in various cases; from memory, I think if $n<8$ and $k=n-1$ then there's a smooth volume minimizing hypersurface.

See e.g. the introduction to Generic regularity of homologically area minimizing hypersurfaces in eight dimensional manifolds for some references.