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.



Smallest volume representatives of homology
Finding volume-minimizing representatives of homology classes is one of the major applications of [geometric measure theory](https://en.wikipedia.org/wiki/Geometric_measure_theory). 


It's a theorem of [Federer and Fleming](https://www.jstor.org/stable/1970227) 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](https://en.wikipedia.org/wiki/Current_(mathematics)) 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](http://intlpress.com/site/pub/files/_fulltext/journals/cag/1993/0001/0002/CAG-1993-0001-0002-a002.pdf) for some references. 


