A well known theorem of Cartan states that every free homotopy class of closed paths in a compact Riemannian manifold is represented by a closed geodesic (theorem 2.2 of Do Carmo, chapter 12, for example). This means that every closed path is homotopic to a closed geodesic (through a homotopy that need not fix base points).
I am wondering if there is a higher dimensional generalization of this result. Let us define a free n-homotopy class of $M$ to be a set $L_n$ of continuous maps $S^n \to M$ such that if $f \in L_n$ and $g: S^n \to M$ is any continuous map which is homotopic to $f$ (no base points required) then $g \in L_n$.
Question 1: Can one use the geometry of $M$ to intelligently single out a preferred class in any free n-homotopy class?
The naive guess is that every free n-homotopy class has an area minimizing representative, but my intuition tells me that this is either not true or really hard to prove. This is because the proof in the 1-dimensional case develops the argument from certain continuity properties of arclength that tend to either fail or require tremendous care when generalized to area. But perhaps this idea abstracts the wrong property of geodesics; maybe one should instead look for representatives which extremize some intelligently chosen functional instead. I'm hoping someone has already thought about this and come up with a good answer.
Question 2: Can anything more be said in the presence of negative curvature?
In the 1-dimensional case, I believe that the closed geodesic guaranteed by Cartan is in fact unique if $M$ is compact with strictly negative curvature. If there is an affirmative answer to question 1, I would be curious to know if there is a corresponding uniqueness statement in negative curvature. And if question 1 doesn't seem to have a nice answer in general, maybe negative curvature helps.
Thanks in advance!