# “Small” maps from sphere to sphere

Start with a continuous map $f:S^{n+k} \rightarrow S^n$ (round unit spheres). The graph of $f$ lives in $S^{n+k}\times S^n$ and suppose it has a surface area (as a subspace of co-dimension $n$). Now vary $f$ within its homotopy class with the aim of reducing the surface area.

1) Is the lower bound on the surface area always obtained?

2) Are there examples of particular elements in a particular $\pi_{n+k}(S^n)$ with known non-trivial bounds on these surface areas? Non-trivial constructions of maps in a homotopy class that realize the lower bound?

3) One can also assign a surface area to a homotopy between two elements of a particular homotopy class. Individually minimizing the surface areas of each element in $\pi_{n+k}(S^n)$ very likely has a cost when it comes to the geometry of the homotopies that arise in the multiplication table of $\pi_{n+k}(S^n)$. As an alternative to the scenario above, one could pick representatives and homotopies to lower the total surface areas of all the homotopies associated to entries of the multiplication table of $\pi_{n+k}(S^n)$. Any non-trivial lower bounds available for this story (perhaps when the group has order 2)?

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You probably know: if the radius of the second factor goes to zero, then your $f$ becomes harmonic. There many explicit examples of harmonic maps $f:S^{n+k}\to S^n$. Some of them should be very symmetric and so they have good chance to have minimal graph... – Anton Petrunin Aug 5 '12 at 13:11
– Ryan Budney Aug 5 '12 at 19:18
I don't know anything about this particular problem, but in general it is very difficult to minimize volume in a homotopy class. (Minimizing in a homology class is well understood.) The obvious question would be whether the Hopf fibration is a minimizer in the sense you describe. There is some literature, starting with Gluck-Ziller, giving a variational characterization of the Hopf fibration in terms of the "volume" of the foliation. – Dan Lee Aug 15 '12 at 19:25