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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    $\begingroup$ 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... $\endgroup$ Commented Aug 5, 2012 at 13:11
  • $\begingroup$ Related mathoverflow.net/questions/36108/geometry-of-null-homotopies $\endgroup$ Commented Aug 5, 2012 at 19:18
  • $\begingroup$ 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. $\endgroup$
    – Dan Lee
    Commented Aug 15, 2012 at 19:25

1 Answer 1


Here's an example to show that the infimum is not always attained:

Consider the standard Hopf map $\pi:S^3\to S^2$, which is not null-homotopic, of course, so it follows that the area of the graph in $S^3\times S^2$ of any differentiable map $f:S^3\to S^2$ that is homotopic to $\pi$ is strictly greater than the area of the graph of a constant map, i.e., of $S^3\times\{c\}\subset S^3\times S^2$, where $c\in S^2$ is fixed, namely the volume of $S^3$.

Now, computation shows that, if $\phi_t:S^3\to S^3$ for $0<t\le 1$ is the $1$-parameter family of conformal maps that fix a point $p_0\in S^3$ and its antipode $-p_0\in S^3$, that is the identity for $t=1$ and expands at $p_0$ by a factor of $1/t$ for $t\in (0,1]$, then the area of the graph of $f_t = \pi\circ\phi_t$ in $S^3\times S^2$ converges to the area of $S^3\times\{\pi(p_0)\}\subset S^3\times S^2$. Indeed, the graph of $f_t$ itself converges to $S^3\times\{\pi(p_0)\}$ outside of an open neighborhood of $\{-p_0\}\times S^2$.

Consequently, the infimum of the areas of the graphs of maps $f:S^3\to S^2$ in the homotopy class of $\pi$ is the volume of $S^3$, but this infimum cannot be obtained by anything in the homotopy class.

Note: The graph of $\pi:S^3\to S^2$ in $S^3\times S^2$ is a minimal submanifold, of course, but it is not minimizing, i.e., it is unstable.

Addendum: Actually, it turns out that this computation is just a special case of a much more general phenomenon. Let $\phi_t:S^{n+k}\to S^{n+k}$ for $0<t\le 1$ be the conformal dilation contracting to a fixed point $p_0\in S^{n+k}$ whose differential at $p_0$ is $t$ times the identity on $T_{p_0}S^{n+k}$. (In particular, $\phi_1$ is the identity map.) One then has the following result:

Proposition: If $f:S^{n+k}\to S^n$ is any $C^1$-map and $k>0$, then the areas of the graphs of the homotopic family $f_t = f\circ\phi_t$ in $S^{n+k}\times S^n$ converge to the volume of $S^{n+k}$ as $t$ goes to $0$.

In particular, the graph area infimum in any homotopy class in $\pi_{n+k}(S^n)$ is the same when $k>0$, namely the volume of $S^{n+k}$, and this can actually be attained only for the trivial homotopy class.


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