Adam Grzela - Minimizing energy in rational homotopy classes

In 1998 T. Rivière proved that there exist infinitely many homotopy classes of $\pi_3(\mathbb{S}^2)$ having a minimizing 3-harmonic map. This result is especially surprising taking into account that in $\pi_n(\mathbb{S}^n)$ there are only three homotopy classes (corresponding to the degrees $\{-1,0,1\}$) in which a minimizer exists.

The primary reason for this unusual distinction is analytical in nature — the appropriate energy scales sub-linearly with the Hopf invariant classifying $\pi_3(\mathbb{S}^2)$. This is an important special case of a broader phenomenon connected to quantitative estimates for elements of the so-called rational homotopy groups.

We explore the notion of optimal-exponent estimates in rational homotopy groups and their relation to variational problems for both local and non-local type energies.