Orateur
Description
We consider maps between spheres $\mathbb S^n$ to $\mathbb S^\ell$ that minimize the Sobolev-space energy $W^{s,n/s}$ for some $s \in (0,1)$ in a given homotopy class. The basic question is: in which homotopy class does a minimizer exist? This is a nontrivial question since the energy under consideration is conformally invariant and bubbles can form. Sacks-Uhlenbeck theory tells us that minimizers exist in a set of homotopy classes that generates the whole homotopy group $\pi_{n}(\mathbb S^\ell)$. Explicit examples are known if $n/s = 2$ or $s=1$.
In this talk we are interested in the stability of the above question in dependence of s. We can show that as s varies locally, the set of homotopy classes in which minimizers exist can be chosen stable. We also discuss that the minimum $W^{s,n/s}$-energy in homotopy classes is
continuously depending on $s$.
Joint work with A. Schikorra