Speaker
Description
We present quantitative stability aspects of the class of Möbius transformations of $\mathbb{S}^{n-1}$ among maps in the critical Sobolev space $W^{1,n-1}$. The case of $\mathbb{S}^{n-1}$- and general $\mathbb{R}^n$-valued maps will be addressed. In the latter, more flexible setting, unlike similar in flavour results for maps defined on domains of $\mathbb{R}^n$, not only a conformal deficit is necessary, but also a deficit measuring the distortion of $\mathbb{S}^{n-1}$ under the maps in consideration which is introduced as an associated isoperimetric deficit.
Next, we consider conformal transformations of $\mathbb{S}^2$ of arbitrary degree, those being entire solutions $\omega\in{\dot W}^{1,2}(\mathbb{R}^2;\mathbb{R}^3)$ of the $H$-system
\begin{align}
\Delta\omega=2\omega_x\wedge\omega_y\, \ \text{in } \mathbb{R}^2\,,
\end{align}
usually referred to as bubbles.
Contrary to conjectures raised in the literature, we find that bubbles with degree at least three can be degenerate: the linearized $H$-system around a bubble can admit solutions that are not tangent to the smooth family of bubbles. We then give a characterization of the degenerate bubbles, and present optimal stability estimates in a neighborhood of any single bubble in the form of Lojasiewicz-type inequalities.
The talk will be based on previous works in collaboration with Stephan Luckhaus, Jonas Hirsch, and more recent (also ongoing) ones with Andre Guerra and Xavier Lamy.
