Calculus of Variations and Geometric Measure Theory

K. Zemas

Geometric rigidity estimates for isometric and conformal maps from $\mathbb{S}^{n−1}$ to $\mathbb{R}^n$

created by zemas on 13 Jan 2024


Ph.D. Thesis

Inserted: 13 jan 2024
Last Updated: 13 jan 2024

Year: 2020


In this thesis we study qualitative as well as quantitative stability aspects of isometric and conformal maps from $\mathbb{S}^{n−1}$ to $\mathbb{R}^n$, when $n \geq 2$ and $n \geq 3$ respectively. Starting from the classical theorem of Liouville, according to which the isometry group of $\mathbb{S}^{n−1}$ is the group of its rigid motions and the conformal group of $\mathbb{S}^{n−1}$ is the one of its Möbius transformations, we obtain stability results for these classes of mappings among maps from $\mathbb{S}^{n−1}$ to $\mathbb{R}^n$ in terms of appropriately defined deficits. Unlike classical geometric rigidity results for maps defined on domains of $\mathbb{R}^n$ and mapping into $\mathbb{R}^n$, not only an isometric\conformal deficit is necessary in this more flexible setting, but also a deficit measuring how much the maps in consideration distort $\mathbb{S}^{n−1}$ in a generalized sense. The introduction of the latter is motivated by the classical Euclidean isoperimetric inequality.