Geometric rigidity estimates for isometric and conformal maps from S^(n-1) to R^n
In this thesis we study qualitative as well as quantitative stability aspects of isometric and conformal maps from S^(n-1) to R^n, when n is greater or equal to 2 or 3 respectively. Starting from the classical theorem of Liouville, according to which the isometry group of S^(n-1) is the group of its rigid motions and the conformal group of S^(n-1) is the one of its Möbius transformations, we obtain stability results for these classes of mappings among maps from S^(n-1) to R^n in terms of appropriately defined deficits. Unlike classical geometric rigidity results for maps defined on domains of R^n and mapping into 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 S^(n-1) in a generalized sense. The introduction of the latter is motivated by the classical Euclidean isoperimetric inequality.
rigidity, stability, isoperimetric inequality
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