The purpose of the special course is to give a detailed presentation of the theory of quasiconformal mappings in n-dimensional Euclidean space at the modern level. The presentation is carried out from a geometric point of view, which is based on the use of conformal invariants 鈥� the modulus of the family of curves and the capacitance of a capacitor.
First, we study conformal mappings in space, as well as the foundations of hyperbolic geometry, including its isometries (along with accompanying matrix groups). This quickly leads to the famous Liouville theorem for smooth maps, which is proved in the spirit of Nevanlinna.
Then the concepts of the modulus of the family of curves and the capacitance of a capacitor are introduced, their initial properties are established. After that, we are talking about deeper properties 鈥� continuity, existence and uniqueness of extremal functions, the structure of sets of zero capacity, and behavior under symmetrization. Thanks to this, an effective toolkit appears that allows one to derive the analytical properties of quasiconformal mappings, including the properties of compactness and normality of their families.
The problem of domain mapping is touched upon: an attempt is made to find a multidimensional analogue of the Riemann theorem for quasiconformal mappings. Classical geometric arguments that prevent existence are indicated, but at the same time "positive" results are reported: a proof of a quasiconformal analogue of Schoenflis's theorem is given, and then V盲is盲l盲's theorem is established for cylindrical domains.
An important theorem of Tukia and V盲is盲l盲 on the continuation is highlighted, and the technique developed by Sullivan is used in the proof.
The course ends with Mostov's rigidity theorem, which is one of the most significant applications of the multidimensional theory of quasiconformal mappings. The proof is constructed in such a way as to show the connection of this theory with hyperbolic geometry and modern aspects of geometric group theory. In particular, the concept of quasi-isometry is discussed in detail and isomorphisms of hyperbolic groups are considered.
