Quasiconformal maps on Carnot groups
This survey is a break in the mini-course. It contains results of a metric nature (quasiconformality belongs to metric geometry).
1. Inspiring classical results
Theorem 1 (Liouville 1850) Every smooth conformal map of an open subset of , , is the restriction of a Möbius transformation.
A Möbius transformation belongs to the group generated by translations, rotations and inversions. This is a finite dimensional Lie group. So Liouville’s theorem is a rigidity result : conformal mappings are scarce in higher dimension. In contrast, the Riemann mapping theorem shows that conformal mappings are flexible in 2 dimensions.
Theorem 2 (Riemann 1850, Poincaré, Koebe 1907) Every proper simply connected open subset of is conformally equivalent to the disk.
To recover a bit of flexibility, quasiconformal mappings have been introduced.
Definition 3 (Grötzsch 1928) A homeomorphism between Euclidean domains is -quasiconformal if for any point , the -ball at is taken to a domain containing an ball and contained in an ball, with .
Theorem 4 (Ahlfors, Väisälä, Gehring) Liouville’s theorem extends to -quasiconformal mappings.
In other words, the differentiability assumption in Liouville’s theorem can be removed.
One may wonder wether quasiconformal mappings are somewhat differentiable anyway. The answer is yes. They are absolutely continuous on lines, they admit distributional derivatives belonging to for some , they are almost everywhere differentiable. But not much more in general.
2. SubRiemannian setting
Grötzsch’ definition is purely metric so it makes sense for subRiemannian manifolds. Interest in the subject in subRiemannian geometry started with the following observation.
Theorem 5 (Koranyi, Reimann 1985) The Heisenberg group admits non Möbius quasiconformal mappings.
In fact, they construct an infinite dimensional space of smooth vectorfields whose flows consist of quasiconformal mappings. They check that smooth contact transformations are locally quasiconformal. They use Paulette Libermann’s parametrization of infinitesimal contact transformations by functions
Definition 6 Let be a Carnot group. Say that is rigid if local contact mappings form a finite dimensional Lie group.
Example 1 Euclidean spaces and Heisenberg groups are not rigid.
Later, people have accumulated rigidity results.
Theorem 7 (Yamaguchi, Pansu, Cowling, De ?, Koranyi, Reimann, Ricci, Ottazzi, Warhurst,…) -type groups, Iwasawa unipotent factors, free nilpotent Lie groups, Hessenberg manifolds… are rigid.
Until recently, only jet spaces had been shown to be non rigid (Warhurst 2005).
Theorem 8 (Ottazzi 2008) Let be a Carnot group. Assume that there exists a horizontal vector such that has rank . Then is non rigid.
Example 2 Engel’s group is non rigid.
2.2. Towards a characterization of rigid Carnot groups ?
The converse of Theorem 7 is not far from being true. First, go to the complexified Lie algebra . Second, there remains a regularity issue to solve.
Theorem 9 (Ottazzi, Warhurst) Let be a Carnot group. Assume that there exists a complex horizontal vector such that has rank . Then is non rigid.
Conversely, assume that there exist no complex horizontal vectors such that has rank . Then quasiconformal mappings of class form a finite dimensional Lie group.
Theorem 9 is related to a classical fact for -structures. Indeed, there exists a complex horizontal vector such that has rank if and only if has a rank graded derivation which vanishes on commutators.
2.3. Generalized Liouville theorem
Theorem 10 (Ottazzi, Warhurst) Let be a Carnot group different from . Then -quasiconformal maps on form a finite dimensional Lie group.
We rely on a recent regularity theorem.
Theorem 11 (Capogna, Cowling 2006) Let be a Carnot group. Then -quasiconformal maps on are smooth.
The proof goes in two steps.
- Show that infinitesimal conformal mappings form a finite dimensional Lie algebra. This relies on Tanaka prolongation.
- A smooth conformal mapping defines an automorphism of this algebra. Show that the automorphism uniquely determines the mapping up to a left translation.