By Armen N. Sergeev
This booklet is predicated on a lecture path given by way of the writer on the academic middle of Steklov Mathematical Institute in 2011. it's designed for a one semester path for undergraduate scholars, acquainted with easy differential geometry, advanced and practical analysis.
The common Teichmüller area T is the quotient of the gap of quasisymmetric homeomorphisms of the unit circle modulo Möbius variations. the 1st a part of the booklet is dedicated to the examine of geometric and analytic homes of T. it truly is an infinite-dimensional Kähler manifold which incorporates all classical Teichmüller areas of compact Riemann surfaces as complicated submanifolds and is the reason the identify “universal Teichmüller space”. except classical Teichmüller areas, T includes the distance S of diffeomorphisms of the circle modulo Möbius differences. The latter area performs an incredible position within the quantization of the idea of tender strings. The quantization of T is gifted within the moment a part of the publication. against this with the case of diffeomorphism house S, which are quantized in frames of the normal Dirac scheme, the quantization of T calls for a fully varied process in response to the noncommutative geometry methods.
The e-book concludes with a listing of 24 difficulties and routines which might be used through the examinations.
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D 0 if simultaneously ; Á 2 WC or ; Á 2 W . 4) is the orthogonal direct sum with respect to the Hermitian inner product h ; i. 4) this inner product is given by the formula h ; Ái D i !. where ˙ NC/ C; Á i !. ; ÁN / 2 V C (resp. Á 2 V C ) onto W˙ . (resp. 2 Definition in terms of harmonic functions. Denote by D the Dirichlet space consisting of harmonic functions h W ! z/j2 dxdy D 1 2 Z ˇ ˇ2 ˇ ˇ2 !
Any Cauchy sequence in T converges. Assertion 3. The space T is contractible. Assertion 4. The space T is not a topological group. In other words, the operation of composition of normalized quasisymmetric homeomorphisms S 1 ! S 1 is not continuous in the Teichmüller metric. 2 Schwarzian derivative. To define a complex structure on the space T , we need to recall the main properties of Schwarzian derivative. log f 0 /0 Â 3 f 00 2 f0 Ã2 : Here are the properties of Schwarzian derivative to be used later (try to prove these properties by yourself).
It is sufficient to prove this assertion for the normalized homeomorphism w , since it differs from w only by a fractional-linear transformation. C/ H) w B g D g B w holds. 2 Lecture VII. g /0 B w @w : Dividing the first relation by the second one, we obtain . 1) is necessary for the G-invariance almost everywhere on C. of w and w. By reversing all implications we see that this condition is also sufficient for the G-invariance of w and w. The given definition and the proved proposition extend immediately to quasiconformal homeomorphisms, defined in the unit disk .