28 research outputs found
The Green function of Teichm\"uller spaces with applications
We describe briefly a new approach to some problems related to Teichm\"uller
spaces, invariant metrics, and extremal quasiconformal maps. This approach is
based on the properties of plurisubharmonic functions, especially of the
plurisubharmonic Green function. The main theorem gives an explicit
representation of the Green function for Teichm\"uller spaces by the
Kobayashi-Teichm\"uller metric of these spaces. This leads to various
applications. In particular, this gives a new characterization of extremal
quasiconformal maps.Comment: 5 page
Milin's coefficients, complex geometry of Teichm\"{u}ller spaces and variational calculus for univalent functions
We investigate the invariant metrics and complex geodesics in the universal
Teichm\"{u}ller space and Teichm\"{u}ller space of the punctured disk using
Milin's coefficient inequalities. This technique allows us to establish that
all non-expanding invariant metrics in either of these spaces coincide with its
intrinsic Teichm\"{u}ller metric.
Other applications concern the variational theory for univalent functions
with quasiconformal extension. It turns out that geometric features caused by
the equality of metrics and connection with complex geodesics provide deep
distortion results for various classes of such functions and create new
phenomena which do not appear in the classical geometric function theory
Hyperbolic distances, nonvanishing holomorphic functions and Krzyz's conjecture
The goal of this paper is to prove the conjecture of Krzyz posed in 1968 that
for nonvanishing holomorphic functions in the unit
disk with , we have the sharp bound for all , with equality only for the function
and its rotations. The problem was considered by many researchers, but only
partial results have been established. The desired estimate has been proved
only for .
Our approach is completely different and relies on complex geometry and
pluripotential features of convex domains in complex Banach spaces
Hyperbolic metrics, homogeneous holomorphic functionals and Zalcman's conjecture
We show, using the Kobayashi and Caratheodory metrics on special holomorphic
disks in the universal Teichmuller space, that a wide class of holomorphic
functionals on the space of univalent functions in the disk is maximized by the
Koebe function or by its root transforms; their extremality is forced by
hyperbolic features. As consequences, this implies the proofs of the famous
Zalcman and Bieberbach conjectures.Comment: This is a revised and improved version of arXiv:0907.3623. Comments
are welcom
The Zalcman conjecture and related problems
At the end of 1960's, Lawrence Zalcman posed a conjecture that the
coefficients of univalent functions
on the unit disk satisfy the sharp inequality ,
with equality only for the Koebe function. This remarkable conjecture implies
the Bieberbach conjecture, investigated by many mathematicians, and still
remains a very difficult open problem for all n > 3; it was proved only in
certain special cases.
We provide a proof of Zalcman's conjecture based on results concerning the
plurisubharmonic functionals and metrics on the universal Teichm\"uller space.
As a corollary, this implies a new proof of the Bieberbach conjecture. Our
method gives also other new sharp estimates for large coefficients.Comment: This paper has been replaced with a revised version arXiv:1109.4646v
A new look at Krzyz's conjecture
Recently the author has presented a new approach to solving extremal problems
of geometric function theory. It involves the Bers isomorphism theorem for
Teichmuller spaces of punctured Riemann surfaces.
We show here that this approach, combined with quasiconformal theory, can be
also applied to nonvanishing holomorphic functions from . In
particular this gives a proof of an old open Krzyz conjecture for such
functions and of its generalizations.
The unit ball of is naturally embedded into the
universal Teichmuller space, and the functions are regarded
as the Schwarzian derivatives of univalent functions in the unit disk.Comment: arXiv admin note: text overlap with arXiv:1908.0518
Extremal quasiconformality vs rational approximation
We show that on most of the hyperbolic simply connected domains the weighted
bounded rational approximation in a natural sup norm is possible only for a
very sparse set of holomorphic functions (in contrast to integral
approximation). The obstructions are caused by the features of extremal
quasiconformality
A general coefficient theorem for univalent functions
Using the Bers isomorphism theorem for Teichmuller spaces of punctured
Riemann surfaces and some of their other complex geometric features, we prove a
general theorem on maximization of homogeneous polynomial (in fact, more
general holomorphic) coefficient functionals on some classes of univalent functions in the unit disk naturally
connected with the canonical class . The given functional is lifted to
the Teichmuller space of the punctured disk which is biholomorphically equivalent to the Bers fiber space over
the universal Teichmuller space. This generates a positive subharmonic function
on the disk with
attaining this maximal value only on the boundary circle, which correspond to
rotations of the Koebe function.
This theorem implies new sharp distortion estimates for univalent functions
giving explicitly the extremal functions, and creates a new bridge between
Teichm\"{u}ller space theory and geometric complex analysis. In particular, it
provides an alternate and direct proof of the Bieberbach conjecture
A question of Kuhnau
It is well known that the square is not a Strebel's point (i.e., its extremal
Beltrami coefficient is not Teichmuller). Many years ago, Reiner Kuhnau posed
the question: does there exist in the case of a "long" rectangle the
corresponding holomorphic quadratic differential?
We prove that the answer is negative for any bounded convex quadrilateral and
establish a stronger result for rectangles.Comment: arXiv admin note: text overlap with arXiv:1806.0276
Quasiconformal features and Fredholm eigenvalues of convex polygons
An important open problem in geometric complex analysis is to find algorithms
for explicit determination of basic functionals intrinsically connected with
conformal and quasiconformal maps, such as their Teichmuller and Grunsky norms,
Fredholm eigenvalues and the quasireflection coefficient. This has not been
solved even for convex polygons. This case has intrinsic interest in view of
the connection of such polygons with the geometry of the universal Teichmuller
space.
We provide a new approach, based on affine transformations of univalent
functions