542 research outputs found
The Theory of Quasiconformal Mappings in Higher Dimensions, I
We present a survey of the many and various elements of the modern
higher-dimensional theory of quasiconformal mappings and their wide and varied
application. It is unified (and limited) by the theme of the author's
interests. Thus we will discuss the basic theory as it developed in the 1960s
in the early work of F.W. Gehring and Yu G. Reshetnyak and subsequently explore
the connections with geometric function theory, nonlinear partial differential
equations, differential and geometric topology and dynamics as they ensued over
the following decades. We give few proofs as we try to outline the major
results of the area and current research themes. We do not strive to present
these results in maximal generality, as to achieve this considerable technical
knowledge would be necessary of the reader. We have tried to give a feel of
where the area is, what are the central ideas and problems and where are the
major current interactions with researchers in other areas. We have also added
a bit of history here and there. We have not been able to cover the many recent
advances generalising the theory to mappings of finite distortion and to
degenerate elliptic Beltrami systems which connects the theory closely with the
calculus of variations and nonlinear elasticity, nonlinear Hodge theory and
related areas, although the reader may see shadows of this aspect in parts
Stream lines, quasilines and holomorphic motions
We give a new application of the theory of holomorphic motions to the study
the distortion of level lines of harmonic functions and stream lines of ideal
planar fluid flow. In various settings, we show they are in fact quasilines -
the quasiconformal images of the real line. These methods also provide quite
explicit global estimates on the geometry of these curves.Comment: 10 pages, 3 figure
n-Harmonic mappings between annuli
The central theme of this paper is the variational analysis of homeomorphisms
h\colon \mathbb X \onto \mathbb Y between two given domains . We look for the extremal mappings in the
Sobolev space which minimize the energy
integral Because of the
natural connections with quasiconformal mappings this -harmonic alternative
to the classical Dirichlet integral (for planar domains) has drawn the
attention of researchers in Geometric Function Theory. Explicit analysis is
made here for a pair of concentric spherical annuli where many unexpected
phenomena about minimal -harmonic mappings are observed. The underlying
integration of nonlinear differential forms, called free Lagrangians, becomes
truly a work of art.Comment: 120 pages, 22 figure
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