40 research outputs found
Schwarzian Derivatives and Flows of Surfaces
This paper goes some way in explaining how to construct an integrable
hierarchy of flows on the space of conformally immersed tori in n-space. These
flows have first occured in mathematical physics -- the Novikov-Veselov and
Davey-Stewartson hierarchies -- as kernel dimension preserving deformations of
the Dirac operator. Later, using spinorial representations of surfaces, the
same flows were interpreted as deformations of surfaces in 3- and 4-space
preserving the Willmore energy. This last property suggest that the correct
geometric setting for this theory is Moebius invariant surface geometry. We
develop this view point in the first part of the paper where we derive the
fundamental invariants -- the Schwarzian derivative, the Hopf differential and
a normal connection -- of a conformal immersion into n-space together with
their integrability equations. To demonstrate the effectivness of our approach
we discuss and prove a variety of old and new results from conformal surface
theory. In the the second part of the paper we derive the Novikov-Veselov and
Davey-Stewartson flows on conformally immersed tori by Moebius invariant
geometric deformations. We point out the analogy to a similar derivation of the
KdV hierarchy as flows on Schwarzian's of meromorphic functions. Special
surface classes, e.g. Willmore surfaces and isothermic surfaces, are preserved
by the flows
Lie geometry of flat fronts in hyperbolic space
We propose a Lie geometric point of view on flat fronts in hyperbolic space
as special omega-surfaces and discuss the Lie geometric deformation of flat
fronts
On Conformal Gauss Maps
We characterise the maps into the space of -spheres in that are the
conformal Gauss maps of conformal immersions of a surface. In particular, we
give an invariant formulation and efficient proof of a characterisation, due to
Dorfmeister--Wang \cites{DorWan13,DorWan}, of the harmonic maps that are
conformal Gauss maps of Willmore surfaces.Comment: 5 pages. v2: minor correction
Quartic differentials and Harmonic maps in conformal surface geometry
Weconsider codimension2 spherec ongruences in pseudo-conformal geometry that are harmonic with respect to the conformal structure of an or-thogonal surface. We characterise the orthogonal surfaces of such congruences as either S-Willmore surfaces, quasi-umbilical surfaces, constant mean curvature surfaces in 3-dimensional space forms or surfaces of constant lightcone mean curvature in 3-dimensional lightcones. We then investigate Bryant’s quartic differential in this context and show that generically this is divergence free if and only if the surface under consideration is either superconformal or orthogonal to a harmonic congruence of codimension 2 spheres. We may then apply the previous result to characterise surfaces with such a property
Quartic differentials and harmonic maps in conformal surface geometry
We consider codimension 2 sphere congruences in pseudo-conformal geometry
that are harmonic with respect to the conformal structure of an orthogonal
surface. We characterise the orthogonal surfaces of such congruences as either
-Willmore surfaces, quasi-umbilical surfaces, constant mean curvature
surfaces in 3-dimensional space forms or surfaces of constant lightlike mean
curvature in 3-dimensional lightcones. We then investigate Bryant's quartic
differential in this context and show that generically this is divergence free
if and only if the surface under consideration is either superconformal or
orthogonal to a harmonic congruence of codimension 2 spheres. We may then apply
the previous result to characterise surfaces with such a property.Comment: 15 page