190 research outputs found
Harmonic maps in unfashionable geometries
We describe some general constructions on a real smooth projective 4-quadric
which provide analogues of the Willmore functional and conformal Gauss map in
both Lie sphere and projective differential geometry. Extrema of these
functionals are characterized by harmonicity of this Gauss map.Comment: plain TeX, uses bbmsl for blackboard bold, 20 page
Periodic discrete conformal maps
A discrete conformal map (DCM) maps the square lattice to the Riemann sphere
such that the image of every irreducible square has the same cross-ratio. This
paper shows that every periodic DCM can be determined from spectral data (a
hyperelliptic compact Riemann surface, called the spectral curve, equipped with
some marked points). Each point of the map corresponds to a line bundle over
the spectral curve so that the map corresponds to a discrete subgroup of the
Jacobi variety. We derive an explicit formula for the generic maps using
Riemann theta functions, describe the typical singularities and give a
geometric interpretation of DCM's as a discrete version of the Schwarzian KdV
equation. As such, the DCM equation is a discrete soliton equation and we
describe the dressing action of a loop group on the set of DCM's. We also show
that this action corresponds to a lattice of isospectral Darboux transforms for
the finite gap solutions of the KdV equation.Comment: 41 pages, 10 figures, LaTeX2
Isothermic surfaces and conservation laws
For CMC surfaces in -dimensional space forms, we relate the moment class of Korevaar--Kusner--Solomon to a second cohomology class arising from the integrable systems theory of isothermic surfaces. In addition, we show that both classes have a variational origin as Noether currents
Isothermic surfaces and conservation laws
For CMC surfaces in -dimensional space forms, we relate the moment class
of Korevaar--Kusner--Solomon to a second cohomology class arising from the
integrable systems theory of isothermic surfaces. In addition, we show that
both classes have a variational origin as Noether currents.Comment: LaTeX, 19 A4 page
Formal conserved quantities for isothermic surfaces
Isothermic surfaces in are characterised by the existence of a pencil
of flat connections. Such a surface is special of type if there
is a family of -parallel sections whose dependence on the
spectral parameter is polynomial of degree . We prove that any
isothermic surface admits a family of -parallel sections which is a
formal Laurent series in . As an application, we give conformally invariant
conditions for an isothermic surface in to be special.Comment: 13 page
Discrete -nets and Guichard nets
We provide a convincing discretisation of Demoulin's -surfaces along
with their specialisations to Guichard and isothermic surfaces with no loss of
integrable structure.Comment: 39 A4 page
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