86 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
Discrete constant mean curvature nets in space forms: Steiner's formula and Christoffel duality
We show that the discrete principal nets in quadrics of constant curvature
that have constant mixed area mean curvature can be characterized by the
existence of a K\"onigs dual in a concentric quadric.Comment: 12 pages, 10 figures, pdfLaTeX (plain pdfTeX source included as bak
file
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
On Guichard's nets and Cyclic systems
In the first part, we give a self contained introduction to the theory of
cyclic systems in n-dimensional space which can be considered as immersions
into certain Grassmannians. We show how the (metric) geometries on spaces of
constant curvature arise as subgeometries of Moebius geometry which provides a
slightly new viewpoint. In the second part we characterize Guichard nets which
are given by cyclic systems as being Moebius equivalent to 1-parameter families
of linear Weingarten surfaces. This provides a new method to study families of
parallel Weingarten surfaces in space forms. In particular, analogs of Bonnet's
theorem on parallel constant mean curvature surfaces can be easily obtained in
this setting.Comment: 25 pages, plain Te
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