169 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
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
Harmonic Riemannian submersions between Riemannian symmetric spaces of noncompact type
We construct harmonic Riemannian submersions that are retractions from symmetric spaces of noncompact type onto their rank-one totally geodesic subspaces. Among the consequences, we prove the existence of a non-constant, globally defined complex-valued harmonic morphism from the Riemannian symmetric space associated to a split real semisimple Lie group. This completes an affirmative proof of a conjecture of Gudmundsson
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
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 submanifolds of symmetric -spaces
We extend the classical theory of isothermic surfaces in conformal 3-space,
due to Bour, Christoffel, Darboux, Bianchi and others, to the more general
context of submanifolds of symmetric -spaces with essentially no loss of
integrable structure.Comment: 35 pages, 3 figures. v2: typos and other infelicities corrected
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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