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 $3$-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 $3$-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 $S^n$ are characterised by the existence of a pencil $\nabla^t$ of flat connections. Such a surface is special of type $d$ if there is a family $p(t)$ of $\nabla^t$-parallel sections whose dependence on the spectral parameter $t$ is polynomial of degree $d$. We prove that any isothermic surface admits a family of $\nabla^t$-parallel sections which is a formal Laurent series in $t$. As an application, we give conformally invariant conditions for an isothermic surface in $S^3$ to be special.Comment: 13 page

Discrete Ω\Omega-nets and Guichard nets

We provide a convincing discretisation of Demoulin's $\Omega$-surfaces along with their specialisations to Guichard and isothermic surfaces with no loss of integrable structure.Comment: 39 A4 page