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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 $S$-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., 15 pages

We investigate curved flats in Lie sphere geometry. We show that in this setting curved flats are in one-to-one correspondence with pairs of Demoulin families of Lie applicable surfaces related by Darboux transformation., Comment: 7 pages

We discuss results for the Ribaucour transformation of curves or of higher dimensional smooth and discrete submanifolds. In particular, a result for the reduction of the ambient dimension of a submanifold is proved and the notion of Ribaucour coordinates is derived using a Bianchi permutability result. Further, we discuss smoothing of semi-discrete curvature line nets and an interpolation by Ribaucour transformations., Comment: 11 pages, 5 figures

Discrete linear Weingarten surfaces in space forms are characterized as special discrete $\Omega$-nets, a discrete analogue of Demoulin's $\Omega$-surfaces. It is shown that the Lie-geometric deformation of $\Omega$-nets descends to a Lawson transformation for discrete linear Weingarten surfaces, which coincides with the well-known Lawson correspondence in the constant mean curvature case., Comment: v2: 18 pages, improved exposition

We characterise the maps into the space of $2$-spheres in $S^n$ 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., 5 pages. v2: minor corrections

Using the gauge theoretic approach for Lie applicable surfaces, we characterise certain subclasses of surfaces in terms of polynomial conserved quantities. These include isothermic and Guichard surfaces of conformal geometry and $L$-isothermic surfaces of Laguerre geometry. In this setting one can see that the well known transformations available for these surfaces are induced by the transformations of the underlying Lie applicable surfaces. We also consider linear Weingarten surfaces in this setting and develop a new B\"{a}cklund-type transformation for these surfaces.

We describe the gauge-theoretic approach to transformations in integrable geometry through discussion of two classical examples: surface of constant negative Gauss curvature and isothermic surfaces.

There is a one-to-one correspondence between associated families of generic conformally flat (local-)hypersurfaces in 4-dimensional space forms and conformally flat 3-metrics with the Guichard condition. In this paper, we study the space of conformally flat 3-metrics with the Guichard condition: for a conformally flat 3-metric with the Guichard condition in the interior of the space, an evolution of orthogonal (local-) Riemannian $2$-metrics with constant Gauss curvature $-1$ is determined; for a $2$-metric belonging to a certain class of orthogonal analytic $2$-metrics with constant Gauss curvature $-1$, a one-parameter family of conformally flat 3-metrics with the Guichard condition is determined as evolutions issuing from the $2$-metric., 31 pages. v2: acknowledgement of support added; metadata corrected. v3: minor changes after referee's report

We show how linear Weingarten surfaces appear as special Ω -surfaces and give a characterization of those linear Weingarten surfaces that allow a Weierstrass type representation.

An order four automorphism of a Lie algebra gives rise to an integrable system discussed by Terng. We show that solutions of this system may be identified with certain vertically harmonic twistor lifts of conformal maps of surfaces in a Riemannian symmetric space. Specialising to 4-dimensional target, we find that surfaces with holomorphic mean curvature in 4-dimensional spaces with constant sectional or holomorphic sectional curvatures constitute an integrable system as do Hamiltonian stationary Lagrangian surfaces in a 4-dimensional Hermitian symmetric space (this last being a result of Helein-Romon)., Comment: 9 pages. v2: corrected typo in metadata

We discuss special isothermic nets of type $$N$$ , a new class of discrete isothermic nets, generalizing isothermic nets with constant mean curvature in spaceforms. In the case $$N=2$$ these are the discrete analogues of Bianchi’s special isothermic surfaces that can be regarded as the origin of the rich transformation theory of isothermic surfaces. Accordingly, special isothermic nets come with Backlund transformations and a Lawson correspondence. The notion of complementary nets naturally occurs and sheds further light on the relation between geometry and integrability.

A Darboux transformation for polarized space curves is introduced and its properties are studied, in particular, Bianchi permutability. Semi-discrete isothermic surfaces are described as sequences of Darboux transforms of polarized curves in the conformal n-sphere and their transformation theory is studied. Semi-discrete surfaces of constant mean curvature are studied as an application of the transformation theory., Comment: plain TeX, 12 pages

We discuss the Ribaucour transformation of Legendre maps in Lie sphere geometry. In this context, we give a simple conceptual proof of Bianchi's original Permutability Theorem and its generalisation by Dajczer--Tojeiro. We go on to formulate and prove a higher dimensional version of the Permutability Theorem. It is shown how these theorems descend to the corresponding results for submanifolds in space forms., v2: Introduction expanded and references added. 20 pages, 4 Postscript figures

We prove that a primitive harmonic map is equivariant if and only if it admits a holomorphic potential of degree one. We investigate when the equivariant harmonic map is periodic, and as an application discuss constant mean curvature cylinders with screw motion symmetries. In this paper, we study the simplest case of this theory: that of equivariant harmonic maps where the underlying PDE reduces to an ODE. We show that such maps are characterised as those with a holomorphic potential of the simplest kind: the 1-form has simple poles at zero and infinity and satisfies a natural reality condition. Along the way, we show that equivariance is a property which is preserved under spectral deformation. A pleasant application of the foregoing theory lies in the fact that several types of surface of classical geometric interest are characterised by harmonicity of an appropriate Gauss map (3, 7, 17). In particular, the classical theory of constant mean curvature surfaces in R 3 amounts to the study of harmonic maps to a 2-sphere with the link between the loop group approach and the classical surfaces being provided by the Sym-Bobenko formula. We apply our general theory to this case which means that we study constant mean curva- ture surfaces with screw motion symmetry. We find a very simple proof of a result of Do Carmo-Dajczer (8) which asserts that these surfaces are precisely those in the associated family of a Delaunay surface (that is, a constant mean curvature surface of revolution). For this, we provide an interpretation of the Sym-Bobenko formula in terms of a homomorphism from a loop group to the Euclidean group which may be of independent interest. We then turn to a detailed study of the period problem for constant mean curvature surfaces with screw motion symmetry. Armed with our knowledge of the holomorphic potential, we are able to explicitly compute, in terms of elliptic functions, the corresponding map into the loop group (usually, this involves solving a Riemann-Hilbert problem). With this in hand, we can prove the existence of infinitely many non-congruent cylinders in the associated family of each Delaunay surface. Otherwise said, in each associated family of equivariant

We investigate the local geometry of a class of K\"ahler submanifolds $M \subset \R^n$ which generalize surfaces of constant mean curvature. The role of the mean curvature vector is played by the $(1,1)$-part (i.e. the $dz_id\bar z_j$-components) of the second fundamental form $\alpha$, which we call the pluri-mean curvature. We show that these K\"ahler submanifolds are characterized by the existence of an associated family of isometric submanifolds with rotated second fundamental form. Of particular interest is the isotropic case where this associated family is trivial. We also investigate the properties of the corresponding Gauss map which is pluriharmonic., Comment: Plain TeX, 21 pages

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., plain TeX, uses bbmsl for blackboard bold, 20 pages

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., 13 pages

Uhlenbeck introduced an invariant, the (minimal) uniton number, of harmonic 2-spheres in a Lie group G and proved that when G=SU(n) the uniton number cannot exceed n-1. In this paper, using new methods inspired by Morse Theory, we explain this result and extend it to an arbitrary compact group G. The same methods also yield Weierstrass formulae for these harmonic maps and simple proofs of most of the known classification theorems for harmonic 2-spheres in symmetric spaces.

We use the dressing method to construct transformations of constrained Willmore surfaces in arbitrary codimension. An adaptation of the Terng--Uhlenbeck theory of dressing by simple factors to this context leads us to define B\"acklund transforms of these surfaces for which we prove Bianchi permutability. Specialising to codimension 2, we generalise the Darboux transforms of Willmore surfaces via Riccati equations, due to Burstall-Ferus-Leschke-Pedit-Pinkall, to the constrained Willmore case and show that they amount to our B\"acklund transforms with real spectral parameter., Comment: 31 A4 pages. v2: metadata adjusted. v3: references added; minor clarifications made

on every compact subdomain of M . Harmonic maps arise in many different contexts in Geometry and Physics (for an overview, see [16,17]) but the setting of concern to us is the following: take M to be 2-dimensional and N to be a Riemannian symmetric space of compact type. In this case, the energy is conformally invariant so that we may take the domain to be a Riemann surface and the methods of complex analysis may be brought to bear. Moreover, the symmetric nature of the target allows us to reformulate the harmonic map equations in a gauge-theoretic way so that harmonic maps may be viewed as simple analogues of Yang–Mills fields.

We show that an inner symmetric space with a compatible Hermitian structure is necessarily Hermitian symmetric, and the Hermitian structure must be invariant. This last result was known for some of the spaces of classical type and was conjectured to be true for all compact Hermitian symmetric spaces in an earlier publication by the first and the last author

We determine the structure of the zero-set of the Nijenhuis tensor of the natural almost complex structure J1 on the total space of the bundle J(G=K; g) of Hermitian structures on the tangent spaces of any even-dimensional Riemannian symmetric space G=K of compact or non-compact type.

The special isothermic surfaces, discovered by Darboux in connection with deformations of quadrics, admit a simple explanation via the gauge-theoretic approach to isothermic surfaces. We find that they fit into a heirarchy of special classes of isothermic surface and extend the theory to arbitrary codimension., 21 pages

We define a transformation on harmonic maps from a Riemann surface into the 2-sphere which depends on a complex parameter, the so-called mu-Darboux transformation. In the case when the harmonic map N is the Gauss map of a constant mean curvature surface f and the parameter is real, the mu-Darboux transformation of -N is the Gauss map of a classical Darboux transform f. More generally, for all complex parameter the transformation on the harmonic Gauss map of f is induced by a (generalized) Darboux transformation on f. We show that this operation on harmonic maps coincides with simple factor dressing, and thus generalize results on classical Darboux transforms of constant mean curvature surfaces: every mu-Darboux transform is a simple factor dressing, and vice versa., Comment: 20 pages

We propose a Lie geometric point of view on flat fronts in hyperbolic space as special Ω -surfaces and discuss the Lie geometric deformation of flat fronts.

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 $R$-spaces with essentially no loss of integrable structure., Comment: 35 pages, 3 figures. v2: typos and other infelicities corrected.

4.1 Quaternionic Vector Bundles 4.2 Complex Quaternionic Bundles 4.3 Holomorphic Quaternionic Bundles

10.1 Surfaces in S3 10.2 Hyperbolic 2-Planes 10.3 Willmore Surfaces in S3 and Minimal Surfaces in Hyperbolic 4-Space

12.1 Ricati equations 12.2 Constant mean curvature surfaces in \({\mathbb{R}}^3\) 12.3 Darboux transforms of Willmore surfaces

5.1 S-Theory 5.2 The Mean Curvature Sphere 5.3 Hopf Fields 5.4 The Conformal Gauss Map

3.1 Projective Spaces and Affine Coordinates 3.2 Metrics on \({\mathbb{H}}P^n\) 3.3 Moebius Transformations on \({\mathbb{H}}P^1\) 3.4 Two-Spheres in S4

7.1 Surfaces in Euclidean Space 7.2 The Mean Curvature Sphere in Affine Coordinates 7.3 The Willmore Condition in Affine Coordinates

We show how pairs of isothermic surfaces are given by curved flats in a pseudo Riemannian symmetric space and vice versa. Calapso's fourth order partial differential equation is derived and, using a solution of this equation, a M\"obius invariant frame for an isothermic surface is built., Comment: 9 pages, AmSLaTeX 1.1, no figures

Over the past few years significant progress has been made in the understanding of various completely integrable nonlinear partial differential equations (soliton equations) and their relationship to classical problems in differential geometry. It has been shown in a series of recent papers [42, 32, 21, 25, 15, 8, 12] that constant mean and Gauss curvature surfaces, Willmore surfaces, minimal surfaces in spheres and projective spaces and generally harmonic maps from a Riemann surface M into various homogeneous spaces may be described as solutions to various soliton equations (see [7]). Moreover, these solutions are algebraic in the sense that they are obtained by integrating ordinary differential equations of Lax type which linearise on the Jacobian of an appropriate algebraic curve.

We study the harmonic map equations for maps of a Riemann surface into a Riemannian symmetric space of compact type from the point of view of soliton theory. There is a well-known dressing action of a loop group on the space of harmonic maps and we discuss the orbits of this action through particularly simple harmonic maps called {\em vacuum solutions}. We show that all harmonic maps of semisimple finite type (and so most harmonic $2$-tori) lie in such an orbit. Moreover, on each such orbit, we define an infinite-dimensional hierarchy of commuting flows and characterise the harmonic maps of finite type as precisely those for which the orbit under these flows is finite-dimensional., Comment: 28 pages, AmSLaTeX 1.1