A Bianchi-Calo method for Bryant type surfaces
We present a Bianchi-Calo type construction method for Bryant type linear Weingarten surfaces in hyperbolic space.
math.DG↗
arXiv subjects
F. E. Burstall
Publications and source records attributed to F. E. Burstall.
Bonnet pairs, isothermic surfaces and the retraction form
We give a modern account of the classical theory of Bianchi \cite{Bia03} (see also \cite{KamPedPin98}) relating isothermic surfaces to Bonnet pairs. The main novelty is to identify the derivatives of the Bonnet pair with a component of the retraction form of the isothermic surface.
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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.
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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.
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Discrete $Ω$-nets and Guichard nets via discrete Koenigs nets
We provide a convincing discretisation of Demoulin's $Ω$-surfaces along with their specialisations to Guichard and isothermic surfaces with no loss of integrable structure.
math.DG↗
On conformal Gauss maps
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.
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On canonical elements
We characterise the canonical elements, in the sense of Burstall--Rawnsley \cite{BurRaw90}, of a compact semisimple Lie algebra and discuss the case of $\mathfrak{so}(n)$ in detail. In so doing, we correct two errors in Burstall et al. \cite{BurEscFerTri04}.
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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.
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Darboux transforms and simple factor dressing of constant mean curvature surfaces
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.
math.DG↗
Special isothermic surfaces of type $d$
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.
math.DG↗
Isothermic submanifolds of symmetric $R$-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 $R$-spaces with essentially no loss of integrable structure.
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Equivariant harmonic cylinders
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.
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The Ribaucour transformation in Lie sphere geometry
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.
math.DG↗
Isothermic surfaces: conformal geometry, Clifford algebras and integrable systems
We give an account of the classical and integrable geometry of isothermic surfaces in arbitrary co-dimension. We show that the classical transformation theory of Darboux, Bianchi and Calapso goes through unchanged in arbitrary co-dimension as does the connection with the "curved flats" of Ferus and Pedit. Moreover, we identify Darboux transformations with the dressing action of "simple factors" in the sense of Terng and Uhlenbeck. In so doing, we advertise the use of Vahlen's Clifford algebra matrices as an efficient computational tool in conformal geometry.
math.DG↗
Kaehler submanifolds with parallel pluri-mean curvature
We investigate the local geometry of a class of Kähler 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 $α$, which we call the pluri-mean curvature. We show that these Kähler 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.
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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.
math.DG↗
Dressing orbits of harmonic maps
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.
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