John Bolton

Almost complex curves and Hopf hypersurfaces in the nearly Kähler 6-sphere

We characterize Hopf hypersurfaces inS6 as open parts of geodesic hyperspheres or of tubes around almost complex curves ofS6.

From surfaces in the 5-sphere to 3-manifolds in complex projective 3-space

In a previous paper it was shown how to associate with a Lagrangian submanifold satisfying Chens equality in 3-dimensional complex projective space, a minimal surface in the 5-sphere with ellipse of curvature a circle. In this paper we focus on the reverse construction.

The affine Toda equations and minimal surfaces

In this article we consider geometrical interpretations of the two-dimensional affine Toda equations for a compact simple Lie group G. These equations originated from the work of Toda [33],[34] over 25 years ago on vibrations of lattices, and they have received considerable attention from both pure and applied mathematicians particularly over the last 15 years. (For the original context of the ideas the reader is referred to [35], [27] and [1] and for a survey of recent work to [29] and [21]).

LINEARLY FULL HARMONIC 2-SPHERES IN S4 OF AREA 20π

We give an explicit description of all harmonic 2-spheres of area 20π in the round 4-sphere S4 in terms of their branch points and umbilics. This is obtained by finding canonical forms for the twistor lifts of such maps into .

Higher Singularities and the Twistor Fibration π: CP3 → S4

We use the Klein correspondences to write down an explicit relationship between two holomorphic curves, namely the directrix curve and the twistor lift, associated to a superminimal map from a Riemann surface to the 4-sphere. We also explain how the singularities of the corresponding osculating curves are related and show that they occur at the branch points or the umbilics of the corresponding superminimal map.

Toda Equations and Plücker Formulae

An extension is obtained to certain maps into full flag manifolds of compact simple Lie groups of the classical Pluecker formulae for holomorphic curves in complex projectuive space.