F.E. Burstall, J.F. Dorfmeister, K. Leschke and A. Quintino,
Darboux transforms and simple factor dressing of constant mean
curvature surfaces, manuscripta mathematica (to appear).
arXiv:1009.5274
[math.DG].
There are many ways to transform a constant mean curvature surface: for example, one could dress the harmonic Gauss map; view it as an isothermic surface and take a Darboux transform; view it as a constrained Willmore surface and dress that. Here we show that, under reality conditions on the spectral parameter, all these procedures coincide.
F.E. Burstall and D.M.J. Calderbank,
Conformal submanifold geometry I–III,
arXiv:1006.5700
[math.DG].
Classical submanifold geometry meets Lie algebra homology and Bernstein–Gelfand–Gelfand operators! The Gauss–Codazzi–Ricci equations of conformal submanifold geometry and much, much more.
F.E. Burstall and S. Santos,
Special isothermic surfaces of type d, J. London
Math. Soc. (2012). DOI: 10.1112/jlms/jdr050.
Free access reprint available here.
The special isothermic surfaces of Bianchi and Darboux, which puzzled me for years, have a simple explanation and generalisation via the magic of polynomial conserved quantities.
F.E. Burstall, U. Hertrich-Jeromin and W. Rossman, Lie geometry of flat fronts in hyperbolic space, C. R. Acad. Sci. Paris, Ser. I 348 (2010) 661-664. DOI: 10.1016/j.crma.2010.04.018
Flat fronts are omega-surfaces. First taster of an on-going project on smooth and discrete omega surfaces.
F.E. Burstall, N.M. Donaldson, F. Pedit and U. Pinkall, Isothermic submanifolds of symmetric R-spaces, J. reine angew. Math. 2011 no. 660 (2011) 191-243. DOI: 10.1515/CRELLE.2011.075.
Longer in the making than Gone With The Wind, we export the entire theory of isothermic surfaces to the more general context of submanifolds of symmetric R-spaces. Absolutely everything goes through with, in many cases, cleaner arguments than those available classically.
F.E. Burstall and I. Khemar, Twistors, 4-symmetric spaces and integrable systems, Math. Ann. 344 (2009) 451-461. DOI:10.1007/s00208-008-0313-5
The 4-symmetric spaces that Helein and Romon see explained via twistor space. As a corollary, surfaces with holomorphic mean curvature vector in 4-dimensional space forms (real or complex) are an integrable system.
Still available on the arXiv:arXiv:0804.4235 [math.DG].
F. Burstall, U. Hertrich-Jeromin, W. Rossman and S. Santos, Discrete surfaces of constant mean curvature, arXiv:0804.2707 [math.DG].
Adventures in discrete differential geometry: we use discrete gauge theory to make a convincing definition of constant mean curvature net that makes sense for any ambient curvature.
A slight return to harmonic maps: equivariant harmonic maps have blindingly simple holomorphic potentials from which you can read off the symmetry.
Available online from the QJM.
More "unfashionable" geometry: a transparent treatment of Ribaucour transformations and their permutability via Lie sphere geometry.
Preprint version available as math.DG/0407244 on the arXiv.
Everything I know About Isothermic Surfaces. An expanded version of some lectures I gave at the National Centre for Theoretical Science, Tsing Hua University, Taiwan in January 1999.
Shiny reprints available on request or get the preprint version math.DG/0003096 from the e-print arXiv.
Sneak preview of my work with David Calderbank on the Gauss-Codazzi-Ricci equations for submanifolds in conformal and other parabolic geometries. Lie algebra homology and Bernstein-Gelfand-Gelfand differential operators play a surprisingly central role.
Shiny reprints available on request or get the preprint version as pdf.
Higher dimensional analogues of constant mean curvature surfaces.
Shiny reprints available on request or get the preprint version math.DG/0111217 from the e-print arXiv.
Low-tech formulation of the Gauss-Codazzi-Ricci equations of conformal surface geometry along with conformally invariant formulations of the Novikov-Veselov and Davey-Stewartson flows on surfaces.
Reprints are available on request and the preprint version is available as math.DG/0111169 from the e-print arXiv.
My first adventure in either projective or Lie sphere geometry.
Shiny reprints are available on request and the preprint version is still available as math.DG/0103162 from the e-print arXiv.
The adventures of the H-seminar at SFB288, TU Berlin in 1998.
Apparently Springer will make this available on-line for free (for a while). In the meantime, an old version (no Bäcklund or Darboux transforms) is available as math.DG/0002075 from the e-print arXiv.
From the definition of a manifold to the Bishop Volume Comparison Theorem in 29 pages! Based on breathless lectures given to an ICMS Instructional Conference in April 1998.
Available as dvi, PostScript or pdf.
This started out as an expanded version of a talk I gave at the Workshop on Geometry and Analysis in Augsburg in July 1998 and, under the title Isothermic surfaces: conformal geometry, Clifford algebras and integrable systems, became an abstract for a similar talk given at the autumn 1998 meeting of the Japanese Mathematical Society in Osaka.
Available as dvi, PostScript or pdf.
Optimal bounds on the uniton number for harmonic 2-spheres in a Lie group among other things.
Still available in the e-print arXiv as dg-ga/9606002 or send me mail if you want a shiny reprint.
Where isothermic surfaces began for me. All my later out-pourings on the subject were a result of my attempt to understand this paper!
The arXiv version is quite different from what eventually got published so ask for a reprint or get the on-line version from Springer Verlag if you subscribe to their Link service.
Available as dg-ga/9410001 from the e-print arXiv. Reprints are also available on request.
A complete account of harmonic 2-tori in spheres or complex projective spaces. The punchline: they all come from twistor theory or (after a prolongation) integrable systems theory.
The preprint version is still available as dvi or PostScript. High quality reprints are also available on request.
Sadly, this book is now out of print (and pulped!). However, thanks to John Wood, the article survives in PostScript form as does the whole book!