# Some preprints, reprints and more emphemeral pieces

Here are pointers to various papers of which I am a (co-)author.

1. F.E. Burstall, J. Cho, U. Hertrich-Jeromin, M. Pember and W. Rossman, Discrete $\Omega$-nets and Guichard nets.

A convincing discretisation of $\Omega$-nets and Guichard nets that replicates the classical smooth theory in almost every detail. A thoroughgoing use of a discrete exterior calculus keeps the computations to a minimum. We also lay the foundations for a theory of discrete curved flats.

ArXiv version: arXiv:2008.01447 [math.DG]

2. Francis Burstall and Mason Pember, Lie applicable surfaces and curved flats.

Curved flats in the Grassmannian of $(2,2)$-planes in $\mathbb{R}^{n+1,2}$ are the same as Demoulin families of Lie applicable maps related by Darboux transforms.

ArXiv version: arXiv:2007.11947 [math.DG]

3. Francis Burstall, Emilio Musso and Mason Pember, Quartic differentials and harmonic maps in conformal surface geometry.

We characterise surfaces in a sphere orthogonal to a harmonic congruence of codimension-$2$ spheres. Surfaces for which Bryant's quartic differential is holomorphic generically fall into this class and so we recover an old result of Voss and extend it to arbitrary codimension and signature.

ArXiv version: arXiv:2007.03992 [math.DG]

4. F.E. Burstall and J.C. Wood, A correction to "The construction of harmonic maps into complex Grassmannians".

Too small for the arXiv to take, we correct here a mistake in an old paper of ours that recently burned some of our friends (sorry, Rui!).

Available as PDF or HTML.

5. On conformal Gauss maps Bull. London Math. Soc. 51 (2019) 989-994. DOI: 10.1112/blms.12293.

Another tiny paper: this one characterises conformal Gauss maps among all maps into the Grassmannian of $2$-spheres in $S^n$. In contrast to the Willmore case, a second order condition is required.

ArXiv version: arXiv:1904.02574 [math.DG]

6. On canonical elements, arXiv:1811.12041 [math.DG].

A tiny paper that characterises the canonical elements of a compact semisimple Lie algebra and works out the case of $\mathfrak{so}(n)$. Written at the urging of John C. Wood to correct inaccuracies in this old paper of Eschenburg, Ferreira, Tribuzy and I.

7. F. Burstall, U. Hertrich-Jeromin and M. Lara Miro, Ribaucour coordinates, Beiträge zur Algebra und Geometrie 60 (2019) 39-55. DOI: 10.1007/s13366-018-0391-9.

Elementary arguments about Ribaucour transforms. Among other things, we find a very simple proof of the result of Dajczer, Florit and Tojeiro that any submanifold with flat normal bundle is an iterated Ribaucour transform of a sphere.

ArXiv version: arXiv:1711.04605 [math.DG]

8. F. Burstall, U. Hertrich-Jeromin, M. Pember and W. Rossman, Polynomial conserved quantities for Lie applicable surfaces, manuscripta math. 158 (2019) 505-546. DOI: 10.1007/s00229-018-1033-0.

Lie applicable surfaces in Lie sphere geometry account for isothermic, Guichard and L-isothermic surfaces via the magic of linear conserved quantities.

ArXiv version: arXiv:1707.01713 [math.DG].

9. F. Burstall, U. Hertrich-Jeromin and W. Rossman, Discrete linear Weingarten surfaces, Nagoya Math. 231 (2018) 55-887. DOI: 10.1017/nmj.2017.11.

Further adventures in discrete Lie sphere geometry: discrete linear Weingarten surfaces are omega and characterised among omega surfaces by conserved quantities just like the smooth case.

ArXiv version: arxiv:1406.1293 [math.DG].

10. Notes on transformations in integrable geometry, in Special Metrics and Group Actions in Geometry (Chiossi, Fino, Musso, Podestà, Vezzoni, eds.), Springer, (2017), 59-80. DOI: 10.1007/978-3-319-67519-0_3.

The gauge-theoretic approach to transformations described through the examples of K-surfaces and isothermic surfaces.

ArXiv version: arXiv:1511.04216 [math.DG].

11. F. Burstall, U. Hertrich-Jeromin, C. Müller and W. Rossman, Semi-discrete isothermic surfaces, Geom. Dedicata 183 (2016) 43-58, MR3523116. DOI:10.1007/s10711-016-0143-7.

Semi-discrete surfaces are a nice playground for seeing the interaction between discrete and smooth theories. Here we unleash the gauge-theoretic machine in this setting and rapidly arrive at a complete theory of semi-discrete isothermic surfaces.

ArXiv version: arxiv:1506.04730 [math.DG].

12. F. Burstall, U. Hertrich-Jeromin, W. Rossman and S. Santos, Discrete special isothermic surfaces, Geom. Dedicata 174 (2015) 1-11, MR3303037. DOI: 10.1007/s10711-014-0001-4.

The shockingly efficient gauge-theoretic formalism is used to transfer my work with Susana Santos on special isothermic surfaces to the discrete setting.

13. F.E. Burstall and A.C. Quintino, Dressing transformations of constrained Willmore surfaces, Comm. Anal. Geom. 22 (2014) 469-518, MR3228303. DOI: 10.4310/CAG.2014.v22.n3.a4.

Does What It Says on the Tin: we make dressing transformations for constrained Willmore surfaces in arbitrary codimension using a frame-free gauge-theoretic formulation. As usual, codimension 2 is special and, as a particular case, we recover the Darboux transformations of Willmore surfaces via Riccati equations from the Yellow Book and generalise these to the constrained case.

ArXiv version: arXiv:1307.2077 [math.DG].

14. F.E. Burstall and S.D. Santos, Formal conserved quantities for isothermic surfaces, Geom. Dedicata 172 (2014) 191-205, MR3253778. DOI: 10.1007/s10711-013-9915-5.

All isothermic surfaces admit a globally defined formal conserved quantity away from the zeros of the holomorphic quadratic differential. This gives characterisations of special isothermic surfaces in terms of (slightly horrific) differential equations on their conformal invariants.

ArXiv version: arXiv:1301.0447 [math.DG].

15. F. Burstall, U. Hertrich-Jeromin, W. Rossman and S. Santos, Discrete surfaces of constant mean curvature, RIMS Kyôkûroku Bessatsu 1880 (2014) 133-179.

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.

ArXiv: arXiv:0804.2707 [math.DG].

16. F.E. Burstall, J.F. Dorfmeister, K. Leschke and A. Quintino, Darboux transforms and simple factor dressing of constant mean curvature surfaces, manuscripta math. 140 (2013) 213-236, MR3016491. DOI: 10.1007/s00229-012-0537-2

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.

ArXiv version: arXiv:1009.5274 [math.DG].

17. F.E. Burstall, U. Hertrich-Jeromin and W. Rossman, Lie geometry of linear Weingarten surfaces, C. R. Acad. Sci. Paris, Ser. I 350 (2012) 413-416, MR2922095. DOI: 10.1016/j.crma.2012.03.018.

More Lie geometry of omega surfaces: linear Weingarten surfaces are omega surfaces characterised by the demand that the isothermic sphere congruences each lie in a linear complex.

18. F.E. Burstall and S. Santos, Special isothermic surfaces of type d, J. London Math. Soc. 85 (2012) 571–591, MR2901079. 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.

19. 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, MR2855825. 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.

20. 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.

21. 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, MR2652493. 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.

22. F.E. Burstall and I. Khemar, Twistors, 4-symmetric spaces and integrable systems, Math. Ann. 344 (2009) 451-461, MR2495778. 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].

23. F.E. Burstall, M. Kilian, Equivariant harmonic cylinders, Quarterly J. Math 57 (2006) 449-468, MR2277594. DOI:10.1093/qmath/hal005

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.

24. F.E. Burstall, U. Hertrich-Jeromin, The Ribaucour transformation in Lie sphere geometry, Differential Geom. Appl. 24 (2006) 503-520, MR2254053. DOI:10.1016/j.difgeo.2006.04.007.

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.

25. Isothermic surfaces: conformal geometry, Clifford algebras and integrable systems in Integrable systems, Geometry and Topology (ed. C.-L. Terng) AMS/IP Studies in Advanced Math. vol. 36 (2006) 1-82, MR2222512.

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.

26. F.E. Burstall, D.M.J. Calderbank, Submanifold geometry in generalized flag manifolds, Rend. Circ. Mat. Palermo (2) Suppl. 72 (2004) 13-41 MR2069394 .

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.

27. F.E. Burstall, J.-H. Eschenburg, M.J. Ferreira, R. Tribuzy, Kähler submanifolds with parallel pluri-mean curvature, Differential Geom. Appl. 20 (2004) 46-66 MR2030166. DOI:10.1016/S0926-2245(03)00055-X.

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.

28. F.E. Burstall, F. Pedit, U. Pinkall, Schwarzian Derivatives and Flows of Surfaces in Differential Geometry and Integrable Systems, (eds. M.A. Guest, R. Miyaoka, Y. Ohnita) Contemp. Math. vol 308, A. M. S. (2002) 39-61 MR1955628.

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.

29. F.E. Burstall, U. Hertrich-Jeromin, Harmonic maps in unfashionable geometries, manuscripta math. 108 (2002) 171-189 MR1918585. DOI: 10.1007/s002290200269.

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.

30. F. Burstall, D. Ferus, K. Leschke, F. Pedit, U. Pinkall, Conformal Geometry of Surfaces in the 4-Sphere and Quaternions, Lecture Notes in Mathematics vol 1772, Springer-Verlag, 2002 MR1887131.

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.

31. Basic Riemannian Geometry in Spectral Theory and Geometry, (eds. E.B. Davies and Y. Safarov), L.M.S. Lect. Note Series 273, C.U.P. (1999), pages 1-29, MR1736864.

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.

32. Isothermic surfaces in arbitrary co-dimension in Atti del Congresso Internazionale in onore di Pasquale Calapso, Rendiconti del Sem. Mat. di Messina (2001) 57-68

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.

33. F.E. Burstall, M.A. Guest Harmonic two-spheres in compact symmetric spaces, revisited, Math. Ann. 309 (1997) 541-572 MR 99f:58046. DOI:10.1007/s002080050127

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.

34. F. Burstall, U. Hertrich-Jeromin, F. Pedit, U. Pinkall Curved Flats and Isothermic Surfaces Math. Z. 225 (1997) 199-209 MR 98j:53004 . DOI:10.1007/PL00004308

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.

35. F. E. Burstall, F. Pedit, Dressing orbits of harmonic maps Duke Math. J. 80 (1995) 353-382 MR 97e:58052 . DOI:10.1215/S0012-7094-95-08015-6

Available as dg-ga/9410001 from the e-print arXiv. Reprints are also available on request.

36. Harmonic tori in spheres and complex projective spaces J. reine u. angew. Math. 469 (1995) 149-177 MR 96m:58053 .

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 pdf. High quality reprints are also available on request.

37. F.E. Burstall, F. Pedit Harmonic maps via Adler-Kostant-Symes theory in Harmonic maps and integrable systems (A. Fordy and J.C. Wood, eds), Aspects of Math. 23, Vieweg (1994) 221-272.

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!

### Still want more?

Then have a look in my collection of rather older (pre-1994) papers.
Fran Burstall <feb@maths.bath.ac.uk>