I1671-N26: Transformations and Singularities

Joint Project between Austria (FWF) and Japan (JSPS)

Abstract. Transformations of surfaces allow to construct new (families of) surfaces of the same kind. The concept of transformation is rather far-reaching: for example, the Weierstrass representation of minimal surfaces can be interpreted as a special case. Moreover, discrete (facetted) surfaces of the same kind naturally occur from repeated transformations.

Singularities are, most generally, the "bad points" of a theory, points at which the methods of a theory fail. This already constitutes a strong motivation for their study: the employed methods require extension, the studied objects reconsideration, and new viewpoints on the theory arise.

The aim of this project is to study the interplay between transformations and singularities. More precisely: we aim to

• understand how transformations of particular surfaces behave (or fail to behave) at singularities; and
• study how those transformations create (or annihilate) singularities, and what the nature of the occurring singularities is.

A good understanding of the interplay between transformations and singularities will shed further light on either theory. Moreover, new results and phenomena will arise from this interplay, for example, regarding "singularities" of facetted or paneled surfaces.

Publications.

F Burstall, U Hertrich-Jeromin, C Müller, W Rossman (2016)

Semi-discrete isothermic surfaces; Geom Dedicata 183, 43-58 (2016) DOI:10.1007/s10711-016-0143-7

Abstract

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.

A Honda (2016)

Weakly complete wave fronts with one principal curvature constant; Kyushu J Math 70, 217-226 (2016) DOI:10.2206/kyushujm.70.217

Abstract

Murata and Umehara gave a classification of complete flat fronts in the Euclidean 3-space and proved their orientability. Here, a flat front is a flat surface (i.e., a surface where one of the principal curvatures is identically zero) with admissible singularities. In this paper,we investigate wave fronts where one of the principal curvatures is a non-zero constant. Although they are orientable in the regular surface case, there exist non-orientable examples. We classify weakly complete ones and derive their orientability.

V Branding, W Rossman (2017)

Magnetic geodesics on surfaces with singularities; Pac J Math Ind 9, 3 (2017) DOI:10.1186/s40736-017-0028-1

Abstract

We focus on the numerical study of magnetic geodesics on surfaces, including surfaces with singularities. In addition to the numerical investigation, we give restrictive necessary conditions for tangency directions of magnetic geodesics passing through certain types of singularities.

W Carl (2017)

On semidiscrete constant mean curvature surfaces and their associated families; Monatsh Math 182, 537-563 (2017) DOI:10.1007/s00605-016-0929-6

Abstract

The present paper studies semidiscrete surfaces in three-dimensional Euclidean space within the framework of integrable systems. In particular, we investigate semidiscrete surfaces with constant mean curvature along with their associated families. The notion of mean curvature introduced in this paper is motivated by a recently developed curvature theory for quadrilateral meshes equipped with unit normal vectors at the vertices, and extends previous work on semidiscrete surfaces. In the situation of vanishing mean curvature, the associated families are defined via a Weierstrass representation. For the general cmc case, we introduce a Lax pair representation that directly defines associated families of cmc surfaces, and is connected to a semidiscrete sinh-Gordon equation. Utilizing this theory we investigate semidiscrete Delaunay surfaces and their connection to elliptic billiards.

A Honda, M Koiso, M Kokubu, M Umehara, K Yamada (2017)

Mixed type surfaces with bounded mean curvature in 3-dimensional space-times; Differ Geom Appl 52, 64-77 (2017) DOI:10.1016/j.difgeo.2017.03.009

Abstract

In this paper, we shall prove that space-like surfaces with bounded mean curvature functions in real analytic Lorentzian 3-manifolds can change their causality to time-like surfaces only if the mean curvature functions tend to zero. Moreover, we shall show the existence of such surfaces with non-vanishing mean curvature and investigate their properties.

U Hertrich-Jeromin, A Honda (2017)

Minimal Darboux transformations; Beitr Alg Geom 58, 81-91 (2017) DOI:10.1007/s13366-016-0301-y, EPrint arXiv:1602.06682

Abstract

We derive a permutability theorem for the Christoffel, Goursat and Darboux transformations of isothermic surfaces. As a consequence we obtain a simple proof of a relation between Darboux pairs of minimal surfaces in Euclidean space, curved flats in the 2-sphere and flat fronts in hyperbolic space.

C Müller, M Yasumoto (2017)

Semi-discrete constant mean curvature surfaces of revolution in Minkowski space; in Mladenov, Meng, Yoshioka (eds), Proc International Conference on Geometry, Integrability and Quantization 18, 191-202 (2017) DOI:10.7546/giq-18-2017-191-202

Abstract

In this paper we describe semi-discrete isothermic constant mean curvature surfaces of revolution with smooth profile curves in Minkowski three-space. Unlike the case of semi-discrete constant mean curvature sur- faces in Euclidean three-space, they might have certain types of singularities in a sense defined by the second author in a previous work. We analyze the singularities of such surfaces.

Y Ogata, M Yasumoto (2017)

Construction of discrete constant mean curvature surfaces in Riemannian spaceforms and applications; Differ Geom Appl 54, 264-281 DOI:10.1016/j.difgeo.2017.04.010

Abstract

In this paper we give a construction for discrete constant mean curvature surfaces in Riemannian spaceforms in terms of integrable systems techniques, which we call the discrete DPW method for discrete constant mean curvature surfaces. Using this construction, we give several examples, and analyze singularities of the parallel constant Gaussian curvature surfaces.

F Burstall, U Hertrich-Jeromin, W Rossman (2018)

Discrete linear Weingarten surfaces; Nagoya Mathematical Journal 231, 55-88 (2018) DOI:10.1017/nmj.2017.11

Abstract

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.

F Burstall, U Hertrich-Jeromin, Y Suyama (2018)

Curvilinear coordinates on generic conformally flat hypersurfaces and constant curvature 2-metrics; J Math Soc Japan 70(2), 617-649 (2018); DOI:10.2969/jmsj/07027420

Abstract

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.

S Fujimori, U Hertrich-Jeromin, M Kokubu, M Umehara, K Yamada (2018)

Quadrics and Scherk towers; Monatsh Math 186, 249-279 (2018) DOI:10.1007/s00605-017-1075-5

Abstract

We investigate the relation between quadrics and their Christoffel duals on the one hand, and certain zero mean curvature surfaces and their Gauss maps on the other hand. To study the relation between timelike minimal surfaces and the Christoffel duals of 1-sheeted hyperboloids we introduce para-holomorphic elliptic functions. The curves of type change for real isothermic surfaces of mixed causal type turn out to be aligned with the real curvature line net.

S Fujimori (2018)

Triply periodic zero mean curvature surfaces in Lorentz-Minkowski 3-space; Adv Stud Pure Math 78, 201-219 (2018) DOI:10.2969/aspm/07810201

Abstract

We construct triply periodic zero mean curvature surfaces of mixed type in the Lorentz-Minkowski 3-space $\mathbb{L}^3$, with the same topology as the triply periodic minimal surfaces in the Euclidean 3-space $\mathbb{R}^3$, called Schwarz rPD surfaces.

M Pember, G Szewieczek (2018)

Channel surfaces in Lie sphere geometry; Beitr Algebra Geom 59, 779-796 (2018) DOI:10.1007/s13366-018-0394-6

Abstract

We discuss channel surfaces in the context of Lie sphere geometry and characterise them as certain $\Omega_0$-surfaces. Since $\Omega_0$-surfaces possess a rich transformation theory, we study the behaviour of channel surfaces under these transformations. Furthermore, by using certain Dupin cyclide congruences, we characterise Ribaucour pairs of channel surfaces.

F Burstall, U Hertrich-Jeromin, M Pember, W Rossman (2019)

Polynomial conserved quantities of Lie applicable surfaces; manuscripta math 158, 505-546 (2019) DOI:10.1007/s00229-018-1033-0

Abstract

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 Bäcklund-type transformation for these surfaces.

A Fuchs (2019 PhD thesis)

Transformations and singularities of isothermic surfaces; PhD thesis, TU Wien (2019)

Abstract

We determine the limiting behaviour of Darboux and Calapso transforms of polarized curves, where the polarization has a pole of first or second order. We then study the analogous problem for isothermic surfaces. We consider those isothermic surfaces for which their Hopf differential factorizes into a real function and a meromorphic quadratic differential. Upon restriction to a simply connected patch, away from the zeros and poles of this differential, the Darboux and Calapso transformations yield new isothermic surfaces. We investigate the limiting behaviour of these transformed patches as the zeros and poles of the meromorphic quadratic differential are approached and determine whether they are continuous around those points.

A Fuchs (2019)

Transformations and singularities of polarized curves; Ann Glob Anal Geom 55, 529-553 (2019) DOI:10.1007/s10455-018-9639-8

Abstract

We study the limiting behaviour of Darboux and Calapso transforms of polarized curves in the conformal $n$-dimensional sphere when the polarization has a pole of first or second order at some point. We prove that for a pole of first order, as the singularity is approached, all Darboux transforms converge to the original curve and all Calapso transforms converge. For a pole of second order, a generic Darboux transform converges to the original curve while a Calapso transform has a limit point or a limit circle, depending on the value of the transformation parameter. In particular, our results apply to Darboux and Calapso transforms of isothermic surfaces when a singular umbilic with index $1\over 2}$ or $1$ is approached along a curvature line.

U Hertrich-Jeromin, W Rossman, G Szewieczek (2020)

Discrete channel surfaces; Math Z 294, 747-767 (2020) DOI:10.1007/s00209-019-02389-4

Abstract

We present a definition of discrete channel surfaces in Lie sphere geometry, which reflects several properties for smooth channel surfaces. Various sets of data, defined at vertices, on edges or on faces, are associated with a discrete channel surface that may be used to reconstruct the underlying particular discrete Legendre map. As an application we investigate isothermic discrete channel surfaces and prove a discrete version of Vessiot's Theorem.

G Szewieczek (2021)

A duality for Guichard nets; manuscripta math 164, 193-221 (2021) https://doi.org/10.1007/s00229-020-01181-7 ti so DOI:10.1007/s00229-020-01181-7

Abstract

In this paper we study G-surfaces, a rather unknown surface class originally defined by Calapso, and show that the coordinate surfaces of a Guichard net are G-surfaces. Based on this observation, we present distinguished Combescure transformations that provide a duality for Guichard nets. Another class of special Combescure transformations is then used to construct a Bäcklund-type transformation for Guichard nets. In this realm a permutability theorem for the dual systems is proven.