P28427-N35: Non-rigidity and Symmetry breaking

Research Project (FWF)

Abstract. Properties or equations used to specify geometric objects possess certain symmetries, such as the specification of a triangle in terms of three angles, which determines a triangle up to similarity, or in terms of three edge lengths, which determines it up to Euclidean motion: these two specifications have different symmetry groups. Various theorems in geometry describe a situation, whereby a "conserved quantity", naturally associated with a geometric object, reduces the symmetry group of its defining properties: the original symmetry is broken.

Vessiot's theorem yields a classical example of symmetry breaking in differential geometry: if a surface can be deformed while preserving all properties relating to angle measurement and, at the same time, envelops a 1-parameter family of spheres, then it is piece of a cone, a cylinder or a surface of revolution. The first two properties only depend on an angle measurement, while being a cone, cylinder or surface of revolution depends on a length measurement. Thus symmetry breaking has occurred.

We will investigate relations between non-rigidity and symmetry breaking, in particular, whether deformability in more than one way invariably leads to symmetry breaking, thus generalizing Vessiot's theorem. However, our main concern will be to detect causes for symmetry breaking rather than just its occurrence, for example, by studying the appearance of the aforementioned "conserved quantities".

Publications.

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.

M Pember, W Rossman, K Saji, K Teramoto (2019)

Characterizing singularities of a surface in Lie sphere geometry; Hokkaido Math J 48(2), 281-308 (2019) DOI:10.14492/hokmj/1562810509, EPrint arXiv:1703.04257

Abstract

The conditions for a cuspidal edge, swallowtail and other fundamental singularities are given in the context of Lie sphere geometry. We then use these conditions to study the Lie sphere transformations of a surface.

M Pember (2020)

G-deformations of maps into projective space; Boll Unione Mat Ital 13, 275-292 (2020) DOI:10.1007/s40574-020-00218-9, EPrint arXiv:1712.06945

Abstract

$G$-deformability of maps into projective space is characterised by the existence of certain Lie algebra valued 1-forms. This characterisation gives a unified way to obtain well known results regarding deformability in different geometries.

M Pember (2020)

Weierstrass-type representations; Geom Dedicata 204, 299-309 (2020) DOI:10.1007/s10711-019-00456-y

Abstract

Weierstrass-type representations have been used extensively in surface theory to create surfaces with special curvature properties. In this paper we give a unified description of these representations in terms of classical transformation theory of $\Omega$-surfaces.

M Pember (2020)

Lie applicable surfaces; Commun Anal Geom 28, 1407-1450 (2020) DOI:10.4310/CAG.2020.v28.n6.a5, EPrint arXiv:1606.07205

Abstract

We give a detailed account of the gauge-theoretic approach to Lie applicable surfaces and the resulting transformation theory. In particular, we show that this approach coincides with the classical notion of $\Omega$- and $\Omega_0$-surfaces of Demoulin.

F Burstall, M Pember (2022)

Lie applicable surfaces and curved flats; manuscripta math 168, 525-533 (2022) DOI:10.1007/s00229-021-01304-8

Abstract

We investigate curved flats in Lie sphere geometry. We show that in this setting curved flats are in one-to-one correspondence with pairs of Demoulin families of Lie applicable surfaces related by Darboux transformation.

A Fuchs, U Hertrich-Jeromin, M Pember (2022)

Symmetry breaking in geometry; EPrint arXiv:2206.13401 (2022)

Abstract

A geometric mechanism that may, in analogy to similar notions in physics, be considered as "symmetry breaking" in geometry is described, and several instances of this mechanism in differential geometry are discussed: it is shown how spontaneous symmetry breaking may occur, and it is discussed how explicit symmetry breaking may be used to tackle certain geometric problems. A systematic study of symmetry breaking in geometry is proposed, and some preliminary thoughts on further research are formulated.

F Burstall, J Cho, U Hertrich-Jeromin, M Pember, W Rossman (2023)

Discrete $\Omega$-nets and Guichard nets via discrete Koenigs nets; Proc London Math Soc 126(2), 790-836 (2023) DOI:10.1112/plms.12499

Abstract

We provide a convincing discretisation of Demoulin's $\Omega$-surfaces along with their specialisations to Guichard and isothermic surfaces with no loss of integrable structure.

J Cho, M Pember, G Szewieczek (2023)

Constrained elastic curves and surfaces with spherical curvature lines; Indiana Univ Math J 72(5), 2059-2099 (2023) DOI:10.1512/iumj.2023.72.9487, EPrint arXiv:2104.11058

Abstract

In this paper, we consider surfaces with one or two families of spherical curvature lines. We show that every surface with a family of spherical curvature lines can locally be generated by a pair of initial data: a suitable curve of Lie sphere transformations and a spherical Legendre curve. We the provide conditions on the initial data for which such a surface is Lie applicable, an integrable class of of surfaces that includes cmc and pseudospherical surfaces. In particular, we show that a Lie applicable surface with exactly one family of spherical curvature lines must be generated by the lift of a constrained elastic curve in some space form. In view of this goal, we give a Lie sphere geometric characterisation of constrained elastic curves via polynomial conserved quantities of a certain family of connections.

U Hertrich-Jeromin, M Pember, D Polly (2023)

Channel linear Weingarten surfaces in space forms; Beitr Algebra Geom 64, 969-1009 (2023) DOI:10.1007/s13366-022-00664-w, EPrint arXiv:2105.00702

Abstract

Channel linear Weingarten surfaces in space forms are investigated in a Lie sphere geometric setting, which allows for a uniform treatment of different ambient geometries. We show that any channel linear Weingarten surface in a space form is isothermic and, in particular, a surface of revolution in its ambient space form. We obtain explicit parametrisations for channel surfaces of constant Gauss curvature in space forms, and thereby for a large class of linear Weingarten surfaces up to parallel transformation.

M Pember, D Polly, M Yasumoto (2023)

Discrete Weierstrass-Type Representations; Discrete Comput Geom 70, 816-844 (2023) DOI:10.1007/s00454-022-00439-z, EPrint arXiv:2105.06774

Abstract

Discrete Weierstrass-type representations yield a construction method in discrete differential geometry for certain classes of discrete surfaces. We show that the known discrete Weierstrass-type representations of certain surface classes can be viewed as applications of the $\Omega$-dual transform to lightlike Gauss maps in Laguerre geometry. From this construction, further Weierstrass-type representations arise. As an application of the techniques we develop, we show that all discrete linear Weingarten surfaces of Bryant or Bianchi type locally arise via Weierstrass-type representations from discrete holomorphic maps.