On the Combinatorics of Demoulin Transforms and (Discrete) Projective Minimal Surfaces

McCarthy, Alan J.; Schief, W. K.

doi:10.1007/s00454-016-9827-x

Cited by 3 publications

(9 citation statements)

References 12 publications

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“…It turns out that the notion of asymptotic correspondence gives rise to a privileged class of surfaces. Thus, the definition proposed in [21] is adopted.…”

Section: Geometric Classification Of Projective Minimal Surfacesmentioning

confidence: 99%

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Discrete projective minimal surfaces

McCarthy

Schief

2018

Advances in Mathematics

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We propose a natural discretisation scheme for classical projective minimal surfaces. We follow the classical geometric characterisation and classification of projective minimal surfaces and introduce at each step canonical discrete models of the associated geometric notions and objects. Thus, we introduce discrete analogues of classical Lie quadrics and their envelopes and classify discrete projective minimal surfaces according to the cardinality of the class of envelopes. This leads to discrete versions of Godeaux-Rozet, Demoulin and Tzitzéica surfaces. The latter class of surfaces requires the introduction of certain discrete line congruences which may also be employed in the classification of discrete projective minimal surfaces. The classification scheme is based on the notion of discrete surfaces which are in asymptotic correspondence. In this context, we set down a discrete analogue of a classical theorem which states that an envelope (of the Lie quadrics) of a surface is in asymptotic correspondence with the surface if and only if the surface is either projective minimal or a Q surface. Accordingly, we present a geometric definition of discrete Q surfaces and their relatives, namely discrete counterparts of classical semi-Q, complex, doubly Q and doubly complex surfaces. arXiv:1801.08428v1 [math.DG]

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“…It turns out that the notion of asymptotic correspondence gives rise to a privileged class of surfaces. Thus, the definition proposed in [21] is adopted.…”

Section: Geometric Classification Of Projective Minimal Surfacesmentioning

confidence: 99%

“…Projective minimal surfaces may be classified in terms of the number of distinct (additional) envelopes of the Lie quadrics [10,20,21]. If two envelopes are the same then the projective minimal surface Σ is of Godeaux-Rozet type.…”

mentioning

confidence: 99%

Discrete projective minimal surfaces

McCarthy

Schief

2018

Advances in Mathematics

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“…We now state a classical theorem [14,19] which lies at the heart of the geometric definition and analysis of discrete projective minimal surfaces and adopt a definition proposed in [18]. Definition 2.7.…”

Section: )mentioning

confidence: 99%

“…It turns out that any discrete asymptotic net gives rise to a one-parameter family of lattices of Lie quadrics [16,18]. Thus, if Q is a quadric associated with a quadrilateral = [r, r 1 , r 2 , r 12 ], then the C 1 condition uniquely determines quadrics Q 1 and Q 2 associated with the quadrilaterals 1 and 2 , respectively.…”

Section: Discrete Surfaces In Pmentioning

confidence: 99%

“…The latter have been used extensively [15] as discretisations of asymptotic nets on hyperbolic surfaces in discrete differential geometry. They naturally give rise to a discretisation of classical Lie quadrics [17,18] to which the discrete canonical frame is adapted. It is observed that this frame does not appear to have an analogue in the continuous setting.…”

Section: Introductionmentioning

confidence: 99%

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Surface theory in discrete projective differential geometry. I. A canonical frame and an integrable discrete Demoulin system

Schief

Szereszewski

2018

Proc. R. Soc. A.

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We present the first steps of a procedure which discretises surface theory in classical projective differential geometry in such a manner that underlying integrable structure is preserved. We propose a canonical frame in terms of which the associated projective Gauss-Weingarten and Gauss-Mainardi-Codazzi equations adopt compact forms. Based on a scaling symmetry which injects a parameter into the linear Gauss-Weingarten equations, we set down an algebraic classification scheme of discrete projective minimal surfaces which turns out to admit a geometric counterpart formulated in terms of discrete notions of Lie quadrics and their envelopes. In the case of discrete Demoulin surfaces, we derive a Bäcklund transformation for the underlying discrete Demoulin system and show how the latter may be formulated as a two-component generalisation of the integrable discrete Tzitzéica equation which has originally been derived in a different context. At the geometric level, this connection leads to the retrieval of the standard discretisation of affine spheres in affine differential geometry.

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Discrete projective minimal surfaces

McCarthy

Schief

2018

Preprint

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On the Combinatorics of Demoulin Transforms and (Discrete) Projective Minimal Surfaces

Cited by 3 publications

References 12 publications

Discrete projective minimal surfaces

Discrete projective minimal surfaces

Surface theory in discrete projective differential geometry. I. A canonical frame and an integrable discrete Demoulin system

Discrete projective minimal surfaces

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