Cauchy problems for discrete affine minimal surfaces

Marcos Craizer; Thomas Lewiner; Ralph Teixeira

Archivum Mathematicum (2012)

  • Volume: 048, Issue: 1, page 1-14
  • ISSN: 0044-8753

Abstract

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In this paper we discuss planar quadrilateral (PQ) nets as discrete models for convex affine surfaces. As a main result, we prove a necessary and sufficient condition for a PQ net to admit a Lelieuvre co-normal vector field. Particular attention is given to the class of surfaces with discrete harmonic co-normals, which we call discrete affine minimal surfaces, and the subclass of surfaces with co-planar discrete harmonic co-normals, which we call discrete improper affine spheres. Within this classes, we show how to solve discrete Cauchy problems analogous to the Cauchy problems for smooth analytic improper affine spheres and smooth analytic affine minimal surfaces.

How to cite

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Craizer, Marcos, Lewiner, Thomas, and Teixeira, Ralph. "Cauchy problems for discrete affine minimal surfaces." Archivum Mathematicum 048.1 (2012): 1-14. <http://eudml.org/doc/247062>.

@article{Craizer2012,
abstract = {In this paper we discuss planar quadrilateral (PQ) nets as discrete models for convex affine surfaces. As a main result, we prove a necessary and sufficient condition for a PQ net to admit a Lelieuvre co-normal vector field. Particular attention is given to the class of surfaces with discrete harmonic co-normals, which we call discrete affine minimal surfaces, and the subclass of surfaces with co-planar discrete harmonic co-normals, which we call discrete improper affine spheres. Within this classes, we show how to solve discrete Cauchy problems analogous to the Cauchy problems for smooth analytic improper affine spheres and smooth analytic affine minimal surfaces.},
author = {Craizer, Marcos, Lewiner, Thomas, Teixeira, Ralph},
journal = {Archivum Mathematicum},
keywords = {discrete differential geometry; discrete affine minimal surfaces; discrete conjugate nets; PQ meshes; discrete differential geometry; discrete affine minimal surface; discrete conjugate nets; PQ meshes; Cauchy problems; affine improper spheres},
language = {eng},
number = {1},
pages = {1-14},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Cauchy problems for discrete affine minimal surfaces},
url = {http://eudml.org/doc/247062},
volume = {048},
year = {2012},
}

TY - JOUR
AU - Craizer, Marcos
AU - Lewiner, Thomas
AU - Teixeira, Ralph
TI - Cauchy problems for discrete affine minimal surfaces
JO - Archivum Mathematicum
PY - 2012
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 048
IS - 1
SP - 1
EP - 14
AB - In this paper we discuss planar quadrilateral (PQ) nets as discrete models for convex affine surfaces. As a main result, we prove a necessary and sufficient condition for a PQ net to admit a Lelieuvre co-normal vector field. Particular attention is given to the class of surfaces with discrete harmonic co-normals, which we call discrete affine minimal surfaces, and the subclass of surfaces with co-planar discrete harmonic co-normals, which we call discrete improper affine spheres. Within this classes, we show how to solve discrete Cauchy problems analogous to the Cauchy problems for smooth analytic improper affine spheres and smooth analytic affine minimal surfaces.
LA - eng
KW - discrete differential geometry; discrete affine minimal surfaces; discrete conjugate nets; PQ meshes; discrete differential geometry; discrete affine minimal surface; discrete conjugate nets; PQ meshes; Cauchy problems; affine improper spheres
UR - http://eudml.org/doc/247062
ER -

References

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  9. Craizer, M., da Silva Moacyr, A. H. B., Teixeira, R. C., 10.1137/080714610, SIAM J. Imaging Sci. 1 (3) (2008), 209–227. (2008) MR2486030DOI10.1137/080714610
  10. Matsuura, N., A discrete analogue of the affine Backlund transformation, Fukuoka Univ. Sci. Rep. 40 (2) (2010), no. 2, 163–173. (2010) Zbl1227.39005MR2766406
  11. Matsuura, N., Urakawa, H., 10.1016/S0393-0440(02)00134-1, J. Geom. Phys. 45 (1–2) (2003), 164–183. (2003) Zbl1035.53022MR1949349DOI10.1016/S0393-0440(02)00134-1
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