| Title:
|
Cauchy problems for discrete affine minimal surfaces (English) |
| Author:
|
Craizer, Marcos |
| Author:
|
Lewiner, Thomas |
| Author:
|
Teixeira, Ralph |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
48 |
| Issue:
|
1 |
| Year:
|
2012 |
| Pages:
|
1-14 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
discrete differential geometry |
| Keyword:
|
discrete affine minimal surfaces |
| Keyword:
|
discrete conjugate nets |
| Keyword:
|
PQ meshes |
| MSC:
|
39A12 |
| MSC:
|
52C99 |
| MSC:
|
53A15 |
| idMR:
|
MR2915845 |
| DOI:
|
10.5817/AM2012-1-1 |
| . |
| Date available:
|
2012-03-15T18:04:28Z |
| Last updated:
|
2013-09-19 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/142085 |
| . |
| Reference:
|
[1] Aledo, J. A., Chaves, R. M. B., Gálvez, J. A.: The Cauchy problem for improper affine spheres and the Hessian one equation.Trans. Amer. Math. Soc. 359 (9) (2007), 4183–4208. Zbl 1121.53008, MR 2309181, 10.1090/S0002-9947-07-04378-4 |
| Reference:
|
[2] Aledo, J. A., Martínez, A., Milán, F.: The affine Cauchy problem.J. Math. Anal. Appl. 351 (2009), 70–83. Zbl 1161.53012, MR 2472921, 10.1016/j.jmaa.2008.09.055 |
| Reference:
|
[3] Bobenko, A. I., Schief, W. K.: Affine spheres: Discretization via duality relations.Experiment. Math. 8 (1999), no. 3, 261–280. Zbl 0972.53012, MR 1724159, 10.1080/10586458.1999.10504404 |
| Reference:
|
[4]
: Discrete Differential Geometry.Oberwolfach Seminars, vol. 38, Birkhauser, 2008. Zbl 1185.68876, MR 2407724 |
| Reference:
|
[5] Bobenko, A. I., Suris, Y. B.: Discrete Differential Geometry: Integrable Structure.Graduate Studies in Mathematics, Vol. 98, AMS, 2008. Zbl 1158.53001, MR 2467378 |
| Reference:
|
[6] Calabi, E.: Hypersurfaces with maximal affinely invariant area.Amer. J. Math. 104 (1982), 91–126. Zbl 0501.53037, MR 0648482, 10.2307/2374069 |
| Reference:
|
[7] Calabi, E.: Convex affine maximal surfaces.Results Math. 13 (1988), 199–223. Zbl 0653.53006, MR 0941331, 10.1007/BF03323241 |
| Reference:
|
[8] Craizer, M., Anciaux, H., Lewiner, T. M.: Discrete affine minimal surfaces with indefinite metric.Differential Geom. Appl. (2009). MR 2594460, 10.1016/j.difgeo.2009.07.004 |
| Reference:
|
[9] Craizer, M., da Silva Moacyr, A. H. B., Teixeira, R. C.: Area distances of convex plane curves and iImproper affine spheres.SIAM J. Imaging Sci. 1 (3) (2008), 209–227. MR 2486030, 10.1137/080714610 |
| Reference:
|
[10] Matsuura, N.: A discrete analogue of the affine Backlund transformation.Fukuoka Univ. Sci. Rep. 40 (2) (2010), no. 2, 163–173. Zbl 1227.39005, MR 2766406 |
| Reference:
|
[11] Matsuura, N., Urakawa, H.: Discrete improper affine spheres.J. Geom. Phys. 45 (1–2) (2003), 164–183. Zbl 1035.53022, MR 1949349, 10.1016/S0393-0440(02)00134-1 |
| Reference:
|
[12] Nomizu, K., Sasaki, T.: Affine Differential Geometry.Cambridge University Press, 1994. Zbl 0834.53002, MR 1311248 |
| . |
