Geodesic
Weierstrass Representation for Discrete Isotropic Surfaces in $\mathbb{R}^{2,1}$, $\mathbb{R}^{3,1}$, and $\mathbb{R}^{2,2}$
Funkcionalʹnyj analiz i ego priloženiâ, Tome 45 (2011) no. 1, pp. 31-40 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

Using an integrable discrete Dirac operator, we construct a discrete version of the Weierstrass representation for hyperbolic surfaces parameterized along isotropic directions in $\mathbb{R}^{2,1}$, $\mathbb{R}^{3,1}$, and $\mathbb{R}^{2,2}$. The corresponding discrete surfaces have isotropic edges. We show that any discrete surface satisfying a general monotonicity condition and having isotropic edges admits such a representation.
Keywords: integrable system, discretization, discrete differential geometry. 
@article{FAA_2011_45_1_a2,
     author = {D. V. Zakharov},
     title = {Weierstrass {Representation} for {Discrete} {Isotropic} {Surfaces} in $\mathbb{R}^{2,1}$, $\mathbb{R}^{3,1}$, and~$\mathbb{R}^{2,2}$},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
     pages = {31--40},
     year = {2011},
     volume = {45},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FAA_2011_45_1_a2/}
}
TY  - JOUR
AU  - D. V. Zakharov
TI  - Weierstrass Representation for Discrete Isotropic Surfaces in $\mathbb{R}^{2,1}$, $\mathbb{R}^{3,1}$, and $\mathbb{R}^{2,2}$
JO  - Funkcionalʹnyj analiz i ego priloženiâ
PY  - 2011
SP  - 31
EP  - 40
VL  - 45
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/FAA_2011_45_1_a2/
LA  - ru
ID  - FAA_2011_45_1_a2
ER  - 
%0 Journal Article
%A D. V. Zakharov
%T Weierstrass Representation for Discrete Isotropic Surfaces in $\mathbb{R}^{2,1}$, $\mathbb{R}^{3,1}$, and $\mathbb{R}^{2,2}$
%J Funkcionalʹnyj analiz i ego priloženiâ
%D 2011
%P 31-40
%V 45
%N 1
%U http://geodesic.mathdoc.fr/item/FAA_2011_45_1_a2/
%G ru
%F FAA_2011_45_1_a2
D. V. Zakharov. Weierstrass Representation for Discrete Isotropic Surfaces in $\mathbb{R}^{2,1}$, $\mathbb{R}^{3,1}$, and $\mathbb{R}^{2,2}$. Funkcionalʹnyj analiz i ego priloženiâ, Tome 45 (2011) no. 1, pp. 31-40. http://geodesic.mathdoc.fr/item/FAA_2011_45_1_a2/

[1] B. G. Konopelchenko, I. A. Taimanov, Generalized Weierstrass formulae, soliton equations and Willmore surfaces. I. Tori of revolution and the mKDV equation, arXiv: dg-ga/9506011

[2] I. A. Taimanov, “Konechnozonnye resheniya modifitsirovannykh uravnenii Veselova–Novikova, ikh spektralnye svoistva i prilozheniya”, Sib. matem. zhurn., 40:6 (1999), 1352–1363 | MR | Zbl

[3] I. A. Taimanov, “Predstavlenie Veiershtrassa zamknutykh poverkhnostei v $\mathbb{R}^3$”, Funkts. analiz i ego pril., 32:4 (1998), 49–62 | DOI | MR | Zbl

[4] B. G. Konopelchenko, “Induced surfaces and their integrable dynamics”, Stud. Appl. Math., 96:1 (1996), 9–52 | DOI | MR

[5] I. A. Taimanov, “Dvumernyi operator Diraka i teoriya poverkhnostei”, UMN, 61 (2006), 85–164 | DOI | MR | Zbl

[6] B. G. Konopelchenko, G. Landolfi, “Induced surfaces and their integrable dynamics. II. Generalized Weierstrass representations in 4D spaces and deformations via DS hierarchy”, Stud. Appl. Math., 104:2 (2000), 129–169 | DOI | MR | Zbl

[7] B. G. Konopelchenko, G. Landolfi, “Generalized Weierstrass representation for surfaces in multidimensional Riemann spaces”, J. Geom. Phys., 29:4 (1999), 319–333 ; http://arxiv.org/abs/math/9804144 | DOI | MR | Zbl

[8] B. G. Konopelchenko, “Weierstrass representations for surfaces in 4D spaces and their integrable deformations via DS hierarchy”, Ann. Global Anal. Geom., 18:1 (2000), 61–74 | DOI | MR | Zbl

[9] A. I. Bobenko, Yu. B. Suris, Discrete differential geometry. Integrable structure, Graduate Studies in Math., 98, Amer. Math. Soc., Providence, RI, 2008 | DOI | MR | Zbl

[10] D. V. Zakharov, A discrete analogue of the modified Novikov–Veselov hierarchy, arXiv: 0904.3728