Weierstrass Representation for Discrete Isotropic Surfaces in $\mathbb{R}^{2,1}$, $\mathbb{R}^{3,1}$, and~$\mathbb{R}^{2,2}$

D. V. Zakharov

Geodesic

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Weierstrass Representation for Discrete Isotropic Surfaces in $\mathbb{R}^{2,1}$, $\mathbb{R}^{3,1}$, and~$\mathbb{R}^{2,2}$

D. V. Zakharov

Funkcionalʹnyj analiz i ego priloženiâ, Tome 45 (2011) no. 1, pp. 31-40

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Résumé

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.

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     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},
     publisher = {mathdoc},
     volume = {45},
     number = {1},
     year = {2011},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FAA_2011_45_1_a2/}
}

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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/