Generalized Weierstrass-Enneper representations of Euclidean, spacelike, and timelike surfaces: a unified Lie-algebraic formulation

DOI10.1007/s00022-016-0358-7zbMath1377.53007OpenAlexW2552781618MaRDI QIDQ2410136

Barbara A. Shipman, Patrick D. Shipman, David Packard

Publication date: 17 October 2017

Published in: Journal of Geometry (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1007/s00022-016-0358-7

zbMATH Keywords

minimal surfacemoving framespace-like surfacetime-like surfaceWeierstrass-Enneper representationLie-algebraic formulation

Mathematics Subject Classification ID

Minimal surfaces in differential geometry, surfaces with prescribed mean curvature (53A10) Surfaces in Euclidean and related spaces (53A05) Non-Euclidean differential geometry (53A35) Lie algebras and Lie superalgebras (17B99)

Related Items (4)

Timelike surfaces in the de Sitter space \(\mathbb{S}_1^3(1)\subset \mathbb{R}_1^4\) ⋮ Timelike minimal surfaces in the three-dimensional Heisenberg group ⋮ Weierstrass representations for triply orthogonal and conformal Euclidean and Lorentzian systems ⋮ Explicit solving of the system of natural PDEs of minimal Lorentz surfaces in \(\mathbb{R}_2^4\)

Cites Work

• Maximal surfaces in the 3-dimensional Minkowski space \(L^ 3\)
• The generalized Weierstrass system inducing surfaces of constant and nonconstant mean curvature in Euclidean three space
• Timelike surfaces in Lorentz 3-space with prescribed mean curvature and Gauss map
• Weierstrass formula for surfaces of prescribed mean curvature
• The problem of nontrivial isometries of surfaces preserving principal curvatures
• On the construction of integrable surfaces on Lie groups
• A Weierstrass representation for minimal surfaces in 3-dimensional manifolds
• The Gauss map and spacelike surfaces with prescribed mean curvature in Minkowski 3-space
• Surfaces on Lie groups, on Lie algebras, and their integrability (with an appendix by Juan Carlos Alvarez Paiva)
• An introduction to Lorentz surfaces
• Lorentz-conformal transformations in the plane
• An Elementary Proof of the Existence of Isothermal Parameters on a Surface
• Constant mean curvature surfaces via an integrable dynamical system
• A Weierstrass representation theorem for Lorentz surfaces
• Maximal surfaces of Riemann type in Lorentz-Minkowski space \(\mathbb L^3\)

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