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October 2020 Canonical coordinates and natural equations for minimal time-like surfaces in $\mathbf{R}^4_2$
Georgi Ganchev, Krasimir Kanchev
Kodai Math. J. 43(3): 524-572 (October 2020). DOI: 10.2996/kmj/1605063628

Abstract

We apply the complex analysis over the double numbers $\mathbf{D}$ to study the minimal time-like surfaces in $\mathbf{R}^4_2$. A minimal time-like surface which is free of degenerate points is said to be of general type. We divide the minimal time-like surfaces of general type into three types and prove that these surfaces admit special geometric (canonical) parameters. Then the geometry of the minimal time-like surfaces of general type is determined by the Gauss curvature $K$ and the curvature of the normal connection $\varkappa$, satisfying the system of natural equations for these surfaces. We prove the following: If $(K, \varkappa),\, K^2- \varkappa^2 > 0 $ is a solution to the system of natural equations, then there exists exactly one minimal time-like surface of the first type and exactly one minimal time-like surface of the second type with invariants $(K, \varkappa)$; if $(K, \varkappa),\, K^2- \varkappa^2$ < 0 is a solution to the system of natural equations, then there exists exactly one minimal time-like surface of the third type with invariants $(K, \varkappa)$.

Citation

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Georgi Ganchev. Krasimir Kanchev. "Canonical coordinates and natural equations for minimal time-like surfaces in $\mathbf{R}^4_2$." Kodai Math. J. 43 (3) 524 - 572, October 2020. https://doi.org/10.2996/kmj/1605063628

Information

Published: October 2020
First available in Project Euclid: 11 November 2020

MathSciNet: MR4173164
Digital Object Identifier: 10.2996/kmj/1605063628

Rights: Copyright © 2020 Tokyo Institute of Technology, Department of Mathematics

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Vol.43 • No. 3 • October 2020
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