Singular Minimal Surfaces which are Minimal

Year 2021, Volume: 4 Issue: 4, 136 - 146, 30.12.2021

Muhittin Evren Aydın , Ayla Erdur Kara , Mahmut Ergüt

https://doi.org/10.32323/ujma.984462

Abstract

In the present paper, we discuss the singular minimal surfaces in Euclidean $3-$space $\mathbb{R}^{3}$ which are minimal. Such a surface is nothing but a plane, a trivial outcome. However, a non-trivial outcome is obtained when we modify the usual condition of singular minimality by using a special semi-symmetric metric connection instead of the Levi-Civita connection on $\mathbb{R}^{3}$. With this new connection, we prove that, besides planes, the singular minimal surfaces which are minimal are the generalized cylinders, providing their explicit equations. A trivial outcome is observed when we use a special semi-symmetric non-metric connection. Furthermore, our discussion is adapted to the Lorentz-Minkowski 3-space.

Keywords

Minimal surface, Semi-symmetric connection, Singular minimal surface, Translation surface

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