Singular Minimal Surfaces which are Minimal

Authors : Muhittin Evren Aydin, Ayla Erdur Kara, Mahmut Ergüt

Pages : 136-146

Doi:10.32323/ujma.984462

View : 29 | Download : 9

Publication Date : 2021-12-30

Article Type : Research

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