Singular minimal surfaces which are minimal

TL;DR

This paper explores singular minimal surfaces in Euclidean and Lorentz-Minkowski spaces, revealing new classes of minimal surfaces such as generalized cylinders under modified connection conditions.

Contribution

It introduces a novel approach using semi-symmetric metric connections to identify non-trivial singular minimal surfaces beyond planes.

Findings

Planes are minimal singular surfaces in Euclidean space.

Generalized cylinders are also minimal under semi-symmetric metric connections.

Different connections lead to distinct classes of minimal surfaces.

Abstract

In the present paper, we discuss the singular minimal surfaces in a Euclidean 3-space R^{3} which are minimal. In fact, 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 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.