Ruled minimal surfaces in $\Bbb R^3$ with density $e^z$

TL;DR

This paper classifies ruled minimal surfaces in three-dimensional space with exponential density, showing only cylindrical ruled minimal surfaces exist and all translation minimal surfaces are ruled.

Contribution

It provides a complete classification of ruled minimal surfaces with density e^z, revealing the absence of noncylindrical examples and linking translation minimal surfaces to ruled surfaces.

Findings

No noncylindrical ruled minimal surfaces exist with density e^z.

A family of cylindrical ruled minimal surfaces is identified.

All translation minimal surfaces are ruled.

Abstract

We classify ruled minimal surfaces in $\Bbb R^3$ with density $e^z.$ It is showed that there is no noncylindrical ruled minimal surface and there is a family of cylindrical ruled minimal surfaces in $\Bbb R^3$ with density $e^z.$ It is also proved that all translation minimal surfaces are ruled.