Periodic minimal surfaces in semidirect products

TL;DR

This paper proves the existence of complete minimal surfaces in certain semidirect product spaces, including the Heisenberg space and Sol3, extending classical minimal surface concepts to these non-Euclidean geometries.

Contribution

It introduces new minimal surfaces in semidirect product spaces, generalizing Scherk surfaces to non-Euclidean geometries like Heisenberg and Sol3.

Findings

Existence of complete minimal surfaces in Heisenberg space.

Existence of such surfaces in Sol3 with a family of metrics.

Surfaces resemble classical doubly and singly periodic Scherk surfaces.

Abstract

In this paper we prove existence of complete minimal surfaces in some metric semidirect products. These surfaces are similar to the doubly and singly periodic Scherk minimal surfaces in $\mathbb R^3$. In particular, we obtain these surfaces in the Heisenberg space with its canonical metric, and in Sol3 with a one-parameter family of non-isometric metrics.