Higher dimensional Schwarz's surfaces and Scherk's surfaces

TL;DR

This paper constructs higher-dimensional analogs of classical minimal surfaces, specifically Schwarz's and Scherk's surfaces, as complete embedded periodic minimal hypersurfaces in Euclidean spaces.

Contribution

It introduces new higher-dimensional periodic minimal hypersurfaces generalizing well-known classical surfaces.

Findings

Constructed higher-dimensional Schwarz's P- and D-surfaces.

Developed higher-dimensional Scherk's second surface.

Established these surfaces as complete embedded periodic hypersurfaces.

Abstract

Higher dimensional generalizations of Schwarz's $P$-surface, Schwarz's $D$-surface and Scherk's second surface are constructed as complete embedded periodic minimal hy- persurfaces in $\mathbb R^n$.