An example of a minimal surface of genus one with two catenoid ends and one Enneper end

TL;DR

This paper constructs a specific complete minimal surface of genus one with three ends, combining catenoid and Enneper types, using advanced mathematical techniques to solve the period problem.

Contribution

It provides a new explicit example of a minimal surface with unique end configurations and genus, expanding the catalog of known minimal surfaces.

Findings

Constructed a genus one minimal surface with two catenoid ends and one Enneper end.

Demonstrated the existence of this surface using Weierstrass representation and elliptic functions.

Solved the period problem explicitly for this surface.

Abstract

In this paper we construct an example of a complete immersed minimal surface in $\mathbb{R}^3$ of genus one with two embedded catenoid-type ends, one Enneper-type end and total Gauss curvature $-16\pi.$ The proof of the existence of this example, was obtained using the Weierstrass representation, the theory of elliptic functions and explicitly solving the period problem.