Minimal Surfaces with Catenoid Ends

TL;DR

This paper proves the existence of new non-periodic symmetric minimal surfaces with finite total curvature and embedded catenoid ends, expanding the understanding of minimal surface structures.

Contribution

It introduces a conjugate surface construction method to establish the existence of symmetric minimal surfaces with positive genus and catenoid ends, extending prior classifications.

Findings

Existence of non-periodic symmetric minimal surfaces with catenoid ends

Surfaces have finite total curvature and positive genus

Maintains symmetry similar to genus-zero counterparts

Abstract

In this paper, we use the conjugate surface construction to prove the existence of certain non-periodic symmetric immersed minimal surfaces. These surfaces have finite total curvature and embedded catenoid ends, and they have positive genus yet maintain the symmetry of their genus-zero counterparts constructed by Jorge-Meeks and Xu.