Non-compact families of complete, properly immersed minimal immersions with fixed topology via desingularization

TL;DR

This paper constructs large families of complete, properly immersed minimal surfaces in three-dimensional space with fixed topology, demonstrating their existence for all large genus and their degeneration to the plane at certain scales.

Contribution

It introduces a method to generate families of minimal surfaces with fixed topology and symmetry, expanding the known catalog of such surfaces and their degenerations.

Findings

Existence of minimal surfaces for all large genus

Surfaces exhibit dihedral symmetries

Surfaces degenerate to the plane at certain scales

Abstract

For fixed large genus, we construct families of complete immersed minimal surfaces in R3 with four ends and dihedral symmetries. The families exist for all large genus and at an appropriate scale degenerate to the plane.