Numerically Destabilizing Minimal Discs

Abstract

When calculating the index of a minimal surface, the set of smooth functions on a domain with compact support is the standard setting to describe admissible variations. We show that the set of admissible variations can be widened in a geometrically meaningful manner leading to a more general notion of index. This allows us to produce explicit examples of destabilizing perturbations for the fundamental Scherk surface. For the dihedral Enneper surfaces we show that both the classical and modified index can be explicitly determined.

Keywords:

• Stability of minimal surfaces
• numerical spectral techniques
• bounded index

Acknowledgments

The authors thank the Dept. of Mathematics at Cal. State Fullerton for encouraging undergraduate student research, as well as Davi Maximo and David Wiygul for their interest and helpful comments.

Declaration of interest

No potential conflict of interest was reported by the authors.

Notes

1 The subscript b in the notation denotes boundary, where the subscript c in ${\text{ind}}_c(\Sigma )$ stands for compact.