Numerically destabilizing minimal discs

TL;DR

This paper introduces a generalized notion of index for minimal surfaces by expanding the set of admissible variations, enabling explicit destabilizing perturbations for certain classical minimal surfaces.

Contribution

It proposes a new, geometrically meaningful way to widen admissible variations for minimal surface index calculations, with explicit examples and computations.

Findings

Explicit destabilizing perturbations for Scherk and dihedral Enneper surfaces.

Both classical and new indices can be explicitly computed for dihedral Enneper surfaces.

The new index provides a broader understanding of minimal surface stability.

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 by considering the difference of area functional, leading to a more general notion of index. This allows us to produce explicit examples of destabilizing perturbations for the fundamental Scherk surface and dihedral Enneper surfaces. In the case of dihedral Enneper surfaces we show that both the classical and our modified index can be explicitly determined.