Barrier methods for minimal submanifolds in the Gibbons-Hawking ansatz

TL;DR

This paper develops a barrier method to study compact minimal submanifolds within hyperkähler 4-manifolds constructed via the Gibbons-Hawking ansatz, advancing classification efforts in this geometric context.

Contribution

It introduces a barrier argument for minimal submanifolds in multi-Eguchi-Hanson and multi-Taub-NUT spaces, and establishes a converse relating stability to convexity of the distance function.

Findings

Barrier method effectively constrains minimal submanifolds.

Results contribute to classification of minimal submanifolds.

Proves a converse linking stability and convexity.

Abstract

We describe a barrier argument for compact minimal submanifolds in the multi-Eguchi-Hanson and in the multi-Taub-NUT spaces, which are hyperkaehler 4-manifolds given by the Gibbons-Hawking ansatz. This approach is used to obtain results towards a classification of compact minimal submanifolds in this setting. We also prove a converse of Tsai and Wang's result that relates the strong stability condition to the convexity of the distance function.