We describe a barrier argument for compact minimal submanifolds in the multi-Eguchi-Hanson and in the multi-Taub-NUT spaces, which are hyperkähler 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.

The author wishes to thank his supervisor Jason D. Lotay for suggesting this project and for his enormous help in its development. The author would also like to thank the referee for the useful comments. This work was supported by the Oxford-Thatcher Graduate Scholarship.