  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.
