Quaternionic maps and minimal surfaces

Chen, Jingyi; Li, Jiayu

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 4 (2005) no. 3, pp. 375-388.

Let $(M,J^{\alpha },\alpha =1,2,3)$ and $(N,{𝒥}^{\alpha },\alpha =1,2,3)$ be hyperkähler manifolds. We study stationary quaternionic maps between $M$ and $N$. We first show that if there are no holomorphic 2-spheres in the target then any sequence of stationary quaternionic maps with bounded energy subconverges to a stationary quaternionic map strongly in $W^{1,2}(M,N)$. We then find that certain tangent maps of quaternionic maps give rise to an interesting minimal 2-sphere. At last we construct a stationary quaternionic map with a codimension-3 singular set by using the embedded minimal ${𝕊}^2$ in the hyperkähler surface ${\tilde{M}}_2^0$ studied by Atiyah-Hitchin.

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