Quaternionic maps and minimal surfaces

Chen, Jingyi; Li, Jiayu

Chen, Jingyi; Li, Jiayu

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 4 (2005) no. 3, p. 375-388

Abstract

Let $(M, J^{α}, α = 1, 2, 3)$ and $(N, 𝒥^{α}, α = 1, 2, 3)$ 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.

MR 2185957 | Zbl 1170.53312

Classification: 53C26, 53C43, 58E12, 58E20

@article{ASNSP_2005_5_4_3_375_0,
     author = {Chen, Jingyi and Li, Jiayu},
     title = {Quaternionic maps and minimal surfaces},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 4},
     number = {3},
     year = {2005},
     pages = {375-388},
     zbl = {1170.53312},
     mrnumber = {2185957},
     language = {en},
     url = {http://www.numdam.org/item/ASNSP_2005_5_4_3_375_0}
}

Chen, Jingyi; Li, Jiayu. Quaternionic maps and minimal surfaces. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 4 (2005) no. 3, pp. 375-388. http://www.numdam.org/item/ASNSP_2005_5_4_3_375_0/

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