Quaternionic maps and minimal surfaces

Jingyi Chen[1]; Jiayu Li[2]

  • [1] Department of Mathematics The University of British Columbia Vancouver, BC, Canada V6T 1Z2
  • [2] Math. Group The abdus salam ICTP Trieste 34100 Italy and Academy of Mathematics and Systems Sciences Chinese Academy of Sciences Beijing 100080, P. R. of China

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (2005)

  • Volume: 4, Issue: 3, page 375-388
  • ISSN: 0391-173X

Abstract

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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 M ˜ 2 0 studied by Atiyah-Hitchin.

How to cite

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Chen, Jingyi, and Li, Jiayu. "Quaternionic maps and minimal surfaces." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 4.3 (2005): 375-388. <http://eudml.org/doc/84564>.

@article{Chen2005,
abstract = {Let $(M, J^\alpha , \alpha =1,2,3)$ and $(N, \{\mathcal \{J\}\}^\alpha , \alpha =1,2,3)$ 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 $\{\mathbb \{S\}\}^2$ in the hyperkähler surface $\widetilde\{M\}^0_2$ studied by Atiyah-Hitchin.},
affiliation = {Department of Mathematics The University of British Columbia Vancouver, BC, Canada V6T 1Z2; Math. Group The abdus salam ICTP Trieste 34100 Italy and Academy of Mathematics and Systems Sciences Chinese Academy of Sciences Beijing 100080, P. R. of China},
author = {Chen, Jingyi, Li, Jiayu},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
language = {eng},
number = {3},
pages = {375-388},
publisher = {Scuola Normale Superiore, Pisa},
title = {Quaternionic maps and minimal surfaces},
url = {http://eudml.org/doc/84564},
volume = {4},
year = {2005},
}

TY - JOUR
AU - Chen, Jingyi
AU - Li, Jiayu
TI - Quaternionic maps and minimal surfaces
JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY - 2005
PB - Scuola Normale Superiore, Pisa
VL - 4
IS - 3
SP - 375
EP - 388
AB - Let $(M, J^\alpha , \alpha =1,2,3)$ and $(N, {\mathcal {J}}^\alpha , \alpha =1,2,3)$ 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 ${\mathbb {S}}^2$ in the hyperkähler surface $\widetilde{M}^0_2$ studied by Atiyah-Hitchin.
LA - eng
UR - http://eudml.org/doc/84564
ER -

References

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