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**Minimal surfaces and the Calabi-Penrose construction.
(Surfaces minimales et la construction de Calabi-Penrose.)**
*(French)*
Zbl 0734.53044

Sémin. Bourbaki, 36e année, Vol. 1983/84, Exp. No. 624, Astérisque 121-122, 197-211 (1985).

[For the entire collection see Zbl 0542.00005.]

In one of his important works, E. Calabi has described a method to study branched minimal immersions from \(S^ 2\) into \(S^ n(1)\) [see H. Rossi, Topics in complex manifolds, Suivi d’un texte d’Eugenio Calabi (1968; Zbl 0192.44001)]. His method was clarified by the introduction of an associated holomorphic curve by S. S. Chern that allowed a deeper understanding of such minimal immersions [S. S. Chern, Stud. and Essays Presented to Yu-Why Chen on his 60-th Birthday, 137–150 (1970; Zbl 0212.26402); the reviewer, Trans. Am. Math. Soc. 210, 75–106 (1975; Zbl 0328.53042)]. Recently, Calabi’s method was considered in connection with the construction introduced by Penrose in general relativity and has been used and extended to some other situations by mathematicians and physicists. In this work, the author fully describes the Calabi-Penrose construction and presents its application to obtain the important results of R. Bryant about the existence of minimal immersions of compact surfaces into \(S^ 4(1)\) and \(S^ 6(1)\) [R. L. Bryant, J. Differ. Geom. 17, 185–232 (1982; Zbl 0526.53055); ibid. 17, 455–473 (1982; Zbl 0498.53046)]. He also shows how the method has been applied to construct minimal maps from \(S^ 2\) into \(P^ n(c)\). The work is a very clear account of the subject. After it appeared, some new results related to the subject have been produced [e.g. S. S. Chern and J. Wolfson, Proc. Natl. Acad. Sci. USA 82, 2217–2219 (1985; Zbl 0601.58023)].

In one of his important works, E. Calabi has described a method to study branched minimal immersions from \(S^ 2\) into \(S^ n(1)\) [see H. Rossi, Topics in complex manifolds, Suivi d’un texte d’Eugenio Calabi (1968; Zbl 0192.44001)]. His method was clarified by the introduction of an associated holomorphic curve by S. S. Chern that allowed a deeper understanding of such minimal immersions [S. S. Chern, Stud. and Essays Presented to Yu-Why Chen on his 60-th Birthday, 137–150 (1970; Zbl 0212.26402); the reviewer, Trans. Am. Math. Soc. 210, 75–106 (1975; Zbl 0328.53042)]. Recently, Calabi’s method was considered in connection with the construction introduced by Penrose in general relativity and has been used and extended to some other situations by mathematicians and physicists. In this work, the author fully describes the Calabi-Penrose construction and presents its application to obtain the important results of R. Bryant about the existence of minimal immersions of compact surfaces into \(S^ 4(1)\) and \(S^ 6(1)\) [R. L. Bryant, J. Differ. Geom. 17, 185–232 (1982; Zbl 0526.53055); ibid. 17, 455–473 (1982; Zbl 0498.53046)]. He also shows how the method has been applied to construct minimal maps from \(S^ 2\) into \(P^ n(c)\). The work is a very clear account of the subject. After it appeared, some new results related to the subject have been produced [e.g. S. S. Chern and J. Wolfson, Proc. Natl. Acad. Sci. USA 82, 2217–2219 (1985; Zbl 0601.58023)].

Reviewer: J. L. Marques Barbosa (MR 87a:58043)

### MSC:

53C42 | Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) |

58E12 | Variational problems concerning minimal surfaces (problems in two independent variables) |