Celebratio Mathematica

incollection S.-S. Chern and J. G. Wolfson: “A simple proof of Frobenius theorem,” pp. 67–69 in Manifolds and Lie groups: Papers in honor of Yozô Matsushima. Edited by J.-I. Hano. Progress in Mathematics 14. Birkhäuser (Boston, MA), 1981. MR 642852 Zbl 0481.58007

Jun-Ichi Hano	Related
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article S. S. Chern and J. G. Wolfson: “Minimal surfaces by moving frames,” Am. J. Math. 105 : 1 (February 1983), pp. 59–83. MR 692106 Zbl 0521.53050

In this paper we will develop, by moving frames, the basic formulas and some results on minimal surfaces, both in a Riemannian and in a Kählerian manifold. When the ambient Riemannian manifold is of constant curvature, we will show that a quartic form defined in a natural way from the second fundamental form is holomorphic, relative to the underlying complex structure of the surface. This will be applied to the study of minimal surfaces on the unit 4-sphere \( S^4(1) \). We will show how to derive by this approach a theorem of R. Bryant that a compact Riemann surface can be conformally immersed as a minimal surface in \( S^4(1) \).

When the ambient space is a Kählerian manifold, the first-order jet of the surface defines a scalar invariant, which is the Kähler form divided by the element of area of the surface. This will be used to normalize the frames attached to the surface. When the surface is minimal, an invariant cubic form appears naturally. We will show that it is holomorphic, if the ambient Kählerian manifold is of constant holomorphic sectional curvature. This will be applied to the complex projective plane \( P_2(\mathbf{C}) \) (with the Fubini–Study metric), to give a family of minimal surfaces in \( P_2(\mathbf{C}) \), and all its minimal two-spheres. A determination of all the minimal 2-spheres in \( P_n(\mathbf{C}) \) method has been carried out by the second author [1982]. This latter problem was first studied by Din–Zakrewski [1980] and a mathematician’s treatment was given by Eels–Wood [1983].

article S. S. Chern and J. Wolfson: “Harmonic maps of \( S^2 \) into a complex Grassmann manifold,” Proc. Nat. Acad. Sci. U.S.A. 82 : 8 (April 1985), pp. 2217–2219. Part II was published in Ann. Math. 125:2 (1987). MR 788858 Zbl 0601.58023

Let \( G(k,n) \) be the Grassmann manifold of all \( C_k \) in \( C_n \), the complex spaces of dimensions \( k \) and \( n \), respectively, or, what is the same, the manifold of all projective spaces \( P_{k-1} \) in \( P_{n-1} \), so that \( G(1,n) \) is the complex projective space \( P_{n-1} \) itself. We study harmonic maps of the two-dimensional sphere \( S^2 \) into \( G(k,n) \). The case \( k=1 \) has been the subject of investigation by several authors [see, for example, Din, A. M. & Zakrzewski, W. J. (1980) Nucl. Phys. B 174, 397–406; Eells, J. & Wood, J. C. (1983) Adv. Math. 49, 217–263; and Wolfson, J. G. Trans. Am. Math. Soc., in press]. The harmonic maps \( S^2\to G(2,4) \) have been studied by Ramanathan [Ramanathan, J. (1984) J. Differ. Geom. 19, 207–219]. We shall describe all harmonic maps \( S^2\to G(2,n) \). The method is based on several geometrical constructions, which lead from a given harmonic map to new harmonic maps, in which the image projective spaces are related by “fundamental collineations.” The key result is the degeneracy of some fundamental collineations, which is a global consequence, following from the fact that the domain manifold is \( S^2 \). The method extends to \( G(k,n) \).

article S.-S. Chern and J. G. Wolfson: “Harmonic maps of the two-sphere into a complex Grassmann manifold, II,” Ann. Math. (2) 125 : 2 (March 1987), pp. 301–335. Part I was published in Proc. Nat. Acad. Sci. U.S.A. 82:8 (1985). MR 881271 Zbl 0627.58017

incollection J. Wolfson: “Chern, some recollections,” pp. 268–270 in S. S. Chern: A great geometer of the twentieth century. Edited by S.-T. Yau. Monographs in geometry and topology. International Press (Hong Kong), 1992. MR 1201367

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