Celebratio Mathematica — Simons

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Works connected to Shiing-Shen Chern

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S.-S. Chern and J. Simons: “Some cohomology classes in principal fiber bundles and their application to Riemannian geometry,” Proc. Natl. Acad. Sci. U.S.A. 68 : 4 (April 1971), pp. 791–794. MR 0279732 Zbl 0209.25401 article

H. B. Lawson, Jr. and J. Simons: “On stable currents and their application to global problems in real and complex geometry,” Ann. Math. (2) 98 : 3 (November 1973), pp. 427–450. Dedicated to S. S. Chern on his 60th birthday. MR 0324529 Zbl 0283.53049 article

In this paper we use some techniques from the calculus of variations to study the topology and geometry of submanifolds of the sphere and the geometry of complex projective space.

The basic idea is to use the existence theorems of Federer and Fleming to represent integral (or \( \mathbb{Z}_p \)) homology classes by rectifiable currents of least mass, and then to show that under various geometric assumptions one can construct deformations of these currents which are mass decreasing. In the case of an immersed submanifold of \( \mathbb{S}^n \) we show that appropriate assumptions on its extrinsic geometry imply the vanishing of a given homology group. In the case of \( \mathbb{P}^n(\mathbb{C}) \) we show that the only stable integral currents are the formal sums of algebraic varieties.

H. Blaine Lawson, Jr.	Related
S. S. Chern	Related	Volume

S.-S. Chern and J. Simons: “Characteristic forms and geometric invariants,” Ann. Math. (2) 99 : 1 (January 1974), pp. 48–69. MR 0353327 Zbl 0283.53036 article

J. Simons: “My interaction with S. S. Chern,” pp. 176–178 in S. S. Chern: A great geometer of the twentieth century (Los Angeles, 1990). Edited by S.-T. Yau. International Press (Hong Kong), 1992. MR 1201357 incollection

S. S. Chern	Related	Volume
Shing-Tung Yau	Related