Celebratio Mathematica — Simons

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Works connected to H. Blaine Lawson, Jr.

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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

J.-P. Bourguignon, H. B. Lawson, and J. Simons: “Stability and gap phenomena for Yang–Mills fields,” Proc. Natl. Acad. Sci. U.S.A. 76 : 4 (April 1979), pp. 1550–1553. MR 526178 Zbl 0408.53023 article

It is shown that any weakly stable Yang–Mills field of type \( \mathit{SU}(2) \) or \( \mathit{SU}(3) \) on the four-sphere must be self-dual or anti-self-dual. Any Yang–Mills field on \( \mathbb{S}^n \), \( n \geq 5 \), is unstable. Examples of stable fields on \( \mathbb{S}^4 \) and \( \mathbb{S}^n/\Gamma \) for \( n \geq 5 \) and \( \Gamma \neq \{e\} \) are given. It is also shown that, for any Yang–Mills field \( R \) on \( \mathbb{S}^4 \), the pointwise condition \( \| R^-\|^2 < 3 \) (or \( \| R^+\|^2 < 3 \)) implies that \( R^- = 0 \) (or respectively that \( R^+ = 0 \)). In general, any Yang–Mills field \( R \) on \( \mathbb{S}^n \), \( n \geq 3 \), that satisfies the pointwise condition \( \| R\|^2 < \frac{1}{2}{2 \choose n} \) is trivial. If \( n = 3 \) or 4, the condition \( \| R\|^2 < \frac{1}{2}{2 \choose n} \) implies that either \( R \) is the trivial field or it is the direct sum of a trivial field with a field of tangent spinors carrying the standard connection.

H. Blaine Lawson, Jr.	Related
Jean Pierre Bourguignon	Related