Sharp lower bound on the curvatures of ASD connections over the cylinder. (English) Zbl 1304.53018

Tsukamoto, Masaki

J. Math. Soc. Japan 66, No. 3, 951-956 (2014).

Let \(X =\mathbb{R}\times S^3\) be the cylinder with the product metric and \(\pi:E = X \times \mathrm {SU}_2 \to X\) the trivial \(\mathrm {SU}_2\)-principal bundle. Let \(A\) be an anti-selfdual connection in \(\pi\) and \(F : \Lambda^2(TX) \to \mathfrak{su}_2 \) its curvature. The author defines the norm of \(F\) at a point \(p\) as \[ |F_p|= \sup\{ \sum_{1\leq i<j<4} a_{ij}F(e_i,e_j)_p | a_{ij}\in \mathbb R,\, \sum_{1\leq i<j<4} a_{ij}=1 \} \] where \(e_i\) is an orthonormal basis of \(T_pX\) and the norm \(|F|\) of \(F\) as the supremum of \(|F|_p\) over \(p \in X\). The main result is the following theorem:
The minimum of the norms \(|F|\) over all non-flat anti-selfdual connections \(A\) in \(\pi\) is equal to \(\frac{1}{\sqrt{2}}\).
It is proven that the \(\mathrm {SU}_2\) instanton \[ A:= \mathrm{Im}(\frac{\bar x dx}{1 + |x|^2}) \] in \(\mathbb{R}^4 =\mathbb{H}\) restricted to \[ \mathbb{R}\times S^3 = \{ (t,u) \} \subset \mathbb{H}^* =\{e^t u ,\, |u|=1 \} \] attains this minimum.

Reviewer: Dmitri Alekseevsky (Moscow)

MSC:

53C07	Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills)
53B50	Applications of local differential geometry to the sciences

Keywords:

anti-selfdual connection; norm of curvature for an ASD connection

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