Quadratic closed G2$\mathrm{G}_2$‐structures

Abstract

This article studies closed G 2 $\mathrm{G}_2$ -structures satisfying the quadratic condition, a second-order PDE system introduced by Bryant involving a parameter λ $\lambda$ . For certain special values of λ $\lambda$ the quadratic condition is equivalent to the Einstein condition for the metric induced by the closed G 2 $\mathrm{G}_2$ -structure (for λ = 1 / 2 $\lambda = 1/2$ ), the extremally Ricci-pinched (ERP) condition (for λ = 1 / 6 $\lambda =1/6$ ), and the condition that the closed G 2 $\mathrm{G}_2$ -structure be an eigenform for the Laplace operator (for λ = 0 $\lambda = 0$ ). Prior to the work in this article, solutions to the quadratic system were known only for λ = 1 / 6 $\lambda = 1/6$ , − 1 / 8 $-1/8$ , and 2 / 5 $2/5$ , and for these values only a handful of solutions were known. In this article, we produce infinitely many new examples of ERP G 2 $\mathrm{G}_2$ -structures, including the first example of a complete inhomogeneous ERP G 2 $\mathrm{G}_2$ -structure and a new example of a compact ERP G 2 $\mathrm{G}_2$ -structure. We also give a classification of homogeneous ERP G 2 $\mathrm{G}_2$ -structures. We provide the first examples of quadratic closed G 2 $\mathrm{G}_2$ -structures for λ = − 1 $\lambda = -1$ , 1 / 3 $1/3$ and 3 / 4 $3/4$ , as well as infinitely many new examples for λ = − 1 / 8 $\lambda = -1/8$ and 2 / 5 $2/5$ . Our constructions involve the notion of special torsion for closed G 2 $\mathrm{G}_2$ -structures, a new concept that is likely to have wider applicability. In the final section of the article, we provide two large families of inhomogeneous complete steady gradient solitons for the Laplacian flow, the first known such examples.