Communications in Analysis and Geometry
Volume 30 (2022)
Extremally Ricci pinched $G_2$-structures on Lie groups
Pages: 1355 – 1389
Only two examples of extremally Ricci pinched $G_2$-structures can be found in the literature and they are both homogeneous. We study in this paper the existence and structure of such very special closed $G_2$-structures on Lie groups. Strong structural conditions on the Lie algebra are proved to hold. As an application, we obtain three new examples of extremally Ricci pinched $G_2$-structures and that they are all necessarily steady Laplacian solitons. The deformation and rigidity of such structures are also studied.
The authors’ research was partially supported by grants from CONICET, FONCYT and Universidad Nacional de Córdoba.
Received 27 January 2018
Accepted 27 December 2019
Published 26 April 2023