Extremally Ricci pinched $G_2$-structures on Lie groups

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.

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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