A class of eternal solutions to the G$_2$-Laplacian flow

TL;DR

This paper explicitly constructs a class of solutions to the G$_2$-Laplacian flow on 7-manifolds, demonstrating their eternal existence and preservation of extremally Ricci-pinched properties, with broader applicability and examples.

Contribution

It provides explicit eternal solutions to the G$_2$-Laplacian flow starting from extremally Ricci-pinched structures and extends the results to more general cases with constant torsion norm.

Findings

Solutions exist for all time and remain extremally Ricci-pinched.

The results apply to any 7-manifold with constant torsion norm.

Various explicit examples are discussed.

Abstract

We explicitly describe the solution of the G$_2$-Laplacian flow starting from an extremally Ricci-pinched closed G$_2$-structure on a compact 7-manifold and we investigate its properties. In particular, we show that the solution exists for all real times and that it remains extremally Ricci-pinched. This result holds more generally on any 7-manifold whenever the intrinsic torsion of the extremally Ricci-pinched G$_2$-structure has constant norm. We also discuss various examples.