Positive Hermitian Curvature Flow on special linear groups and perfect solitons

TL;DR

This paper investigates invariant solutions to the Positive Hermitian Curvature Flow on complex Lie groups, revealing instability of certain metrics and discovering new non-algebraic solitons on perfect Lie groups, thus challenging existing conjectures.

Contribution

It demonstrates the dynamical instability of canonical metrics on special linear groups and constructs new non-algebraic solitons, providing counterexamples to prior conjectures.

Findings

Canonical metrics on SL(n,C) are dynamically unstable.

Existence of non-algebraic, homogeneous solitons on perfect Lie groups.

First example of a geometric flow with non-algebraic solitons besides G2-Laplacian flow.

Abstract

We study invariant solutions to the Positive Hermitian Curvature Flow, introduced by Ustinovskiy, on complex Lie groups. We show in particular that the canonical scale-static metrics on the special linear groups, arising from the Killing form, are dynamically unstable. This disproves a conjecture of Ustinovskiy. We also construct certain perfect Lie groups that admit at least two distinct invariant solitons for the flow, only one of which is algebraic. This is the second known example of a geometric flow with non-algebraic, homogeneous solitons. The first being the G2-Laplacian flow.