Session S23 - Group actions in Differential Geometry

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Positive Hermitian Curvature Flow on complex Lie groups

James Stanfield

The University of Queensland, Australia   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

The success of the (Kähler)-Ricci flow sparks a natural desire to seek suitable generalisations to non-Kähler Hermitian geometry. In this presentation, we will focus on one such generalisation. Namely, the Positive Hermitian Curvature Flow introduced by Ustinovskiy as part of a family of Hermitian Curvature Flows originally studied by Streets and Tian. This evolution equation is of interest as it preserves many natural curvature positivity conditions for Hermitian manifolds.

We consider the Positive Hermitian Curvature Flow on the space of left-invariant Hermitian metrics on a complex Lie group G, a large class of Hermitian manifolds which are typically non-Kähler. Specifically, we will study the asymptotic behaviour when G is nilpotent or almost-abelian. Time permitting, we will also discuss results regarding soliton solutions in these settings and the case where G is simple.

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