On the hypersymplectic flow
Organizers
Speaker
Amanda Petcu
Time
Tuesday, May 26, 2026 3:00 PM - 4:00 PM
Venue
A3-4-301
Online
Zoom 293 812 9202
(BIMSA)
Abstract
A conjecture of Simon Donaldson is that on a compact $4$-manifold $X^4$ one can flow from a hypersymplectic structure to a hyperk\"ahler structure while remaining in the same cohomology class. To this end the hypersymplectic flow was introduced by Fine--Yao. In this talk we will introduce the notion of a positive triple on $X^4$ to define a hypersymplectic and hyperk\"ahler structure. Given a closed positive triple we will show how one can define either a closed $G_2$ structure or a coclosed $G_2$ structure on $\mathbb{T}^3 \times X^4$. We will evolve the coclosed $G_2$ structure under the $G_2$-Laplacian coflow and show that the coflow descends to a flow of the positive triple on $X^4$, which is again the Fine--Yao hypersymplectic flow.
In this talk we will also discuss some solitons of the hypersymplectic flow. Solitons of a geometric flow are a triple $(\lambda, X, \underline{\omega})$ of a constant $\lambda$, a vector field $X$ and an initial geometric structure $\underline{\omega}$ that give rise to self similar solutions of the flow. Self similar solutions of a geometric flow are solutions which evolve by scalings and diffeomorphisms. We let $X^4 = \mathbb{R}^4 \setminus \{0\}$ with a particular cohomogeneity one action and introduce a hypersymplectic structure built from data invariant under this action. This is used as the initial geometric structure in our soliton triple. We will conclude with some results concerning steady solitons and hyperk\"ahler solitons.
In this talk we will also discuss some solitons of the hypersymplectic flow. Solitons of a geometric flow are a triple $(\lambda, X, \underline{\omega})$ of a constant $\lambda$, a vector field $X$ and an initial geometric structure $\underline{\omega}$ that give rise to self similar solutions of the flow. Self similar solutions of a geometric flow are solutions which evolve by scalings and diffeomorphisms. We let $X^4 = \mathbb{R}^4 \setminus \{0\}$ with a particular cohomogeneity one action and introduce a hypersymplectic structure built from data invariant under this action. This is used as the initial geometric structure in our soliton triple. We will conclude with some results concerning steady solitons and hyperk\"ahler solitons.