Hypersymplectic 4-manifolds, the G2-Laplacian flow, and extension assuming bounded scalar curvature

Abstract

A hypersymplectic structure on a 4-manifold X is a triple ω of symplectic forms which at every point span a maximal positive-definite subspace of Λ 2 for the wedge product. This article is motivated by a conjecture of Donaldson: when X is compact ω can be deformed through cohomologous hypersymplectic structures to a hyperkähler triple. We approach this via a link with G2-geometry. A hypersymplectic structure ω on a compact manifold X defines a natural G2-structure φ on X × T 3 which has vanishing torsion precisely when ω is a hyperkähler triple. We study the G2-Laplacian flow starting from φ, which we interpret as a flow of hypersymplectic structures. Our main result is that the flow extends as long as the scalar curvature of the corresponding G2-structure remains bounded. An application of our result is a lower bound for the maximal existence time of the flow, in terms of weak bounds on the initial data (and with no assumption that scalar curvature is bounded along the flow). Preliminary definitions and short time existenceThe main result of this section is:Proposition 2.1. Let ω be a hypersymplectic structure on a compact 4-manifold X. Then there exists a unique short time solution ω(t) to the hypersymplectic flow (3) starting at ω.This will follow easily from the analogous result of Bryant-Xu for the G 2 -Laplacian flow. Along the way we make explicit the relationship between the natural metric g ω on X induced by a Proof. This is immediate from the uniqueness part of Bryant-Xu's theorem [4, Theorem 0.1]: since the initial data φ(0) is T 3 invariant, so is the ensuing flow.By T 3 -invariance, the 3-form φ(t) on X × T 3 necessarily has the shape∈ Ω 2 (X) and D(t) ∈ Ω 3 (X) are paths of forms on X. Moreover, since dφ(t) = 0, it follows that these forms on X are closed.Lemma 2.7. A(t) = 1.Proof. We know that dA(t) = 0, i.e., that for every t, A(t) is constant. Moreover,Since ∂ t φ is exact, this integral is independent of t, and so A(t) = A(0) = 1.Next we consider the involution ϑ : X × T 3 → X × T 3 given by ϑ(p, t) → (p, −t) and writê φ(t) = −ϑ * φ(t).Lemma 2.8. For all t,φ(t) = φ(t). Hence B i (t) = 0 = D(t) vanish identically, and φ(t) remains of the form (2) for as long as it exists, for a closed triple ω(t) of 2-forms on X.

Cohomogeneity‐one G2‐Laplacian flow on the 7‐torus

“…Let and and similarly , then has uniform lower bound on the injectivity radius by Proposition and uniform lower bound on the volume ratio on all scales, and moreover is uniformly bounded. The same argument as in the proof of [, Theorem 5.1] shows that we can take a Cheeger–Gromov limit where and defines a complete hyper‐Kähler structure on . Let be the meridian geodesic segment connecting and whose length is half of the length of the meridian circle they lie on such that is the distance midpoint of .…”

“…Proof Note the relationship between the hypersymplectic flow on and the ‐Laplacian flow on [, Lemma 2.9], and the relationship between the volume forms of the corresponding metrics on 4‐dimension and 7‐dimension [, Lemma 2.5]. The general evolution equation for the volume form in ‐Laplacian flow [, equation 3.8] gives the stated result.…”

“…The speculated answer for compact is ‘ YES ’, and Donaldson described a general constraint PDE of elliptic type to attack it. In , a geometric flow is introduced to deform each symplectic form in its cohomology class simultaneously and the stationary solution of the flow is a hyper‐Kähler triple. A significant fact is that this flow is the ‘gradient flow’ of some ‘volume functional’, which is bounded from above by topological data and whose critical point (if it exists) is exactly the hyper‐Kähler structure in the same cohomology class (see ).…”

“…Write for the matrix of inner products of the forms , then it follows easily that (cf. [, Lemma 2.4]).…”

“…In general, assuming is uniformly bounded, the long‐time existence of ‐Laplacian flow was obtained by . For the hypersymplectic flow on a compact 4‐manifold, the long‐time existence assuming bounded torsion was obtained in . There is also an interesting reduction of a warped ‐Laplacian flow on to a coupled flow of the ‐structure and the warped function on in .…”