## Duke Mathematical Journal

### Hypersymplectic 4-manifolds, the $G_{2}$-Laplacian flow, and extension assuming bounded scalar curvature

#### Abstract

A hypersymplectic structure on a 4-manifold $X$ is a triple $\underline{\omega}$ of symplectic forms which at every point span a maximal positive definite subspace of $\Lambda^{2}$ for the wedge product. This article is motivated by a conjecture by Donaldson: when $X$ is compact, $\underline{\omega}$ can be deformed through cohomologous hypersymplectic structures to a hyper-Kähler triple. We approach this via a link with $G_{2}$-geometry. A hypersymplectic structure $\underline{\omega}$ on a compact manifold $X$ defines a natural $G_{2}$-structure $\phi$ on $X\times\mathbb{T}^{3}$ which has vanishing torsion precisely when $\underline{\omega}$ is a hyper-Kähler triple. We study the $G_{2}$-Laplacian flow starting from $\phi$, 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 $G_{2}$-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).

#### Article information

Source
Duke Math. J., Volume 167, Number 18 (2018), 3533-3589.

Dates
Received: 8 May 2017
Revised: 29 May 2018
First available in Project Euclid: 6 November 2018

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1541473293

Digital Object Identifier
doi:10.1215/00127094-2018-0040

Mathematical Reviews number (MathSciNet)
MR3881202

Zentralblatt MATH identifier
07009771

#### Citation

Fine, Joel; Yao, Chengjian. Hypersymplectic 4-manifolds, the $G_{2}$ -Laplacian flow, and extension assuming bounded scalar curvature. Duke Math. J. 167 (2018), no. 18, 3533--3589. doi:10.1215/00127094-2018-0040. https://projecteuclid.org/euclid.dmj/1541473293

#### References

• [1] S. Bando, A. Kasue, and H. Nakajima, On a construction of coordinates at infinity on manifolds with fast curvature decay and maximal volume growth, Invent. Math. 97 (1989), 313–349.
• [2] S. Bauer, Almost complex 4-manifolds with vanishing first Chern class, J. Differential Geom. 79 (2008), 25–32.
• [3] R. Bryant, “Some remarks on $G_{2}$-structures” in Proceedings of Gökova Geometry-Topology Conference (Gökova, 2005), International Press, Somerville, Mass., 2006, 75–109.
• [4] R. Bryant and F. Xu, Laplacian flow for closed $G_{2}$-structures: short time behavior, preprint, arXiv:1101.2004v1 [math.DG].
• [5] J. Cheeger, M. Gromov, and M. Taylor, Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds, J. Differential. Geom. 17 (1982), 15–53.
• [6] G. Chen, Shi-type estimates and finite time singularities of flows of $G_{2}$ structures, Q. J. Math. 69 (2018), 779–797.
• [7] X.-X. Chen and B. Wang, On the conditions to extend Ricci flow (III), Int. Math. Res. Not. IMRN 10 (2013), 2349–2367.
• [8] S. K. Donaldson, “Two-forms on four-manifolds and elliptic equations” in Inspired by S. S. Chern, Nankai Tracts. Math. 11, World Scientific, Hackensack, N.J., 2006, 153–172.
• [9] S. K. Donaldson, “Adiabatic limits of co-associative Kovalev–Lefschetz fibrations” in Algebra, Geometry, and Physics In the 21st Century, Progr. Math. 324, Birkhäuser, Cham, 2017, 1–29.
• [10] J. Eells and J. H. Sampson, Harmonic mappings of Riemannian manifolds, Amer. J. Math. 86 (1964), 109–160.
• [11] J. Enders, R. Müller, and P. M. Topping, On type-I singularities in Ricci flow, Comm. Anal. Geom. 19 (2011), 905–922.
• [12] J. Fine, A gauge theoretic approach to the anti-self-dual Einstein equations, preprint, arXiv:1111.5005v2 [math.DG].
• [13] H. Geiges, Symplectic couples on 4-manifolds, Duke Math. J. 85 (1996), 701–711.
• [14] N. Hitchin, The geometry of three-forms in six dimensions, J. Diferential Geom. 55 (2000), 547–576.
• [15] B. Kleiner and J. Lott, Notes on Perelman’s papers, Geom. Topol. 12 (2008), 2587–2855.
• [16] P. B. Kronheimer, A Torelli-type theorem for gravitational instantons, J. Differential Geom. 29 (1989), 685–697.
• [17] J. Lotay and Y. Wei, Laplacian flow for closed $G_{2}$ structures: Shi-type estimates, uniqueness and compactness, Geom. Funct. Anal. 27 (2017), 165–233.
• [18] T. B. Madsen, $\operatorname{Spin}(7)$-manifolds with three-torus symmetry, J. Geom. Phys. 61 (2011), 2285–2292.
• [19] T. B. Madsen and A. Swann, Multi-moment maps, Adv. Math. 229 (2012), 2287–2309.
• [20] R. Müller, Ricci flow coupled with Harmonic map flow, Ann. Sci. Éc. Norm. Supér. (4) 45 (2012), 101–142.
• [21] G. Perelman, The entropy formula for the Ricci flow and its geometric applications, preprint, arXiv:math/0211159v1 [math.DG].
• [22] N. Šešum, Curvature tensor under the Ricci flow, Amer. J. Math. 127 (2005), 1315–1324.
• [23] M. Simon, Some integral curvature estimates for the Ricci flow in four dimensions, preprint, arXiv:1504.02623v1 [math.DG].
• [24] B. Wang, On the conditions to extend Ricci flow, Int. Math. Res. Not. IMRN 2008, Art. ID rnn012.
• [25] B. Wang, On the conditions to extend Ricci flow (II), Int. Math. Res. Not. IMRN 2012, 3192–3223.
• [26] Z. Zhang, Scalar curvature behaviour for finite-time singularity of Kähler–Ricci flow, Michigan Math. J. 59 (2010), 419–433.