Duke Mathematical Journal

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

Joel Fine and Chengjian Yao

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A hypersymplectic structure on a 4-manifold X is a triple ω̲ of symplectic forms which at every point span a maximal positive definite subspace of Λ2 for the wedge product. This article is motivated by a conjecture by Donaldson: when X is compact, ω̲ can be deformed through cohomologous hypersymplectic structures to a hyper-Kähler triple. We approach this via a link with G2-geometry. A hypersymplectic structure ω̲ on a compact manifold X defines a natural G2-structure ϕ on X×T3 which has vanishing torsion precisely when ω̲ is a hyper-Kähler 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).

Article information

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53C44: Geometric evolution equations (mean curvature flow, Ricci flow, etc.)
Secondary: 53D35: Global theory of symplectic and contact manifolds [See also 57Rxx] 53C26: Hyper-Kähler and quaternionic Kähler geometry, "special" geometry

symplectic geometry geometric flows $G_{2}$ manifolds hyper-Kähler manifolds


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

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  • [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.