Abstract
We survey recent progress in the study of flows of isometric G 2-structures on seven-dimensional manifolds, that is, flows that preserve the metric, while modifying the G 2-structure. In particular, heat flows of isometric G 2-structures have been recently studied from several different perspectives, in particular in terms of 3-forms, octonions, vector fields, and geometric structures. We will give an overview of each approach, the results obtained, and compare the different perspectives.
This work was supported by the National Science Foundation [DMS-1811754].
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References
L. Bagaglini, The energy functional of G 2-structures compatible with a background metric. J. Geom. Anal. (2019). https://doi.org/10.1007/s12220-019-00264-6
J. Boling, C. Kelleher, J. Streets, Entropy, stability and harmonic map flow. Trans. Am. Math. Soc. 369(8), 5769–5808 (2017). https://doi.org/10.1090/tran/6949
R.L. Bryant, Metrics with exceptional holonomy. Ann. Math. (2) 126(3), 525–576 (1987). https://doi.org/10.2307/1971360
R.L. Bryant, Some remarks on G 2-structures, in Proceedings of Gökova Geometry-Topology Conference 2005 (GGT), Gökova (2006), pp. 75–109. math/0305124
Y.M. Chen, W.Y. Ding, Blow-up and global existence for heat flows of harmonic maps. Invent. Math. 99(3), 567–578 (1990). https://doi.org/10.1007/BF01234431
Y.M. Chen, M. Struwe, Existence and partial regularity results for the heat flow for harmonic maps. Math. Z. 201(1), 83–103 (1989). https://doi.org/10.1007/BF01161997
T.H. Colding, W.P. Minicozzi II, Generic mean curvature flow I: generic singularities. Ann. Math. (2) 175(2), 755–833 (2012). https://doi.org/10.4007/annals.2012.175.2.7
S. Dwivedi, P. Gianniotis, S. Karigiannis, A gradient flow of isometric G 2 structures. J. Geom. Anal. (2019). 1904.10068. https://doi.org/10.1007/s12220-019-00327-8
S. Grigorian, Short-time behaviour of a modified Laplacian coflow of G2-structures. Adv. Math. 248, 378–415 (2013). 1209.4347. https://doi.org/10.1016/j.aim.2013.08.013
S. Grigorian, G 2-structures and octonion bundles. Adv. Math. 308, 142–207 (2017). 1510.04226. https://doi.org/10.1016/j.aim.2016.12.003
S. Grigorian, Estimates and monotonicity for a heat flow of isometric G 2-structures. Calc. Var. Partial Differ. Equ. 58(5), Art. 175, 37 (2019). https://doi.org/10.1007/s00526-019-1630-0
N.J. Hitchin, The geometry of three-forms in six dimensions. J. Differ. Geom. 55(3), 547–576 (2000). math/0010054. http://projecteuclid.org/euclid.jdg/1090341263
S. Karigiannis, Deformations of G 2 and Spin(7) structures on manifolds. Can. J. Math. 57, 1012 (2005). math/0301218. https://doi.org/10.4153/CJM-2005-039-x
S. Karigiannis, Flows of G 2-structures, I. Q. J. Math. 60(4), 487–522 (2009). math/0702077. https://doi.org/10.1093/qmath/han020
S. Karigiannis, B. McKay, M.-P. Tsui, Soliton solutions for the Laplacian coflow of some G 2-structures with symmetry. Differ. Geom. Appl. 30(4), 318–333 (2012). 1108.2192. https://doi.org/10.1016/j.difgeo.2012.05.003
C. Kelleher, J. Streets, Entropy, stability, and Yang-Mills flow. Commun. Contemp. Math. 18(2), 1550032, 51 (2016). https://doi.org/10.1142/S0219199715500327
E. Loubeau, H.N. Sá Earp, Harmonic flow of geometric structures (2019). 1907.06072
M. Struwe, On the evolution of harmonic maps in higher dimensions. J. Differ. Geom. 28(3), 485–502 (1988). http://projecteuclid.org/euclid.jdg/1214442475
C.M. Wood, An existence theorem for harmonic sections. Manuscripta Math. 68(1), 69–75 (1990). https://doi.org/10.1007/BF02568751
C.M. Wood, Harmonic sections and equivariant harmonic maps. Manuscripta Math. 94(1), 1–13 (1997). https://doi.org/10.1007/BF02677834
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Grigorian, S. (2022). Isometric Flows of G 2-structures. In: Cerejeiras, P., Reissig, M., Sabadini, I., Toft, J. (eds) Current Trends in Analysis, its Applications and Computation. Trends in Mathematics(). Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-87502-2_55
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