Bach flow of simply connected nilmanifolds
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Adam Thompson
Abstract
The Bach flow is a fourth-order geometric flow defined on four-manifolds. For a compact manifold, it is the negative gradient flow for the L2-norm of the Weyl curvature. In this paper, we study the Bach flow on four-dimensional simply connected nilmanifolds whose Lie algebra is indecomposable. We show that the Bach flow beginning at an arbitrary left invariant metric exists for all positive times and after rescaling converges in the pointed Cheeger–Gromov sense to an expanding Bach soliton which is non-gradient. Combining our results with previous results of Helliwell gives a complete description of the Bach flow on simply connected nilmanifolds.
Funding statement: I would also like to thank the Australian Mathematical Sciences Institute for supporting this research
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Communicated by: T. Leistner
Acknowledgements
I am grateful to Ramiro Lafuente for his support and encouragement throughout this research.
References
[1] E. Bahuaud, D. Helliwell, Short-time existence for some higher-order geometric flows. Comm. Partial Differential Equations 36 (2011), 2189–2207. MR2852074 Zbl 1242.5307910.1080/03605302.2011.593015Search in Google Scholar
[2] E. Bahuaud, D. Helliwell, Uniqueness for some higher-order geometric flows. Bull. Lond. Math. Soc. 47 (2015), 980–995. MR3431578 Zbl 1338.5309210.1112/blms/bdv076Search in Google Scholar
[3] A. L. Besse, Einstein manifolds. Springer 1987. MR867684 Zbl 0613.5300110.1007/978-3-540-74311-8Search in Google Scholar
[4] C. Böhm, R. A. Lafuente, Immortal homogeneous Ricci flows. Invent. Math. 212 (2018), 461–529. MR3787832 Zbl 1447.5307810.1007/s00222-017-0771-zSearch in Google Scholar
[5] W. A. de Graaf, Classification of solvable Lie algebras. Experiment. Math. 14 (2005), 15–25. MR2146516 Zbl 1173.1730010.1080/10586458.2005.10128911Search in Google Scholar
[6] A. Derdziński, Self-dual Kähler manifolds and Einstein manifolds of dimension four. Compositio Math. 49 (1983), 405–433. MR707181 Zbl 0527.53030Search in Google Scholar
[7] C. Fefferman, C. R. Graham, Conformal invariants. Astérisque (1985), 95–116. MR837196 Zbl 0602.53007Search in Google Scholar
[8] E. Griffin, Gradient ambient obstruction solitons on homogeneous manifolds. Ann. Global Anal. Geom. 60 (2021), 469–499. MR4304859 Zbl 1486.5304810.1007/s10455-021-09784-3Search in Google Scholar
[9] D. Helliwell, Bach flow on homogeneous products. SIGMA Symmetry Integrability Geom. Methods Appl. 16 (2020), Paper No. 027, 35 pages. MR4082988 Zbl 1436.53078Search in Google Scholar
[10] W. Kühnel, H.-B. Rademacher, Conformal transformations of pseudo-Riemannian manifolds. In: Recent developments in pseudo-Riemannian geometry, 261–298, Eur. Math. Soc., Zürich 2008. MR2436234 Zbl 1155.5303710.4171/051-1/8Search in Google Scholar
[11] J. Lauret, Homogeneous nilmanifolds of dimensions 3 and 4. Geom. Dedicata 68 (1997), 145–155. MR1484561 Zbl 0889.5303210.1023/A:1004936725971Search in Google Scholar
[12] J. Lauret, Ricci soliton homogeneous nilmanifolds. Math. Ann. 319 (2001), 715–733. MR1825405 Zbl 0987.5301910.1007/PL00004456Search in Google Scholar
[13] J. Lauret, The Ricci flow for simply connected nilmanifolds. Comm. Anal. Geom. 19 (2011), 831–854. MR2886709 Zbl 1244.5307710.4310/CAG.2011.v19.n5.a1Search in Google Scholar
[14] J. Lauret, Ricci flow of homogeneous manifolds. Math. Z. 274 (2013), 373–403. MR3054335 Zbl 1272.5305510.1007/s00209-012-1075-zSearch in Google Scholar
[15] J. Lauret, Geometric flows and their solitons on homogeneous spaces. Rend. Semin. Mat. Univ. Politec. Torino 74 (2016), 55–93. MR3772582 Zbl 1440.53061Search in Google Scholar
[16] C. Lopez, Ambient obstruction flow. Trans. Amer. Math. Soc. 370 (2018), 4111–4145. MR3811522 Zbl 1407.5306610.1090/tran/7106Search in Google Scholar
[17] P. Petersen, W. Wylie, Rigidity of homogeneous gradient soliton metrics and related equations. Differential Geom. Appl. 84 (2022), Paper No. 101929, 29 pages. MR4457372 Zbl 1506.5306410.1016/j.difgeo.2022.101929Search in Google Scholar
[18] J. Stanfield, Positive Hermitian curvature flow on nilpotent and almost-abelian complex Lie groups. Ann. Global Anal. Geom. 60 (2021), 401–429. MR4291615 Zbl 1480.5311310.1007/s10455-021-09782-5Search in Google Scholar
[19] J. Streets, The long time behavior of fourth order curvature flows. Calc. Var. Partial Differential Equations 46 (2013), 39–54. MR3016500 Zbl 1258.5307410.1007/s00526-011-0472-1Search in Google Scholar
[20] V. S. Varadarajan, Lie groups, Lie algebras, and their representations. Springer 1984. MR746308 Zbl 0955.2250010.1007/978-1-4612-1126-6Search in Google Scholar
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Articles in the same Issue
- Frontmatter
- Duality related with key varieties of ℚ-Fano threefolds constructed from projective bundles
- Continuous CM-regularity and generic vanishing
- Quotient spaces of K3 surfaces by non-symplectic involutions fixing a curve of genus 8 or more
- A note on polarized varieties with high nef value
- Anisotropic area-preserving nonlocal flow for closed convex plane curves
- New sextics of genus 6 and 10 attaining the Serre bound
- The balanced superelliptic mapping class groups are generated by three elements
- Bach flow of simply connected nilmanifolds
Articles in the same Issue
- Frontmatter
- Duality related with key varieties of ℚ-Fano threefolds constructed from projective bundles
- Continuous CM-regularity and generic vanishing
- Quotient spaces of K3 surfaces by non-symplectic involutions fixing a curve of genus 8 or more
- A note on polarized varieties with high nef value
- Anisotropic area-preserving nonlocal flow for closed convex plane curves
- New sextics of genus 6 and 10 attaining the Serre bound
- The balanced superelliptic mapping class groups are generated by three elements
- Bach flow of simply connected nilmanifolds
