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\(\hbox {G}_2\) Manifolds with Nodal Singularities Along Circles

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This paper finds matching building blocks for the construction of a compact manifold with \(\hbox {G}_2\) holonomy and nodal singularities along circles using twisted connected sum method by solving the Calabi conjecture on certain asymptotically cylindrical manifolds with nodal singularities. However, by comparison to the untwisted connected sum case, it turns out that the obstruction space for the singular twisted connected sum construction is infinite dimensional. By analyzing the obstruction term, there are strong evidences that the obstruction may be resolved if a further gluing is performed in order to get a compact manifold with \(\hbox {G}_2\) holonomy and isolated conical singularities with link \({\mathbb {S}}^3\times {\mathbb {S}}^3\).

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Acknowledgements

The author would like to thank Edward Witten for introducing him to this problem and for many fruitful discussions. The author is also grateful to the referees for detailed suggestions as well as Jeff Cheeger, Xiuxiong Chen, Sir Simon Donaldson, Lorenzo Foscolo, Mark Haskins, Hans-Joachim Hein, Helmut Hofer, Fanghua Lin, Rafe Mazzeo, Johannes Nordström, Song Sun, Akshay Venkatesh, Jeff Viaclovsky and Ruobing Zhang for helpful conversations. This material is based upon work supported by the National Science Foundation under Grant No. 1638352, as well as support from the S. S. Chern Foundation for Mathematics Research Fund.

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    Gao Chen

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Chen, G. \(\hbox {G}_2\) Manifolds with Nodal Singularities Along Circles. J Geom Anal 31, 1360–1414 (2021). https://doi.org/10.1007/s12220-019-00283-3

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