Aslan, Benjamin;
(2022)
Special submanifolds in nearly Kähler 6-manifolds.
Doctoral thesis (Ph.D), UCL (University College London).
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Abstract
This thesis is on J-holomorphic curves and special Lagrangians of nearly Kähler manifolds, with a focus on nearly Kähler CP^3. We consider the following four problems. Firstly, we relate classical geometric properties of surfaces in four-manifolds to properties of their twistor lifts, based on the work of Eells, Salamon and Friedrich. This leads to the construction of deformation invariant quantities for J-holomorphic curves in certain twistor spaces, such as CP^3 or the manifold of complete flags in C^3. We give an example of how the twistor lift of the discriminant locus of a family of quadrics in CP^3 performs a desingularisation. Secondly, we introduce the class of transverse J-holomorphic curves in CP^3, for which we define angle functions. It turns out that the angle functions essentially encode the geometry of the curve, which results in classification results for J-holomorphic curves with special geometric properties. We derive a system of PDEs for the angle functions which enables us to establish a Bonnet-type theorem for transverse J-holomorphic curves. By constructing toric moment-type maps we relate them to the theory of U(1) invariant minimal surfaces in S^4. Thirdly, we consider the deformation problem for J-holomorphic curves in general nearly Kähler manifolds. We turn to infinitesimal deformations and show that they are eigensections of a twisted Dirac operator on the normal bundle of the curve. By solving this equation explicitly we show that homogeneous tori in CP^3 and S^6 are rigid and compute the spectrum of the Dirac operator in these cases. Lastly, we derive the structure equations for special Lagrangians in CP^3. This yields a classification of totally geodesic special Lagrangians. By introducing moment maps we also classify all SU(2) invariant special Lagrangians in CP^3 and provide new homogeneous examples.
| Type: | Thesis (Doctoral) |
|---|---|
| Qualification: | Ph.D |
| Title: | Special submanifolds in nearly Kähler 6-manifolds |
| Open access status: | An open access version is available from UCL Discovery |
| Language: | English |
| Additional information: | Copyright © The Author 2022. Original content in this thesis is licensed under the terms of the Creative Commons Attribution-NonCommercial 4.0 International (CC BY-NC 4.0) Licence (https://creativecommons.org/licenses/by-nc/4.0/). Any third-party copyright material present remains the property of its respective owner(s) and is licensed under its existing terms. Access may initially be restricted at the author’s request. |
| UCL classification: | UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics UCL > Provost and Vice Provost Offices > UCL BEAMS UCL |
| URI: | https://discovery-pp.ucl.ac.uk/id/eprint/10145664 |
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