Thesis
Moduli spaces of compact RCD structures
- Abstract:
-
This thesis investigates RCD spaces, which are metric measure spaces with Ricci curvature bounded below and dimension bounded above in a synthetic sense. We introduce moduli spaces of compact RCD structures and study their topology. In particular, we discuss the results obtained in [MN22] (written in collaboration with Andrea Mondino) and [Nav22].
In Chapter 2, we present the primary tools we use in the thesis. We recall Gromov–Hausdorff type topologies and RCD spaces with their covering and moduli spaces. The main contributions of this chapter are the equivariant measured Gromov–Hausdorff topology and the equivariant theorem (both obtained in [MN22]).
In Chapter 3, we focus on the case of nonnegative curvature. In particular, we obtain topological invariants of RCD(0,N) spaces using the splitting theorem. In addition, we introduce the Albanese and soul maps and prove their continuity. This last result is the most technical part of the chapter and was obtained in [MN22]. Finally, we use the Albanese map to construct examples of moduli spaces with non-trivial higher homotopy groups in every dimension N ≥ 3.
Chapter 4 is devoted to nonnegative curvature in dimension 2 and discusses the results of [Nav22]. We obtain a classification (up to homeomorphism) of the topological spaces that admit an RCD(0, 2) structure. For every space appearing in the classification, we compute the homeomorphism type of the moduli space of RCD(0, 2) structures and show that it is contractible.
Finally, in Chapter 5, we apply Ricci flow techniques to study moduli spaces of RCD(−1, 2) structures. In particular, we show that if a space has a negative Euler characteristic, then its moduli space of RCD(−1, 2) structures is homotopy equivalent to its moduli space of hyperbolic metrics.
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Authors
Contributors
- Institution:
- University of Oxford
- Division:
- MPLS
- Department:
- Mathematical Institute
- Sub department:
- Mathematical Institute
- Research group:
- Geometry and Oxford Centre for Nonlinear PDE
- Oxford college:
- St Hilda's College
- Role:
- Supervisor
- Institution:
- Institut Fourier, Université de Grenoble
- Role:
- Supervisor
- Institution:
- University of Oxford
- Division:
- MPLS
- Department:
- Mathematical Institute
- Sub department:
- Mathematical Institute
- Research group:
- Geometry, Mathematical Physics, and Oxford Centre for Nonlinear PDE
- Role:
- Examiner
- Institution:
- Karlsruher Institut für Technologie
- Role:
- Examiner
- Funder identifier:
- http://dx.doi.org/10.13039/501100000781
- Funding agency for:
- Navarro, D
- Grant:
- ERC Starting Grant CURVATURE, Grant agreement No. 802689
- Programme:
- European Union Horizon 2020 research and innovation programme
- Type of award:
- DPhil
- Level of award:
- Doctoral
- Awarding institution:
- University of Oxford
- Language:
-
English
- Keywords:
- Subjects:
- Deposit date:
-
2023-07-19
Terms of use
- Copyright holder:
- Navaro, D
- Copyright date:
- 2023
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