The Ricci Flow on Spin Manifolds
This first part of this thesis generalizes classical spin geometry to the setting of weighted manifolds (manifolds with density) and provides applications to the Ricci f low. Spectral properties of the naturally associated weighted Dirac operator, introduced by Perelman, and its relationship with th...
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| Format: | Thesis |
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Massachusetts Institute of Technology
2024
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| Online Access: | https://hdl.handle.net/1721.1/155325 |
| _version_ | 1826190740863057920 |
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| author | Baldauf, Julius |
| author2 | Minicozzi II, William P. |
| author_facet | Minicozzi II, William P. Baldauf, Julius |
| author_sort | Baldauf, Julius |
| collection | MIT |
| description | This first part of this thesis generalizes classical spin geometry to the setting of weighted manifolds (manifolds with density) and provides applications to the Ricci f low. Spectral properties of the naturally associated weighted Dirac operator, introduced by Perelman, and its relationship with the weighted scalar curvature are investigated. Further, a generalization of the ADM mass for weighted asymptotically Euclidean (AE) manifolds is defined; on manifolds with nonnegative weighted scalar curvature, it satisfies a weighted Witten formula and thereby a positive weighted mass theorem. Finally, on such manifolds, Ricci flow is the gradient flow of said weighted ADM mass, for a natural choice of weight function. This yields a monotonicity formula for the weighted spinorial Dirichlet energy of a weighted Witten spinor along Ricci flow. This part is joint work with Tristan Ozuch. The second part of this thesis introduces a functional generalizing Perelman’s weighted Hilbert-Einstein action and the Dirichlet energy for spinors. It is well-defined on a wide class of non-compact manifolds; on asymptotically Euclidean manifolds, the functional is shown to admit a unique critical point, which is necessarily of min-max type, and Ricci flow is its gradient flow. The proof is based on variational formulas for weighted spinorial functionals, valid on all spin manifolds with boundary. This part is also joint work with Tristan Ozuch.
The final part of this thesis studies the Ricci flow on closed manifolds admitting harmonic spinors, providing a new definition of Ricci flow. It is shown that Perelman’s Ricci flow entropy can be expressed in terms of the energy of harmonic spinors in all dimensions, and in four dimensions, in terms of the energy of Seiberg-Witten monopoles. Consequently, Ricci flow is the gradient flow of these energies. The proof relies on a weighted version of the monopole equations, introduced here. Further, a sharp parabolic Hitchin-Thorpe inequalityfor simply-connected, spin 4-manifolds is proven. From this, it follows that the normalized Ricci flow on any exotic K3 surface must become singular. |
| first_indexed | 2024-09-23T08:44:38Z |
| format | Thesis |
| id | mit-1721.1/155325 |
| institution | Massachusetts Institute of Technology |
| last_indexed | 2024-09-23T08:44:38Z |
| publishDate | 2024 |
| publisher | Massachusetts Institute of Technology |
| record_format | dspace |
| spelling | mit-1721.1/1553252024-06-28T03:59:03Z The Ricci Flow on Spin Manifolds Baldauf, Julius Minicozzi II, William P. Massachusetts Institute of Technology. Department of Mathematics This first part of this thesis generalizes classical spin geometry to the setting of weighted manifolds (manifolds with density) and provides applications to the Ricci f low. Spectral properties of the naturally associated weighted Dirac operator, introduced by Perelman, and its relationship with the weighted scalar curvature are investigated. Further, a generalization of the ADM mass for weighted asymptotically Euclidean (AE) manifolds is defined; on manifolds with nonnegative weighted scalar curvature, it satisfies a weighted Witten formula and thereby a positive weighted mass theorem. Finally, on such manifolds, Ricci flow is the gradient flow of said weighted ADM mass, for a natural choice of weight function. This yields a monotonicity formula for the weighted spinorial Dirichlet energy of a weighted Witten spinor along Ricci flow. This part is joint work with Tristan Ozuch. The second part of this thesis introduces a functional generalizing Perelman’s weighted Hilbert-Einstein action and the Dirichlet energy for spinors. It is well-defined on a wide class of non-compact manifolds; on asymptotically Euclidean manifolds, the functional is shown to admit a unique critical point, which is necessarily of min-max type, and Ricci flow is its gradient flow. The proof is based on variational formulas for weighted spinorial functionals, valid on all spin manifolds with boundary. This part is also joint work with Tristan Ozuch. The final part of this thesis studies the Ricci flow on closed manifolds admitting harmonic spinors, providing a new definition of Ricci flow. It is shown that Perelman’s Ricci flow entropy can be expressed in terms of the energy of harmonic spinors in all dimensions, and in four dimensions, in terms of the energy of Seiberg-Witten monopoles. Consequently, Ricci flow is the gradient flow of these energies. The proof relies on a weighted version of the monopole equations, introduced here. Further, a sharp parabolic Hitchin-Thorpe inequalityfor simply-connected, spin 4-manifolds is proven. From this, it follows that the normalized Ricci flow on any exotic K3 surface must become singular. Ph.D. 2024-06-27T19:45:05Z 2024-06-27T19:45:05Z 2024-05 2024-05-15T16:20:08.781Z Thesis https://hdl.handle.net/1721.1/155325 In Copyright - Educational Use Permitted Copyright retained by author(s) https://rightsstatements.org/page/InC-EDU/1.0/ application/pdf Massachusetts Institute of Technology |
| spellingShingle | Baldauf, Julius The Ricci Flow on Spin Manifolds |
| title | The Ricci Flow on Spin Manifolds |
| title_full | The Ricci Flow on Spin Manifolds |
| title_fullStr | The Ricci Flow on Spin Manifolds |
| title_full_unstemmed | The Ricci Flow on Spin Manifolds |
| title_short | The Ricci Flow on Spin Manifolds |
| title_sort | ricci flow on spin manifolds |
| url | https://hdl.handle.net/1721.1/155325 |
| work_keys_str_mv | AT baldaufjulius thericciflowonspinmanifolds AT baldaufjulius ricciflowonspinmanifolds |