Limiting behavior of Ricci flows
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2004.
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| Formato: | Tese |
| Idioma: | eng |
| Publicado em: |
Massachusetts Institute of Technology
2006
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| Acesso em linha: | http://hdl.handle.net/1721.1/32244 |
| _version_ | 1826202560073039872 |
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| author | Šešum, Nataša, 1975- |
| author2 | Gang Tian. |
| author_facet | Gang Tian. Šešum, Nataša, 1975- |
| author_sort | Šešum, Nataša, 1975- |
| collection | MIT |
| description | Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2004. |
| first_indexed | 2024-09-23T12:09:25Z |
| format | Thesis |
| id | mit-1721.1/32244 |
| institution | Massachusetts Institute of Technology |
| language | eng |
| last_indexed | 2024-09-23T12:09:25Z |
| publishDate | 2006 |
| publisher | Massachusetts Institute of Technology |
| record_format | dspace |
| spelling | mit-1721.1/322442019-04-10T22:21:47Z Limiting behavior of Ricci flows Å eÅ¡um, NataÅ¡a, 1975- Gang Tian. Massachusetts Institute of Technology. Dept. of Mathematics. Massachusetts Institute of Technology. Dept. of Mathematics. Mathematics. Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2004. Includes bibliographical references (p. 83-85). Consider the unnormalized Ricci flow ...Richard Hamilton showed that if the curvature operator is uniformly bounded under the flow for all times ... then the solution can be extended beyond T. In the thesis we prove that if the Ricci curvature is uniformly bounded under the flow for all times ... then the curvature tensor has to be uniformly bounded as well. In particular, this means that if the Ricci tensor stays uniformly bounded up to a finite time T, a Ricci flow can not develop a singularity at T. We will give two different proofs of that result. One of them relies on Hamilton's estimates on distance changes along the flow and the other one relies on the identities for reduced distances and the monotonicity formula for reduced volumes that has been introduced and proved by Perelman in [29]. Consider the Ricci flow ... on a closed, n-dimensional manifold M. Assume that a solution of the flow exists for all times ... and that the curvatures and the diameters are uniformly bounded along the flow. We will prove that for every sequence ... there exists a subsequence such that g(ti + t) converges to a metric h(t) and h(t) is a Ricci soliton. We will also prove that if one of the limit solitons is integrable, then a soliton that we get in the limit is unique up to diffeomorphisms and the convergence toward it is exponential. (cont.) We will also prove that in an arbitrary dimension, for a given Kähler-Ricci flow with uniformly bounded Ricci curvatures, for every sequence of times ti converging to infinity, there exists a subsequence such that ... and the convergence is smooth outside a singular set (which is a set of codimension at least 4). Moreover, g(t) is a solution of the flow off the singular set. In the case of a complex dimension 2, for any sequence of times converging to infinity we can find a subsequence of times such that we have a convergence toward a Kähler-Ricci soliton, away from finitely many isolated singularities. by NataÅ¡a Å eÅ¡um. Ph.D. 2006-03-29T18:27:14Z 2006-03-29T18:27:14Z 2004 2004 Thesis http://hdl.handle.net/1721.1/32244 56019708 eng M.I.T. theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. See provided URL for inquiries about permission. http://dspace.mit.edu/handle/1721.1/7582 85 p. 3317393 bytes 3315585 bytes application/pdf application/pdf application/pdf Massachusetts Institute of Technology |
| spellingShingle | Mathematics. Šešum, Nataša, 1975- Limiting behavior of Ricci flows |
| title | Limiting behavior of Ricci flows |
| title_full | Limiting behavior of Ricci flows |
| title_fullStr | Limiting behavior of Ricci flows |
| title_full_unstemmed | Limiting behavior of Ricci flows |
| title_short | Limiting behavior of Ricci flows |
| title_sort | limiting behavior of ricci flows |
| topic | Mathematics. |
| url | http://hdl.handle.net/1721.1/32244 |
| work_keys_str_mv | AT aeaumnataaa1975 limitingbehaviorofricciflows |