The moment map and the group of volume-preserving diffeomorphisms: applications to differential geometry
In this work, we use this moment map to relate solutions to certain differential equations to (i) diffeomorphisms of compact Riemann surfaces, (ii) additional Kähler metrics on a given compact Kähler manifold, and (iii) symplectic forms on 4-manifolds.
- Autores:
-
Dorado Toro, Daniel Fernando
- Tipo de recurso:
- Trabajo de grado de pregrado
- Fecha de publicación:
- 2023
- Institución:
- Universidad de los Andes
- Repositorio:
- Séneca: repositorio Uniandes
- Idioma:
- eng
- OAI Identifier:
- oai:repositorio.uniandes.edu.co:1992/69072
- Acceso en línea:
- http://hdl.handle.net/1992/69072
- Palabra clave:
- Differential geometry
Moment map
Symplectic geometry
Manifolds
Kähler manifolds
Infinite-dimensional manifolds
Diffeomorphism groups
Matemáticas
- Rights
- openAccess
- License
- Attribution-NonCommercial-NoDerivatives 4.0 Internacional
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The moment map and the group of volume-preserving diffeomorphisms: applications to differential geometry |
| title |
The moment map and the group of volume-preserving diffeomorphisms: applications to differential geometry |
| spellingShingle |
The moment map and the group of volume-preserving diffeomorphisms: applications to differential geometry Differential geometry Moment map Symplectic geometry Manifolds Kähler manifolds Infinite-dimensional manifolds Diffeomorphism groups Matemáticas |
| title_short |
The moment map and the group of volume-preserving diffeomorphisms: applications to differential geometry |
| title_full |
The moment map and the group of volume-preserving diffeomorphisms: applications to differential geometry |
| title_fullStr |
The moment map and the group of volume-preserving diffeomorphisms: applications to differential geometry |
| title_full_unstemmed |
The moment map and the group of volume-preserving diffeomorphisms: applications to differential geometry |
| title_sort |
The moment map and the group of volume-preserving diffeomorphisms: applications to differential geometry |
| dc.creator.fl_str_mv |
Dorado Toro, Daniel Fernando |
| dc.contributor.advisor.none.fl_str_mv |
Cardona Guio, Alexander |
| dc.contributor.author.none.fl_str_mv |
Dorado Toro, Daniel Fernando |
| dc.contributor.jury.none.fl_str_mv |
Cortissoz Iriarte, Jean Carlos |
| dc.subject.keyword.none.fl_str_mv |
Differential geometry Moment map Symplectic geometry Manifolds Kähler manifolds Infinite-dimensional manifolds Diffeomorphism groups |
| topic |
Differential geometry Moment map Symplectic geometry Manifolds Kähler manifolds Infinite-dimensional manifolds Diffeomorphism groups Matemáticas |
| dc.subject.themes.es_CO.fl_str_mv |
Matemáticas |
| description |
In this work, we use this moment map to relate solutions to certain differential equations to (i) diffeomorphisms of compact Riemann surfaces, (ii) additional Kähler metrics on a given compact Kähler manifold, and (iii) symplectic forms on 4-manifolds. |
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2023 |
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2023-08-02T13:28:27Z |
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2023-08-02T13:28:27Z |
| dc.date.issued.none.fl_str_mv |
2023-06-01 |
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Trabajo de grado - Pregrado |
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eng |
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eng |
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F. Klein, «Vergleichende Betrachtungen über neuere geometrische Forschungen», de, Mathematische Annalen 43, 63-100 (1893). K. Mann, «The Structure of Homeomorphism and Diffeomorphism Groups», en, Notices of the Amer- ican Mathematical Society 68, 1 (2021). S. K. Donaldson, «Moment maps and diffeomorphisms», Surveys in differential geometry 3, 107-127 (2002). S. K. Donaldson, «Moment maps in differential geometry», Surveys in differential geometry 8, 171-189 (2003). D. McDuff and D. Salamon, Introduction to symplectic topology (Oxford University Press, Mar. 2017). R. H. Abraham and J. E. Marsden, Foundations of mechanics, eng, 2. ed., rev., enl., and reset (Perseus Books, Cambridge, Mass, 2002). R. Bryant, «An introduction to Lie groups and symplectic geometry», in Geometry and Quantum Field Theory, Vol. 1, edited by D. S. Freed and K. K. Uhlenbeck, IAS/Park City mathematics series (AMS and IAS/Park City Mathematics Institute, 1995), pp. 5-181. D. Huybrechts, Complex geometry: an introduction, Universitext (Springer, Berlin ; New York, 2005), 309 pp. N. Hitchin, «Hyperkähler manifolds», Séminaire Bourbaki 34, 137-166 (1992). A. Schmeding, An introduction to infinite-dimensional differential geometry, Cambridge Studies in Advanced Mathematics (Cambridge University Press, 2022). H. Amiri, H. Glöckner, and A. Schmeding, «Lie groupoids of mappings taking values in a Lie groupoid», en, Archivum Mathematicum, 307-356 (2020). A. Banyaga, The structure of classical diffeomorphism groups, Mathematics and Its Applications (Springer Verlag, 1997). K.-H. Neeb, Infinite-Dimensional Lie Groups, 2005. J. M. Lee, Introduction to smooth manifolds, 2nd ed, Graduate texts in mathematics 218 (Springer, 2013). H. Hofer and E. Zehnder, Symplectic invariants and Hamiltonian dynamics, Birkhäuser Advanced Texts Basler Lehrbücher (Birkhäuser, Basel, 1994). J. Marsden and A. Weinstein, «Reduction of symplectic manifolds with symmetry», en, Reports on Mathematical Physics 5, 121-130 (1974). S. K. Donaldson and P. Kronheimer, The Geometry of Four-Manifolds, eng, Reprinted, Oxford math- ematical monographs (Clarendon Press, Oxford, 2007). E. Calabi, «Métriques kählériennes et fibrés holomorphes», Annales scientifiques de l'École normale supérieure 12, 269-294 (1979). P. Michor and C. Vizman, «n-transitivity of certain diffeomorphism groups.», Acta Mathematica Universitatis Comenianae. New Series 63, 221-225 (1994). H. Omori, «On Banach-Lie groups acting on finite dimensional manifolds», Tohoku Mathematical Journal 30, 223-250 (1978). J. B. Conway, A Course in Functional Analysis, en, Vol. 96, Graduate Texts in Mathematics (Springer New York, New York, NY, 2007). R. Meise and D. Vogt, Introduction to functional analysis, Oxford graduate texts in mathematics 2 (Clarendon Press ; Oxford University Press, Oxford : New York, 1997). P. W. Michor, Manifolds of differentiable mappings, Shiva mathematics series ; 3 (Shiva Pub, Orpington [Eng.], 1980). L. C. Evans, Partial differential equations, 2nd ed, Graduate studies in mathematics v. 19, OCLC: ocn465190110 (American Mathematical Society, Providence, R.I, 2010). C.-V. Pao, Nonlinear parabolic and elliptic equations, Softcover reprint of the hardcover 1st edition 1992 (Springer Science + Business Media, LLC, New York, 2013). D. H. Phong, J. Song, and J. Sturm, «Complex Monge Ampere Equations», 10.48550/ARXIV.1209. 2203 (2012). X. Chen, «On the lower bound of the Mabuchi energy and its application», International Mathematics Research Notices 2000, 607 (2000). |
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Klein, «Vergleichende Betrachtungen über neuere geometrische Forschungen», de, Mathematische Annalen 43, 63-100 (1893).K. Mann, «The Structure of Homeomorphism and Diffeomorphism Groups», en, Notices of the Amer- ican Mathematical Society 68, 1 (2021).S. K. Donaldson, «Moment maps and diffeomorphisms», Surveys in differential geometry 3, 107-127 (2002).S. K. Donaldson, «Moment maps in differential geometry», Surveys in differential geometry 8, 171-189 (2003).D. McDuff and D. Salamon, Introduction to symplectic topology (Oxford University Press, Mar. 2017).R. H. Abraham and J. E. Marsden, Foundations of mechanics, eng, 2. ed., rev., enl., and reset (Perseus Books, Cambridge, Mass, 2002).R. Bryant, «An introduction to Lie groups and symplectic geometry», in Geometry and Quantum Field Theory, Vol. 1, edited by D. S. Freed and K. K. Uhlenbeck, IAS/Park City mathematics series (AMS and IAS/Park City Mathematics Institute, 1995), pp. 5-181.D. Huybrechts, Complex geometry: an introduction, Universitext (Springer, Berlin ; New York, 2005), 309 pp.N. Hitchin, «Hyperkähler manifolds», Séminaire Bourbaki 34, 137-166 (1992).A. Schmeding, An introduction to infinite-dimensional differential geometry, Cambridge Studies in Advanced Mathematics (Cambridge University Press, 2022).H. Amiri, H. Glöckner, and A. Schmeding, «Lie groupoids of mappings taking values in a Lie groupoid», en, Archivum Mathematicum, 307-356 (2020).A. Banyaga, The structure of classical diffeomorphism groups, Mathematics and Its Applications (Springer Verlag, 1997).K.-H. Neeb, Infinite-Dimensional Lie Groups, 2005.J. M. Lee, Introduction to smooth manifolds, 2nd ed, Graduate texts in mathematics 218 (Springer, 2013).H. Hofer and E. Zehnder, Symplectic invariants and Hamiltonian dynamics, Birkhäuser Advanced Texts Basler Lehrbücher (Birkhäuser, Basel, 1994).J. Marsden and A. Weinstein, «Reduction of symplectic manifolds with symmetry», en, Reports on Mathematical Physics 5, 121-130 (1974).S. K. Donaldson and P. Kronheimer, The Geometry of Four-Manifolds, eng, Reprinted, Oxford math- ematical monographs (Clarendon Press, Oxford, 2007).E. Calabi, «Métriques kählériennes et fibrés holomorphes», Annales scientifiques de l'École normale supérieure 12, 269-294 (1979).P. Michor and C. Vizman, «n-transitivity of certain diffeomorphism groups.», Acta Mathematica Universitatis Comenianae. New Series 63, 221-225 (1994).H. Omori, «On Banach-Lie groups acting on finite dimensional manifolds», Tohoku Mathematical Journal 30, 223-250 (1978).J. B. Conway, A Course in Functional Analysis, en, Vol. 96, Graduate Texts in Mathematics (Springer New York, New York, NY, 2007).R. Meise and D. Vogt, Introduction to functional analysis, Oxford graduate texts in mathematics 2 (Clarendon Press ; Oxford University Press, Oxford : New York, 1997).P. W. Michor, Manifolds of differentiable mappings, Shiva mathematics series ; 3 (Shiva Pub, Orpington [Eng.], 1980).L. C. Evans, Partial differential equations, 2nd ed, Graduate studies in mathematics v. 19, OCLC: ocn465190110 (American Mathematical Society, Providence, R.I, 2010).C.-V. Pao, Nonlinear parabolic and elliptic equations, Softcover reprint of the hardcover 1st edition 1992 (Springer Science + Business Media, LLC, New York, 2013).D. H. Phong, J. Song, and J. Sturm, «Complex Monge Ampere Equations», 10.48550/ARXIV.1209. 2203 (2012).X. 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