Decomposition of the Laplace-Beltrami operator on riemannian manifolds
Ilustraciones
- Autores:
-
Estévez Joya, Lizeth Alexandra
- Tipo de recurso:
- Fecha de publicación:
- 2021
- Institución:
- Universidad Nacional de Colombia
- Repositorio:
- Universidad Nacional de Colombia
- Idioma:
- eng
- OAI Identifier:
- oai:repositorio.unal.edu.co:unal/80138
- Palabra clave:
- 510 - Matemáticas::516 - Geometría
Variables diferenciables
Differential manifolds
Variedades de Riemann
Riemann manifolds
Differential equations
Ecuaciones diferenciales
Laplace-Beltrami operator
Manifolds with indefinite metrics
Existence problems for PDEs
Polar actions
Variedades con métricas indefinidas
Problemas de existencia para EDPs
Operador de Laplace-Beltrami
Acciones polares
- Rights
- openAccess
- License
- Atribución-NoComercial-CompartirIgual 4.0 Internacional
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Decomposition of the Laplace-Beltrami operator on riemannian manifolds |
| dc.title.translated.spa.fl_str_mv |
Descomposición del operador Laplace-Beltrami sobre variedades riemannianas |
| title |
Decomposition of the Laplace-Beltrami operator on riemannian manifolds |
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Decomposition of the Laplace-Beltrami operator on riemannian manifolds 510 - Matemáticas::516 - Geometría Variables diferenciables Differential manifolds Variedades de Riemann Riemann manifolds Differential equations Ecuaciones diferenciales Laplace-Beltrami operator Manifolds with indefinite metrics Existence problems for PDEs Polar actions Variedades con métricas indefinidas Problemas de existencia para EDPs Operador de Laplace-Beltrami Acciones polares |
| title_short |
Decomposition of the Laplace-Beltrami operator on riemannian manifolds |
| title_full |
Decomposition of the Laplace-Beltrami operator on riemannian manifolds |
| title_fullStr |
Decomposition of the Laplace-Beltrami operator on riemannian manifolds |
| title_full_unstemmed |
Decomposition of the Laplace-Beltrami operator on riemannian manifolds |
| title_sort |
Decomposition of the Laplace-Beltrami operator on riemannian manifolds |
| dc.creator.fl_str_mv |
Estévez Joya, Lizeth Alexandra |
| dc.contributor.advisor.none.fl_str_mv |
Becerra Rojas, Edward Samuel |
| dc.contributor.author.none.fl_str_mv |
Estévez Joya, Lizeth Alexandra |
| dc.subject.ddc.spa.fl_str_mv |
510 - Matemáticas::516 - Geometría |
| topic |
510 - Matemáticas::516 - Geometría Variables diferenciables Differential manifolds Variedades de Riemann Riemann manifolds Differential equations Ecuaciones diferenciales Laplace-Beltrami operator Manifolds with indefinite metrics Existence problems for PDEs Polar actions Variedades con métricas indefinidas Problemas de existencia para EDPs Operador de Laplace-Beltrami Acciones polares |
| dc.subject.lemb.none.fl_str_mv |
Variables diferenciables Differential manifolds Variedades de Riemann Riemann manifolds Differential equations Ecuaciones diferenciales |
| dc.subject.proposal.eng.fl_str_mv |
Laplace-Beltrami operator Manifolds with indefinite metrics Existence problems for PDEs Polar actions |
| dc.subject.proposal.spa.fl_str_mv |
Variedades con métricas indefinidas Problemas de existencia para EDPs Operador de Laplace-Beltrami Acciones polares |
| description |
Ilustraciones |
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2021 |
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2021-09-08T21:44:10Z |
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2021-09-08T21:44:10Z |
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2021 |
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Trabajo de grado - Maestría |
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info:eu-repo/semantics/masterThesis |
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info:eu-repo/semantics/acceptedVersion |
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Text |
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acceptedVersion |
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https://repositorio.unal.edu.co/handle/unal/80138 |
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Universidad Nacional de Colombia |
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Repositorio Institucional Universidad Nacional de Colombia |
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https://repositorio.unal.edu.co/ |
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https://repositorio.unal.edu.co/handle/unal/80138 https://repositorio.unal.edu.co/ |
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Universidad Nacional de Colombia Repositorio Institucional Universidad Nacional de Colombia |
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eng |
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eng |
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P. Ahmadi and S. Kashani. Cohomogeneity one Minkowski space R^n_1. Publicationes mathematicae, 78, 02 2011 P. Ahmadi, S. Safari, and M. Hassani. A classification of cohomogeneity one actions on the Minkowski space R. Bulletin of the Iranian Mathematical Society, 10 2020 N. M. Alba, J. Galvis, and E. Becerra. On polar actions invariant solutions of semilinear equations on manifolds. arXiv:1802.08625, 2018 M.M. Alexandrino and R.G. Bettiol. Lie groups and geometric aspects of isometric actions. Springer International Publishing, 2015 J.K. Beem, P. Ehrlich, and K. Easley. Global Lorentzian geometry, second edition. Chapman & Hall/CRC Pure and Applied Mathematics. Taylor & Francis, 1996 J. Berndt, S. Console, and C.E. Olmos. Submanifolds and holonomy. Chapman & Hall/CRC Monographs and Research Notes in Mathematics. CRC Press, second edition, 2016 C. Bär. Lorentzian Geometry. Lecture Notes, Summer Term 2004. Universit at Potsdam, 2020. C. Bär, N. Ginoux, and F. Pf a e. Wave equations on Lorentzian manifolds and quantization. European Mathematical Society., 2007. José Díaz-Ramos. Proper isometric actions. 12 2008. Paul Ehrlich, Yoon-Tae Jung, Jeong-Sik Kim, and Seon-Bu Kim. Jacobians and volume comparison for Lorentzian warped products. Contemporary Mathematics, "Recent Advances in Riemannian and Lorentzian Geometries", pages 39{52, 01 2003. F. Flaherty and M.P. do Carmo. Riemannian geometry. Mathematics: Theory & Applications. Birkh auser Boston, 2013. S. Helgason. Differential operators on homogeneous spaces. Acta Math., 102(3-4):239-299 1959. S. Helgason. Di erential geometry and symmetric spaces. Pure and Applied Mathematics. Elsevier Science, 1962. S. Helgason. A formula for the radial part of the Laplace-Beltrami operator. Journal of Differential Geometry, 6(3):411-419, 1972. S. Helgason. Groups and geometric analysis: integral geometry, invariant di erential operators, and spherical functions. Pure and applied mathematics. Academic Press, 1984. S. Helgason. The Radon transform. Progress in Mathematics. Birkh auser Boston, 1999. J. Jost. Riemannian geometry and geometric analysis. Universitext. Springer Berlin Heidelberg, 2011. J. Lee. Introduction to smooth manifolds. Graduate Texts in Mathematics. Springer New York, 2012. J.M. Lee. Riemannian manifolds: An introduction to curvature. Graduate Texts in Mathematics. Springer New York, 2006. J.M. Lee. Introduction to Riemannian manifolds. Graduate Texts in Mathematics. Springer International Publishing, 2019. B. O'Neill. Semi-Riemannian geometry with applications to relativity. Pure and Applied Mathematics. Elsevier Science, 1983. T. Sakai. Riemannian geometry. Fields Institute Communications. American Mathematical Soc., 1996. R.A. Wijsman. Invariant measures on groups and their use in statistics. IMS Lecture Notes. Institute of Mathematical Statistics, 1990. Joseph A. Wolf. Spaces of constant curvature. sixth edition. AMS, 2011. Hongtao Xue and Xigao Shao. Existence of positive entire solutions of a semilinear elliptic problem with a gradient term. Nonlinear Analysis: Theory, Methods & Applications, 71(7):3113-3118, 2009. Q.S. Zhang. Sobolev inequalities, heat kernels under Ricci flow, and the Poincare conjecture. CRC Press, 2010. |
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Atribución-NoComercial-CompartirIgual 4.0 Internacional |
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ix, 95 páginas |
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Universidad Nacional de Colombia |
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Bogotá - Ciencias - Maestría en Ciencias - Matemáticas |
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Departamento de Matemáticas |
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Facultad de Ciencias |
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Bogotá, Colombia |
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Universidad Nacional de Colombia - Sede Bogotá |
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Atribución-NoComercial-CompartirIgual 4.0 Internacionalhttp://creativecommons.org/licenses/by-nc-sa/4.0/info:eu-repo/semantics/openAccesshttp://purl.org/coar/access_right/c_abf2Becerra Rojas, Edward Samuel9a9d39ad891fd65c4cf1bb2e325e446aEstévez Joya, Lizeth Alexandra51b2b50790f951f77d0ce9839e4871c32021-09-08T21:44:10Z2021-09-08T21:44:10Z2021https://repositorio.unal.edu.co/handle/unal/80138Universidad Nacional de ColombiaRepositorio Institucional Universidad Nacional de Colombiahttps://repositorio.unal.edu.co/IlustracionesThe goal of this work is to study the geometric conditions required on a pseudo-Riemannian manifold M to guarantee the existence of solutions of a second order partial differential equation posed on M. We closely follow the basic ideas in [3], where the considerations are carried out in the presence of a Riemannian metric and it is used the decomposition of the Laplace-Beltrami operator given by Helgason in [14]. In the indefinite case, the geometry of the manifold changes significantly and there are various pitfalls to watch out for. However, conditions for Helgason's results, in a certain sense, can be naturally set. Hence, we consider M as a globally hyperbolic Lorentzian manifold because a one dimensional submanifold \Sigma transversal to the orbits of a given group action is easily recognisable. This means M is endowed with a polar action and thus the equation can be reduced on \Sigma as in the Riemannian case. Then the solutions are obtained in such a way that these are constant along the orbits of the action. Finally, we propose an extension of our considerations to Lorentzian warped products.En este trabajo estudiamos las condiciones geométricas requeridas sobre una variedad pseudo-Riemanniana M para garantizar la existencia de soluciones de una ecuación diferencial parcial de segundo orden planteada sobre M. Seguimos de cerca las ideas principales expuestas en [3], donde las consideraciones se llevan a cabo en presencia de una métrica Riemanniana y se usa la descomposición del operador Laplace-Beltrami dada por Helgason en [14]. A pesar de que en el caso indefinido la geometría de la variedad cambia significativamente y hay varios obstáculos a los que prestar atención, las condiciones para los resultados de Helgason se pueden encontrar naturalmente. En vista de esto, consideramos M como una variedad Lorentziana globalmente hiperbólica, pues en este contexto se puede identificar fácilmente una subvariedad unidimensional \Sigma transversal a las órbitas de una acción de grupo dada. Esto significa que M está dotada de una acción polar y así la ecuación se puede reducir sobre \Sigma como en el caso Riemanniano. Entonces las soluciones se obtienen de tal manera que resultan ser constantes a largo de las órbitas de la acción. Por último, proponemos una extensión de nuestras consideraciones a productos warped Lorentzianos. (Texto tomado de la fuente).MaestríaMagíster en Ciencias - MatemáticasDifferential Geometryix, 95 páginasapplication/pdfengUniversidad Nacional de ColombiaBogotá - Ciencias - Maestría en Ciencias - MatemáticasDepartamento de MatemáticasFacultad de CienciasBogotá, ColombiaUniversidad Nacional de Colombia - Sede Bogotá510 - Matemáticas::516 - GeometríaVariables diferenciablesDifferential manifoldsVariedades de RiemannRiemann manifoldsDifferential equationsEcuaciones diferencialesLaplace-Beltrami operatorManifolds with indefinite metricsExistence problems for PDEsPolar actionsVariedades con métricas indefinidasProblemas de existencia para EDPsOperador de Laplace-BeltramiAcciones polaresDecomposition of the Laplace-Beltrami operator on riemannian manifoldsDescomposición del operador Laplace-Beltrami sobre variedades riemannianasTrabajo de grado - Maestríainfo:eu-repo/semantics/masterThesisinfo:eu-repo/semantics/acceptedVersionTexthttp://purl.org/redcol/resource_type/TMP. Ahmadi and S. Kashani. Cohomogeneity one Minkowski space R^n_1. Publicationes mathematicae, 78, 02 2011P. Ahmadi, S. Safari, and M. Hassani. A classification of cohomogeneity one actions on the Minkowski space R. Bulletin of the Iranian Mathematical Society, 10 2020N. M. Alba, J. Galvis, and E. Becerra. On polar actions invariant solutions of semilinear equations on manifolds. arXiv:1802.08625, 2018M.M. Alexandrino and R.G. Bettiol. Lie groups and geometric aspects of isometric actions. Springer International Publishing, 2015J.K. Beem, P. Ehrlich, and K. Easley. Global Lorentzian geometry, second edition. Chapman & Hall/CRC Pure and Applied Mathematics. Taylor & Francis, 1996J. Berndt, S. Console, and C.E. Olmos. Submanifolds and holonomy. Chapman & Hall/CRC Monographs and Research Notes in Mathematics. CRC Press, second edition, 2016C. Bär. Lorentzian Geometry. Lecture Notes, Summer Term 2004. Universit at Potsdam, 2020.C. Bär, N. Ginoux, and F. Pf a e. Wave equations on Lorentzian manifolds and quantization. European Mathematical Society., 2007.José Díaz-Ramos. Proper isometric actions. 12 2008.Paul Ehrlich, Yoon-Tae Jung, Jeong-Sik Kim, and Seon-Bu Kim. Jacobians and volume comparison for Lorentzian warped products. Contemporary Mathematics, "Recent Advances in Riemannian and Lorentzian Geometries", pages 39{52, 01 2003.F. Flaherty and M.P. do Carmo. Riemannian geometry. Mathematics: Theory & Applications. Birkh auser Boston, 2013.S. Helgason. Differential operators on homogeneous spaces. Acta Math., 102(3-4):239-299 1959.S. Helgason. Di erential geometry and symmetric spaces. Pure and Applied Mathematics. Elsevier Science, 1962.S. Helgason. A formula for the radial part of the Laplace-Beltrami operator. Journal of Differential Geometry, 6(3):411-419, 1972.S. Helgason. Groups and geometric analysis: integral geometry, invariant di erential operators, and spherical functions. Pure and applied mathematics. Academic Press, 1984.S. Helgason. The Radon transform. Progress in Mathematics. Birkh auser Boston, 1999.J. Jost. Riemannian geometry and geometric analysis. Universitext. Springer Berlin Heidelberg, 2011.J. Lee. Introduction to smooth manifolds. Graduate Texts in Mathematics. Springer New York, 2012.J.M. Lee. Riemannian manifolds: An introduction to curvature. Graduate Texts in Mathematics. Springer New York, 2006.J.M. Lee. Introduction to Riemannian manifolds. Graduate Texts in Mathematics. Springer International Publishing, 2019.B. O'Neill. Semi-Riemannian geometry with applications to relativity. Pure and Applied Mathematics. Elsevier Science, 1983.T. Sakai. Riemannian geometry. Fields Institute Communications. American Mathematical Soc., 1996.R.A. Wijsman. Invariant measures on groups and their use in statistics. IMS Lecture Notes. Institute of Mathematical Statistics, 1990.Joseph A. Wolf. Spaces of constant curvature. sixth edition. AMS, 2011.Hongtao Xue and Xigao Shao. Existence of positive entire solutions of a semilinear elliptic problem with a gradient term. Nonlinear Analysis: Theory, Methods & Applications, 71(7):3113-3118, 2009.Q.S. Zhang. Sobolev inequalities, heat kernels under Ricci flow, and the Poincare conjecture. CRC Press, 2010.EstudiantesInvestigadoresPúblico generalORIGINAL1049627183.2021.pdf1049627183.2021.pdfTesis de Maestría en Ciencias - Matemáticasapplication/pdf724232https://repositorio.unal.edu.co/bitstream/unal/80138/4/1049627183.2021.pdf69e5159438fb9cdd745b02bfac1ba318MD54LICENSElicense.txtlicense.txttext/plain; charset=utf-83964https://repositorio.unal.edu.co/bitstream/unal/80138/3/license.txtcccfe52f796b7c63423298c2d3365fc6MD53THUMBNAIL1049627183.2021.pdf.jpg1049627183.2021.pdf.jpgGenerated Thumbnailimage/jpeg3839https://repositorio.unal.edu.co/bitstream/unal/80138/5/1049627183.2021.pdf.jpg44301c13a010088b24c4bc710466819bMD55unal/80138oai:repositorio.unal.edu.co:unal/801382024-07-29 00:00:04.903Repositorio Institucional Universidad Nacional de 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