Final Report Summary - ISOMETRIC (Isometric actions)
The purpose of this project has been to study isometric actions on Riemannian manifolds. More specifically, our aim has been to classify polar and hyperpolar actions on symmetric spaces of noncompact type, study the geometry of their orbits, and characterise them in terms of geometric data.
This project has also been used to fund the research of Miguel Domínguez-Vázquez, a PhD student whose PhD thesis will be defended in 2013.
The main results of our research can be summarized as follows.
Publications (peer reviewed):
- J. Berndt, J. C. Díaz-Ramos, H. Tamaru: Hyperpolar homogeneous foliations on symmetric spaces of noncompact type, J. Differential Geom., 86 (2010) 191 - 235.
- J. C. Díaz-Ramos, M. Domínguez-Vázquez: Non-Hopf real hypersurfaces with constant principal curvatures in complex space forms, to appear in Indiana Univ. Math. J. (preprint arXiv:09113624 [math.DG]).
- J. C. Díaz-Ramos: M. Domínguez-Vázquez, Inhomogeneous isoparametric hypersurfaces in complex hyperbolic spaces, to appear in Math. Z. (preprint arXiv:10115160 [math.DG]).
- J. Berndt, J. C. Díaz-Ramos: Homogeneous polar foliations of complex hyperbolic spaces, to appear in Comm. Anal. Geom. (preprint arXiv:1107.0688 [math.DG]).
- J. C. Díaz-Ramos, A. Kollross: Polar actions with a fixed point, Differential Geom. Appl. 29 (2011) 20-25.
Preprints:
- J. Berndt, J. C. Díaz-Ramos: Polar actions on the complex hyperbolic plane, preprint arXiv:1108.0543 [math.DG].
- J. C. Díaz-Ramos. M. Domínguez-Vázquez: Isoparametric hypersurfaces in Damek-Ricci spaces, preprint arXiv:1111.0264 [math.DG].
- M. Domínguez Vázquez: Isoparametric foliations on complex projective spaces, preprint arXiv:1204.3428v1 [math.DG].
Conference proceedings:
- M. Domínguez Vázquez: Real hypersurfaces with constant principal curvatures in complex space forms, Differential Geom. Appl. 29 (2011), suppl. 1, 65 - 70.
- J. C. Díaz-Ramos: The orbits of cohomogeneity one actions on complex hyperbolic spaces, Differential Geometry. Proceedings of the VIII International Colloquium on Differential Geometry 146-155, World Sci. Publ. Co., Singapore, 2009. ISBN-13 978-981-4261-16-6.
- M. Domínguez Vázquez: On an inhomogeneous isoparametric family of hypersurfaces with constant principal curvatures in the Cayley hyperbolic plane, Proceedings of the XX International Fall Workshop on Geometry and Physics (Madrid 2011), to appear.
Conference and seminar talks:
- Seminar talk, Darmstadt (Germany), 12 December 2011, "Isoparametric hypersurfaces in complex space forms".
- "2nd European Young and Mobile Workshop, Geometric Analysis and Partial Differential Equations", Granada (Spain), 28 - 30 November 2011, invited talk: "Polar actions on noncompact symmetric spaces".
- "20th International Fall Workshop on Geometry and Physics", Madrid (Spain), 31 August - 3 September 2011, poster by M. Domínguez Vázquez: "Some results on isoparametric hypersurfaces in nonflat complex space forms".
- "VI International Meeting on Lorentzian Geometry", Granada (Spain), 6 - 9 September 2011, invited talk: "Polar actions on symmetric spaces of noncompact type". Talk by Miguel Domínguez-Vázquez: "Families of parallel CMC hypersurfaces in noncompact rank-one symmetric spaces".
- Seminar talk, Santiago de Compostela (Spain), 21 July 2011, "Foliaciones polares homogéneas en espacios hiperbólicos complejos".
- Seminar talk, Cologne (Germany), April 2011, talk by Miguel Domínguez-Vázquez: "Homogeneous and isoparametric hypersurfaces in non-compact rank-one symmetric spaces".
- "Conference in Geometry and Global Analysis", Santiago de Compostela, 13 - 17 December 2010, invited talk by Miguel Domínguez Vázquez: "Principal curvatures of isoparametric hypersurfaces in complex hyperbolic spaces".
- "International Meeting on Differential Geometry", Córdoba (Spain), 15 - 17 November 2010, talk by Miguel Domínguez-Vázquez: "On isoparametric hypersurfaces in complex hyperbolic spaces".
- Seminar talk, Cologne (Germany), 16 November 2010, "Non-Hopf real hypersurfaces with constant principal curvatures in complex space forms".
- "Encontro de Xeometría 2010", Ferrol (Spain), 7 October 2010, talk by Miguel Domínguez-Vázquez: "O problema da clasificación das hipersuperficies isoparamétricas".
- DGA 2010 - Differential Geometry and its Applications", Brno (Czech Republic), 27 - 31 August 2010, talk by Miguel Domínguez-Vázquez: "Real hypersurfaces with constant principal curvatures in complex space forms".
- "Workshop on Hypersurface Geometry and Integrable Systems", Sendai (Japan), 24 - 27 August 2010, talk: "Homogeneous hyperpolar foliations on symmetric spaces of noncompact type". Talk by Miguel Domínguez-Vázquez: "Real hypersurfaces with constant principal curvatures in complex projective and hyperbolic spaces".
- Seminar talk, Cologne (Germany), 2 July 2010, "Homogeneous hyperpolar foliations on symmetric spaces of noncompact type".
- "Lectures on Lorentzian Geometry", Granada (Spain), 8 - 11 February 2010, poster: "Hypersurfaces with constant principal curvatures in complex projective and hyperbolic spaces".
- Seminar talk, Augsburg (Germany), 8 January 2010, "Hyperpolar homogeneous foliations on noncompact symmetric spaces".
- "Geometry Meeting 2009", Ferrol (Spain), 29 - 30 October 2009, invited talk: "Acciones hiperpolares en espacios simétricos no compactos".
Organisation of scientific events:
- "Differential Geometry days", Santiago de Compostela, forecoming 27 - 29 June 2012.
- "Conference in Geometry and Global Analysis", Santiago de Compostela, 13 - 17 December 2010.
- "Workshop on Differential Geometry and Relativity", Cedeira (Spain), 31 October - 1 November 2009.
This project has also been used to fund the research of Miguel Domínguez-Vázquez, a PhD student whose PhD thesis will be defended in 2013.
The main results of our research can be summarized as follows.
Publications (peer reviewed):
- J. Berndt, J. C. Díaz-Ramos, H. Tamaru: Hyperpolar homogeneous foliations on symmetric spaces of noncompact type, J. Differential Geom., 86 (2010) 191 - 235.
- J. C. Díaz-Ramos, M. Domínguez-Vázquez: Non-Hopf real hypersurfaces with constant principal curvatures in complex space forms, to appear in Indiana Univ. Math. J. (preprint arXiv:09113624 [math.DG]).
- J. C. Díaz-Ramos: M. Domínguez-Vázquez, Inhomogeneous isoparametric hypersurfaces in complex hyperbolic spaces, to appear in Math. Z. (preprint arXiv:10115160 [math.DG]).
- J. Berndt, J. C. Díaz-Ramos: Homogeneous polar foliations of complex hyperbolic spaces, to appear in Comm. Anal. Geom. (preprint arXiv:1107.0688 [math.DG]).
- J. C. Díaz-Ramos, A. Kollross: Polar actions with a fixed point, Differential Geom. Appl. 29 (2011) 20-25.
Preprints:
- J. Berndt, J. C. Díaz-Ramos: Polar actions on the complex hyperbolic plane, preprint arXiv:1108.0543 [math.DG].
- J. C. Díaz-Ramos. M. Domínguez-Vázquez: Isoparametric hypersurfaces in Damek-Ricci spaces, preprint arXiv:1111.0264 [math.DG].
- M. Domínguez Vázquez: Isoparametric foliations on complex projective spaces, preprint arXiv:1204.3428v1 [math.DG].
Conference proceedings:
- M. Domínguez Vázquez: Real hypersurfaces with constant principal curvatures in complex space forms, Differential Geom. Appl. 29 (2011), suppl. 1, 65 - 70.
- J. C. Díaz-Ramos: The orbits of cohomogeneity one actions on complex hyperbolic spaces, Differential Geometry. Proceedings of the VIII International Colloquium on Differential Geometry 146-155, World Sci. Publ. Co., Singapore, 2009. ISBN-13 978-981-4261-16-6.
- M. Domínguez Vázquez: On an inhomogeneous isoparametric family of hypersurfaces with constant principal curvatures in the Cayley hyperbolic plane, Proceedings of the XX International Fall Workshop on Geometry and Physics (Madrid 2011), to appear.
Conference and seminar talks:
- Seminar talk, Darmstadt (Germany), 12 December 2011, "Isoparametric hypersurfaces in complex space forms".
- "2nd European Young and Mobile Workshop, Geometric Analysis and Partial Differential Equations", Granada (Spain), 28 - 30 November 2011, invited talk: "Polar actions on noncompact symmetric spaces".
- "20th International Fall Workshop on Geometry and Physics", Madrid (Spain), 31 August - 3 September 2011, poster by M. Domínguez Vázquez: "Some results on isoparametric hypersurfaces in nonflat complex space forms".
- "VI International Meeting on Lorentzian Geometry", Granada (Spain), 6 - 9 September 2011, invited talk: "Polar actions on symmetric spaces of noncompact type". Talk by Miguel Domínguez-Vázquez: "Families of parallel CMC hypersurfaces in noncompact rank-one symmetric spaces".
- Seminar talk, Santiago de Compostela (Spain), 21 July 2011, "Foliaciones polares homogéneas en espacios hiperbólicos complejos".
- Seminar talk, Cologne (Germany), April 2011, talk by Miguel Domínguez-Vázquez: "Homogeneous and isoparametric hypersurfaces in non-compact rank-one symmetric spaces".
- "Conference in Geometry and Global Analysis", Santiago de Compostela, 13 - 17 December 2010, invited talk by Miguel Domínguez Vázquez: "Principal curvatures of isoparametric hypersurfaces in complex hyperbolic spaces".
- "International Meeting on Differential Geometry", Córdoba (Spain), 15 - 17 November 2010, talk by Miguel Domínguez-Vázquez: "On isoparametric hypersurfaces in complex hyperbolic spaces".
- Seminar talk, Cologne (Germany), 16 November 2010, "Non-Hopf real hypersurfaces with constant principal curvatures in complex space forms".
- "Encontro de Xeometría 2010", Ferrol (Spain), 7 October 2010, talk by Miguel Domínguez-Vázquez: "O problema da clasificación das hipersuperficies isoparamétricas".
- DGA 2010 - Differential Geometry and its Applications", Brno (Czech Republic), 27 - 31 August 2010, talk by Miguel Domínguez-Vázquez: "Real hypersurfaces with constant principal curvatures in complex space forms".
- "Workshop on Hypersurface Geometry and Integrable Systems", Sendai (Japan), 24 - 27 August 2010, talk: "Homogeneous hyperpolar foliations on symmetric spaces of noncompact type". Talk by Miguel Domínguez-Vázquez: "Real hypersurfaces with constant principal curvatures in complex projective and hyperbolic spaces".
- Seminar talk, Cologne (Germany), 2 July 2010, "Homogeneous hyperpolar foliations on symmetric spaces of noncompact type".
- "Lectures on Lorentzian Geometry", Granada (Spain), 8 - 11 February 2010, poster: "Hypersurfaces with constant principal curvatures in complex projective and hyperbolic spaces".
- Seminar talk, Augsburg (Germany), 8 January 2010, "Hyperpolar homogeneous foliations on noncompact symmetric spaces".
- "Geometry Meeting 2009", Ferrol (Spain), 29 - 30 October 2009, invited talk: "Acciones hiperpolares en espacios simétricos no compactos".
Organisation of scientific events:
- "Differential Geometry days", Santiago de Compostela, forecoming 27 - 29 June 2012.
- "Conference in Geometry and Global Analysis", Santiago de Compostela, 13 - 17 December 2010.
- "Workshop on Differential Geometry and Relativity", Cedeira (Spain), 31 October - 1 November 2009.