Report

Teaser, summary, work performed and final results

Periodic Reporting for period 1 - ISOPARAMETRIC (Geometric and analytic aspects of isoparametric hypersurfaces)

Teaser

Isoparametric hypersurfaces are intriguing geometric objects whose study traces back to works of É. Cartan, B. Segre and T. Levi-Civita in the 30s. In Riemannian geometry, a hypersurface is called isoparametric if it and its nearby equidistant hypersurfaces have constant mean...

Summary

Isoparametric hypersurfaces are intriguing geometric objects whose study traces back to works of É. Cartan, B. Segre and T. Levi-Civita in the 30s. In Riemannian geometry, a hypersurface is called isoparametric if it and its nearby equidistant hypersurfaces have constant mean curvature. Over the last decades, the study of these objects has revealed connections with several areas of mathematics and mathematical physics.

The aim of the project ISOPARAMETRIC was to investigate isoparametric hypersurfaces and some related geometric and analytic concepts by combining already established tools with methods from geometric analysis and partial differential equations. Thus, we have investigated isoparametric hypersurfaces and some related notions in general Riemannian manifolds (WP1), the role of these objects in overdetermined problems of elliptic partial differential equations (WP2), and the geometry and symmetry of certain hypersurfaces with constant mean curvature in noncompact symmetric spaces (WP3). Among the results and conclusions obtained within the framework of this project, we emphasize:

1) The classification of isoparametric hypersurfaces in homogeneous 3-spaces with 4-dimensional isometry group.
2) Under certain conditions, homogeneous manifolds whose isotropy representation induces an isoparametric foliation are symmetric spaces.
3) Compact and asymptotically homogeneous Riemannian manifolds admit solution domains to overdetermined boundary problems of a wide range of semilinear elliptic partial differential equations. The boundary of such domains are isoparametric in the case of harmonic spaces.
4) Ruled real hypersurfaces with constant mean curvature in nonflat complex space forms are minimal, and hence can be classified.
5) We classified isoparametric hypersurfaces in complex hyperbolic spaces. The only compact examples are geodesic spheres.

This project has been implemented within the Geometric Analysis team at ICMAT, in Madrid, under the supervision of one of its main researchers, Alberto Enciso. Several dissemination and public engagement activities were developed during the duration of the action, including invited lectures at international conferences and seminars, various short stays at different research centers, and general public outreach and communication activities. Moreover, the researcher has been involved in teaching tasks and supervision of students.

Work performed

Research results and published articles:

- Classification of isoparametric surfaces in homogeneous 3-manifolds with 4-dimensional isometry group. Joint work with J. M. Manzano.

- Investigation of homogeneous Riemannian manifolds whose isotropy actions produce isoparametric foliations. Joint work with J. C. Díaz-Ramos and A. Kollross.

- Existence of solutions to overdetermined boundary value problems associated with a rather broad family of semilinear elliptic equations in general Riemannian manifolds. Partial symmetry result for the border of such domains in harmonic spaces. Joint work with A. Enciso and D. Peralta-Salas.

- Ruled hypersurfaces with constant mean curvature in nonflat complex space forms are minimal and, therefore, can be classified into three types. Joint work with O. Pérez-Barral.

- Survey article containing an introduction to submanifold geometry in symmetric spaces of noncompact type. Joint work with J. C. Díaz-Ramos and V. Sanmartín-López.

- Classification of isoparametric hypersurfaces in complex hyperbolic spaces. Joint work with J. C. Díaz-Ramos and V. Sanmartín-López.

- Alternative proof to the fact that the principal curvatures of isoparametric hypersurfaces in complex hyperbolic spaces coincide with those of homogeneous hypersurfaces. Joint work with J. C. Díaz-Ramos and V. Sanmartín-López.

- Classification of isoparametric submanifolds in two-dimensional complex space forms. Joint work with J. C. Díaz-Ramos and C. Vidal-Castiñeira.

- Characterization of strongly 2-Hopf and certain austere hypersurfaces in complex projective and hyperbolic planes. Joint work with J. C. Díaz-Ramos and C. Vidal-Castiñeira.

- Almost complete classification of polar foliations on quaternionic projective spaces. Joint work with C. Gorodski.

Training and transfer-of-knowledge activities:

- Supervision of several students at undergraduate, master and PhD levels.

- Invitation of two researchers to learn about totally geodesic Riemannian foliations on symmetric spaces and singular Riemannian foliations on Finsler geometry, respectively.

- Teaching two undergraduate subjects at Universidad Autónoma de Madrid.

- I lectured two research-focused minicourses.

Dissemination activities and research visits:

- I gave talks at 7 conferences held in Argentina, Brazil, Japan, Belgium and Spain, and in 7 seminars in the United States, Brazil and Spain.

- I visited the following centers: Universidad de Murcia (Spain), University of Hiroshima (Japan), Instituto Nacional de Matemática Pura e Aplicada − IMPA (Brazil), Washington University in St. Louis (USA), University of Notre Dame (USA), and University of Oklahoma (USA).

Public engagement activities:

- Hands-on workshop on soap bubbles, minimal surfaces and shortest paths. Aimed at high school students.

- Several interviews in Spanish newspapers.

- Short talk in a meeting of scientists of different areas.

- Participation in a roundtable aimed at PhD students.

- General public talk.

Final results

- Constructing solutions to overdetermined boundary value problems associated to certain semilinear elliptic equations in general Riemannian manifolds, and obtaining some results about the symmetry of such solutions.
- Classifying isoparametric surfaces, surfaces with constant principal curvatures, and homogeneous surfaces in homogeneous 3-manifolds with 4-dimensional isometry group.
- Proving that homogeneous spaces with polar isotropy action must be, under certain circumstances, symmetric spaces.
- Classifying ruled hypersurfaces with constant mean curvature in complex projective and hyperbolic spaces.
- Classifying isoparametric hypersurfaces in complex hyperbolic spaces.

Website & more info

More info: http://verso.mat.uam.es/.