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.
