Isoparametric hypersurfaces in product spaces

GEOMETRY SEMINAR

Conferenciante: João Paulo dos Santos (Universidade de Brasilia)

Abstract: A smooth function \(f\) on a Riemannian manifold \(\widetilde{M}\) is isoparametric if \(|\nabla f|\) and \(\Delta f\) are constant on the level sets of \(f\). A hypersurface \(M \subset \widetilde{M}\) is an isoparametric hypersurface if it is a regular level set of an isoparametric function defined on \(\widetilde{M}\). When the ambient space is a space form \(\mathbb{Q}^{n}_c\), the definition of isoparametric hypersurface is equivalent to saying that the hypersurface has constant principal curvatures. However, in arbitrary ambient spaces of nonconstant sectional curvature, the equivalence between isoparametric hypersurfaces and hypersurfaces with constant principal curvatures may no longer be true. In this talk, it will be presented characterization and classification results on isoparametric hypersurfaces with constant principal curvatures in the product spaces \(\mathbb{Q}_{c_{1}}^2 \times \mathbb{Q}_{c_{2}}^2\), for \(c_{i} \in \{-1,0,1\}\) and \(c_1 \neq c_2\). The talk is based on a joint work with João Batista Marques dos Santos.