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 Jo$\tilde{\rm a}$o Batista Marques dos Santos.

We consider hypersurfaces of products \(M\times \mathbb{R}\) with constant r-th mean curvature — to be called \(H_r\)-hypersurfaces — where \(M\) is an arbitrary Riemannian manifold. We develop a general method for constructing them, and employ it to produce many examples for a variety of manifolds \(M\), including all simply connected space forms and the Hadamard manifolds known as Damek-Ricci spaces. Uniqueness results for complete \(H_r\)-hypersurface of \(\mathbb{H}^n\times\mathbb{R}\) or \(\mathbb{S}^n\times\mathbb{R}\) \((n \geq 3)\) are also obtained. This is a joint work with Ronaldo de Lima (UFRN) and Fernando Manfio (ICMC-USP).

Sala EINSTEIN UGR (virtual)

We consider conformally flat hypersurfaces in four dimensional space forms with their associated Guichard nets and Lamé’s system of equations. We show that the symmetry group of the Lamé’s system, satisfying Guichard condition, is given by translations and dilations in the independent variables and dilations in the dependents variables. We obtain the solutions which are invariant under the action of the 2-dimensional subgroups of the symmetry group. For the solutions which are invariant under translations, we obtain the corresponding conformally flat hypersurfaces and we describe the corresponding Guichard nets. We show that the coordinate surfaces of the Guichard nets have constant Gaussian curvature, and the sum of the three curvatures is equal to zero. Moreover, the Guichard nets are foliated by flat surfaces with constant mean curvature. We prove that there are solutions of the Lamé’s system, given in e terms of Jacobi elliptic functions, which are invariant under translations, that correspond to a new class of conformally flat hypersurfaces.

Presentaremos un estudio de hipersuperficies conformemente llanas en $\mathbb{R}^4$ relacionadas con soluciones de un sistema de equaciones diferenciales parciales invariantes por un grupo de simetría. Asociadas a una clase particular de las soluciones del sistema, obtenemos una clase de superficies llanas en el espacio hiperbólico que son invariantes por un movimiento helicoidal, así que presentaremos también una clasificación de estas superficies en términos de datos holomorfos y una caracterización a través de las primera y segunda forma fundamentales.