Universidade de Brasília

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.

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.

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We will consider the geometric problem of findind immersed surfaces in the upper half-space such that the Gaussian curvatures induced from the ambient euclidean and hyperbolic metrics coincide.
We will show the surprising geometric connection between such immersions and minimal surfaces (in euclidean space and timelike surfaces in Minkowski space). This connection enables the construction of infinitly many non trivial examples, we will discuss some of them in the talk.Joint work with Nilton Barroso.