Recent Teaching Experience

Ricardo Sa Earp

First, I would like to outline two doctoral theses held at Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), under my supervision. In September 2003, my student Marcos Alexandrino did his doctoral dissertation on Singular Riemannian Foliations with Sections and Transnormal Maps. I observe that this thesis had the cooperation of Gudlaugur Thorbergsson (Köln University, Germany). Let me say a few words about it: It is known that examples of singular Riemannian Foliation with Section are the so-called Polar Actions. In fact, it follows from classical Lie Group theory that the orbits of an adjoint action of a compact Lie group intercepts a maximal torus orthogonally. This is an example of a Polar Action. More generally, a compact isometric action is said to be Polar if it admits sections, i.e totally geodesic submanifolds that intercepts the orbits orthogonally. Marcos has obtained interesting results on these singular foliations in connection with Transnormal Maps, which are generalizations of Isoparametric Maps. Two publication have arised from his thesis: Integrable Riemannian submersion with singularities (Geometriae Dedicata, 2004), and Singular Riemannian foliations with sections (Illinois Journal of Mathematics, 2004).

Next, my student Elias M. Guio did his doctoral dissertation entitled A priori gradient estimates, existence and non existence results on a mean curvature equation in hyperbolic space. This work may be summarized as follows: There is a well-known criterion for the solvability of the Dirichlet problem for the equation of constant mean curvature on a bounded C² domains. In fact, there exists some inequality involving the mean curvature and the curvature of the boundary of the domain that ensures some a priori gradient estimates and a posteriori existence for the Dirichlet problem. This inequality is sharp in the following sense: if it does not hold at a point then one can infer non existence for a certain smooth boundary data. This classical result was established by Serrin in 1969. The work carried out on Elias thesis goes into an analogous direction, taking into account a certain Serrin condition type in hyperbolic space raised by his thesis. A novelty value here is the deduction of a solution of the related Dirichlet problem for prescribed mean curvature. Moreover, the sharpness of the result is assured by a non existence result if the condition given in Elias thesis fails at a point. Then first part of his Thesis is published in Communication on Pure and Applied Analysis, 2005.

Now I would like to discuss about some classes taught at Pontifícia Universidade Católica do Rio de Janeiro in the last Years. Now I would like to discuss about some classes taught at Pontifícia Universidade Católica do Rio de Janeiro in the last Years. Since earlier nineteen’s, I have prepared several undergraduate students to pursue their studies in Mathematics, for instance a former student Carolina Araujo is now a Princeton PhD young active mathematician. It is worth mention that I have given several classes on Hyperbolic Geometry, Riemann Surfaces, Minimal Surfaces based on my book with Eric Toubiana entitled Introduction à la géométrie hyperbolique et aux surfaces de Riemann. The second edition of our book is expected to appear soon, edited by Cassini, France. Besides, I have given undergraduate classes of Calculus, Linear Algebra, Real Analysis and Differential Geometry. I have also have given graduate classes of Complex Analysis, Differential Geometry, Differential Equations and Variational Methods.