Nowadays, many important research subjects in Differential Geometry lie in the interface between Geometry and Analysis. As in the far past, when geometrical problems were translated into algebraic problems, the new geometrical challenges are now transformed into partial differential equations questions and its solutions are searched through the applications of specific techniques of both areas. In this context lie great part of the Differential Geometry research developed in Brazil.
Team and Interlocutors
J. L. Barbosa, M. do Carmo, M. Dajczer and their collaborators at UF Ceará, UF Alagoas, UF Piauí, UF Rio de Janeiro, UF Espírito Santo, PUC-Rio, UF São Carlos.
P. Bérard (University of Grenoble), D. Gromoll (State University of New York), H. Rosenberg (University of Paris ? Jussieu)
Immersions with Constant r-mean Curvature
This topic made great progress when it was proved that the positivity of such curvatures implied that the equations that represent it where elliptic. This result was in the basis of many others about existence and uniqueness of such immersions. In fact, this is a topic of great interest, in these days, particularly in which concerns the immersions into the hyperbolic space and the immersions type space into Minkowisky spaces. The study of stability of such immersions, that is an important issue in the subject, is treated through the study of the corresponding Jacobi Equations by the introduction of proper hypothesis to guarantee its elipticity.
Recent work of J. Lucas Barbosa and F. Mercuri provided a new class of hypersurfaces characterized solely by geometrical and topological properties and not by equations, and showed, for such class, the validity of some results that were known and thought to be valid only in the theory of minimal surfaces. Applications of this result are expected and are under investigation for new developments in the theory of minimal surfaces.
Stability and Morse Index of minimal surfaces and of hypersurfaces of constant mean curvature greater than or equal to zero. Investigation on hypersurfaces of Euclidean space of constant scalar curvature. Minimal surfaces and surfaces of constant mean curvature have deep relations with Analysis and Capilarity Theory, and the scalar curvature appears naturally in the equations of Relativity.
These topics are pursued by Manfredo do Carmo and other researchers, especially H. Alencar, F. Brito, M. F. Elbert, K. Frensel, L. Lima, J. Ripoll e W. Santos.
Theory of Submanifolds
M. Dajczer, D. Gromoll, R. Tojeiro, L. A. Florit, F. Yang and C. Olmos form a research team devoted to the study of several aspects of the theory of submanifolds (isometric and conformal immersions), a branch of Differential Geometry. To this point, the group has published more than 35 papers authored by at least two members of the group. We also interact doing research and giving elementary and advanced talks in several other mathematical centers in Brazil like: Fortaleza, Porto Alegre, Belo Horizonte, Campinas, São Paulo, Rio de Janeiro.