A Weingarten surface S in a three-dimensional Riemannian (or semi-Riemannian) manifold is a regular surface whose principal curvatures k1 and k2 verify a relationship (usually polynomial, but not necessarily) P (k1, k2) = 0. Since the mean and Gaussian curvatures H and K determine/are determined by the principal curvatures of S, the relation P can always be rewritten in the form Q (H, K) = 0. Important (and very studied) examples of these surfaces are the constant Gaussian curvature surfaces, the constant mean curvature surfaces and, in particular, the minimal surfaces. Weingarten Surfaces is a classic topic of Differential Geometry that began with Weingarten's work in the 19th century. Although it is an old and very studied topic, much remains to be done. The classification of these surfaces in the Euclidean space, for example, still remains open and, it seems, very far from being obtained. We will begin this project by studying Weingarten surfaces in Euclidean space and, after that, we will move to surfaces in the Hyperbolic space. If the project progresses quickly, the additional time will be used to study Weingarten surfaces in Lorentz-Minkowski space.
News published in Agência FAPESP Newsletter about the scholarship: