Project's purposes
An isoperimetric problem is a minimization problem that consists on finding a configuration of regions of given volumes with the least possible perimeter. They have been usually treated by means of isoperimetric inequalities, that are neither but relations between the perimeter of the configuration and the enclosed volumes. Isoperimetric problems have been intensively studied in Euclidean and hyperbolic spaces, in the sphere, and in more general Riemannian manifolds. They have also been considered in probability spaces like Gauss spaces, and in certain pseudo-Riemannian manifolds such as the Heisenberg group and, more generally, in Carnot-Carathéodory spaces, in which suitable notions of perimeter and volume can be defined. The difficulty of the subject and the numerous potential applications make it one of the most interesting ones in modern Riemannian Geometry.
In this project we intend to develop techniques suitable to face this kind of questions by means of the study of certain selected problems, such as the characterization of isoperimetric regions in the Heisenberg group, the study of the isoperimetric inequality outside a convex set in a Cartan-Hadamard manifold, the investigation of planar isoperimetric configurations enclosing and separating multiple areas, and the study of the topological and geometrical type of isoperimetric solutions inside a convex set.
Research lines
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Isoperimetric inequalities in Riemannian manifolds
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Constant mean curvature surfaces, minimal surfaces
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Heisenberg group, Carnot-Caratheodory groups
