A notion of area for immersed submanifolds in a sub-Riemannian geometry.

A notion of area for immersed submanifolds in a sub-Riemannian geometry.

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Gianmarco Giovannardi, University of Bologna
Fine Hall 401

I will describe the area functional for immersed sub-manifolds in a sub-Riemannian structure. Since we are interested in high codimensional submanifolds, the notion of degree introduced by Magnani and Vittone is a central tool in this setting. We will see that the area functional depends on the degree of the immersed submanifold. Then, it turns out that not all the possible variations are admissible for the area functional but only the ones that preserve the degree. Similar to the case of singular curves in sub-Riemannian geometry, we discovered that some submanifolds are isolated point of the domain of the area functional. Then, we will deal with some first variation area formulasin order to provide a suitable definition for themean curvature operator in simple contexts. This is joint work with my two supervisors G. Citti and M. Ritoré.