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An introduction to varifold geometry and applications

Marco Pozzetta
Università di Pisa
Friday, June 5, 2020 - 14:00
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We will introduce the notion of varifold as a generalization of the concept of smooth immersed manifold in the Euclidean space, and we will focus on the class of integer rectifiable varifolds. Proceeding by analogy with smooth manifolds, we will discuss three aspects: the support of a varifold, the existence of generalized tangent spaces, and the notion of a generalized mean curvature vector.
This will lead us to a classical sequential compactness theorem for sequences of varifolds, that can be seen as a tool for solving minimization problems by direct methods in Calculus of Variations. Time permitting, we will also present a remarkable monotonicity formula for varifolds with square integrable mean curvature.
Then we will present an application of this theory to the minimization of the Willmore energy, that is classically defined on 2-dimensional surfaces as the surface integral of the square of the mean curvature. More precisely, we will address the problem of finding an optimal elastic (i.e., Willmore minimizing) connected compact surface spanning an assigned disconnected boundary. We will discuss the existence of minimizers for such a problem suitably stated in the setting of varifolds. These results are in collaboration with Matteo Novaga.
Prerequisites for attending the seminar are a basic knowledge of measure theory and of the theory of surfaces in the Euclidean space. However, the exposition will be as simple as possible. 

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