An Introduction to Varifold Geometry and Applications

29 ottobre 2019
Versione stampabile

Luogo: Povo Zero, via Sommarive 14 (Povo) - Seminar Room "-1"
Ore: 3.00 PM

"Doc in Progress" is pleased to introduce you to
  • Marco Pozzetta - PhD in Mathematics, University of Pisa

Abstract

In the first part of the seminar we will introduce the notion of (integer rectifiable) varifold as a generalization of the concept of smooth immersed manifold. We will focus on three geometric aspects: the support of a varifold, the existence of a generalized tangent space, and the notion of generalized mean curvature. We will then define the natural notion of convergence in the sense of varifolds and we will discuss the classical sequential compactness theorem for varifolds. Time permitting we will also present the remarkable monotonicity formula for varifolds with square integrable mean curvature. In the weak setting of varifolds it is natural to ask for existence of minimizers of many different variational problems. Hence we will present an application of the varifold theory to the minimization of the Willmore energy, that is classically defined on 2-dimensional surfaces as the surface integral of the squareofthemean curvature.Moreprecisely, we willconsidertheproblemof finding an optimal elastic (i.e. Willmore minimizing) connected compact surface having an assigned disconnected boundary, and we will show the existence of minimizers for such problem suitably stated in the setting of varifolds. Thisresultis in collaboration with Matteo Novaga.