# Isoperimetric problem and minimal surfaces in the Heisenberg group

### (University of Padua)

**Event:** ERC School on Geometric Measure Theory and Real Analysis

**Date:** Sep 30, 2013

**Place:** Centro De Giorgi

**Abstract.** The lecture is an introduction to Geometric Measure Theory, H-perimeter, minimal surfaces, and to the isoperimetric problem in the Heisenberg group.

1. Introduction. The Heisenberg group and its Lie algebra, Carnot-Caratheodory metric, functional spaces and inequalities.

2. H-perimeter. Sets with finite H-perimeter, blow-up and structure theorems, different notions of surface area, area formulas.

3. Isoperimetric problem. Existence of isoperimetric sets, Pansu conjecture, convex, C^2, and symmetric solutions, rearrangements.

4. H-minimal surfaces. Minimal surfaces equations, nonregular minimal surfaces, approximation of minimal boundaries, the regularity problem.