Isoperimetric problem and minimal surfaces in the Heisenberg group

Roberto Monti

(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.