5.th School on
“Analysis and Geometry in Metric Spaces”
Levico Terme, June 24-29, 2007
|
Abstract: Carnot groups are
model spaces for subriemannian geometry in the same
way that euclidean space is the model space for riemannian geometry.
The aim of this course is to explain why 1-quasiconformal mappings of
a subriemannian metric space should come from a
finite-dimensional group of smooth transformations, and to prove this for
mappings of Carnot groups. This involves a
regularity theorem for Q-harmonic functions due to Luca Capogna,
an algebraic regularity theorem due to Capogna and
Cowling, and some differential geometric ideas due to Tanaka. If time permits, more general
rigidity theorems for mappings with geometric properties will be outlined. |
|
Abstract: H. Whitney developed his theory of flat
forms based on duality with respect to polyhedral flat chains. The fact that
flat forms are invariant under bi-Lipschitz
transformations allows one to define the de Rham
theory on Lipschitz manifolds, for example. In
these lectures, I will discuss recent use of Whitney forms in the problem of
(local) bi-Lipschitz parametrization
of metric spaces by Euclidean spaces. In particular, a characterization of
those metric surfaces that admit such local parametrizations
is given in terms of Whitney
forms. We also discuss an extension of Whitney's theory to arbitrary Banach spaces. |
|
Abstract: We describe recent progress in the study of
isoperimetric regions in the Heisenberg group H^1. |