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Titles and Abstracts
Please find a listing of the submitted titles and abstracts below.
A printable collection of the abstracts is available as PDF here.
Luca Capogna: A gradient flow of diffeomorphisms for L^{p} dilation
Motivated by Ahlfors approach to study the Grotsch problem for extremal quasiconformal mappings we derive a nonlinear evolution system of PDE which is the gradient flow for the L^{p} norm of the outer dilation with 1 < p < ∞. We prove short time existence, derive evolution equations for associated geometric quantities and study the asymptotic case p → ∞. This is an ongojng project, joint with Andy Raich (UARK).
Giovanna Citti, University of Bologna
Katrin Fässler: Minimal distortion and modulus of curve families
A fundamental problem in the classical theory of quasiconformal (QC) maps in the complex plane is the identification of extremal mappings. These are homeomorphisms which minimize the L^{∞}norm of the distortion within a given class of QC maps between planar domains. An analogous question can be posed for the larger class of finite distortion maps, where a mean distortion functional has to be minimized instead of the maximal distortion.
We present a method by modulus of curve families to explore such minimization problems in the complex plane and outline how this approach could be used to study extremal QC maps in the Heisenberg group.
This is joint work with Zoltán Balogh and Ioannis Platis.
Bruno Franchi: Maxwell's equations in Carnot groups
In this talk we present a geometric formulation of Maxwell's equations in Carnot groups (connected simply connected
nilpotent Lie group with stratified Lie algebra of step κ) in the setting of the intrinsic complex of differential forms defined by M. Rumin. We show that these equations are invariant under the action of suitably defined Lorentz transformations, and we prove the equivalence of these equations with differential equations "in coordinates"'. Moreover, we analyze the notion of "vector potential"', and we show that it satisfies a new class of 2κth order evolution differential equations.
Valentino Magnani: Sobolev surfaces and contact equations
In the first Heisenberg group,
we discuss the nonexistence of horizontal
Sobolev surfaces. Our approach is based on
the nonexistence of weak solutions to the
corresponding system of contact equations.
This fact along with the areatype formula
for surfaces shows the nonexistence of
Sobolev surfaces whose intrinsic Hausdorff
dimension equals two. Different notions of
Sobolev surface will be discussed.
Roberto Monti: Rearrangements in metric spaces
We discuss some notions of rearrangement for functions and sets in the setting of a metric space with a measure. We also refer to the problem of rearrangement in the Heisenberg group.
Daniele Morbidelli: Classification of CR mappings in a class of real hypersurfaces in C^{n+1}
We discuss the problem of classifying local CR mappings between strictly pseudoconvex hypersurfaces in C^{n+1}. In particular we introduce some tools based on the analysis of classical differential invariants going back to Tanaka, Webster and Chern. Our main application concerns a class of generalized ellipsoids, where we classify all local CR mappings. Joint with R. Monti
Francesco Montefalcone: Local Monotonicity and Isoperimetric Inequality on Hypersurfaces
We generalize to the setting of kstep Carnot groups an
isoperimetric inequality, involving the mean curvature of the
hypersurface, due to Michael and Simon, and Allard, independently.
These results can be found in the recent preprint: 'Isoperimetric,
Sobolev and Poincaré inequalities on hypersurfaces in subRiemannian
Carnot groups', available on Arxiv.
Alessandro Ottazzi: The Liouville theorem for Carnot groups
Using Tanaka prolongation theory, we show that 1qc mappings on a domain of a Carnot group, other than R or R^{2}, all come as restriction of the action of some finite dimensional Lie group. First we interpret the 1qc condition at the infinitesimal level, then we show how the Tanaka construction intervenes to describe the space of 1qc vector fields. The last step is integration to the map level. In particular, we show the calculations in the case of the Heisenberg group. This work is in collaboration with Ben Warhurst.
Pierre Pansu: conformal Hölder exponents
Riemannian geometry suggests to define the conformal Hölder exponent of a map f between metric spaces X and Y as the supremum of a's in [0,1] such that f is C^{a} with respect to some metric in the quasisymmetric gauge of Y. And to find the optimal conformal Hölder exponent of homeomorphisms between Carnot groups. Conformal dimension gives a lower bound, but unsharp.
Kirsi Peltonen: Quasiregular dynamics in the Heisenberg group
We report on recent developments of a higher dimensional real counterpart of the iteration theory of rational functions in the extended complex
plane. A subclass of quasiregular mappings, called uniformly quasiregular
mappings (UQR) have been studied rst in Ivaniec and Martin (1996) acting on the Riemann sphere
and further for example in Peltonen (1999) acting on a compact Riemannian manifold so
that all the iterates of the mapping are Kquasiregular for xed distortion
K, independently of the number of iterates.
We describe some basic constructions that can be extended further to the
subRiemannian setting in the Heisenberg group. Moreover, we construct
a noninjective uniformly quasiregular mapping g acting on the one point
compactication of the Heisenberg group equipped with a subRiemannian
metric. We further show that there exits a measurable horizontal conformal
structure which is equivariant under the semigroup Γ generated by g. This
is equivalent to the existence of a equivariant CR structure. This fact is
interesting also from the point of view of several complex variables since it
was known already for Poincaré (1907) that the only semigroup of CR maps with
respect to the standard CR structure must be restrictions to the sphere of a
subgroup of the conformal automorphisms of the unit ball in the Euclidean
space of two complex variables.
Joint work with Zoltán Balogh and Katrin Fässler
Cornel Pintea: Size of tangencies to noninvolutive distributions
An upper estimate is given for the Hausdorff dimension of the the tangency set of ndimensional C^{2}submanifolds of R^{n+m} relative to noninvolutive distributions of rank n. The work is a continuation of Balogh (2003), in which the author studied the case of the horizontal distribution in the Heisenberg group. Based on a joint work with Zoltán Balogh and Heiner Rohner.
Séverine Rigot: Monge's transport problem in nonriemannian spaces
The classical Monge's transport problem refers to the problem of moving one distribution of mass onto another as efficiently as possible where the efficiency is expressed in terms of the average distance transported. I will present in this talk a solution to Monge's transport problem between two compactly supported Borel probability measures in the Heisenberg group equipped with its CarnotCaratheodory distance assuming that the initial measure is absolutely continuous with respect to the Haar measure of the group (joint work with L. De Pascale). The Heisenberg group is taken here as an illustrative explicit example of nonriemannian space and I shall also discuss extension of the techniques used in this work to other metric spaces.
Jeehyeon Seo: BiLipschitz embeddability of the Grushin plane into Euclidean space
Many subRiemmanian manifolds like the Heisenberg group do not admit biLipschitz embedding into any Euclidean space. In contrast, the Grushin plane admits a biLipschitz embedding into some Euclidean space. This is done by extending a biLipschitz embedding of the singular line,
using a Whitney decomposition of its complement.
Raul Serapioni: Characterizations of rectifiable sets in H^{n}
According to Federer's definition, kdimensional rectifiable sets in R^{n} are contained, up to a negligible set, in the countable union of Lipschitz images of subsets of R^{k}. In Euclidean spaces it is equivalent to use coverings with countable unions of kdimensional C^{1} submanifolds or to require the almost everywhere existence of approximate tangent spaces or of tangent measures.
Also in groups it is possible to follow the pattern of the three definitions, provided we have good intrinsic notions of Lipschitz maps or of C^{1} submanifolds or of approximate tangent subgroups and tangent measures. But the complete equivalence of the three definitions is still an open question.
We discuss some partial equivalence of the three definitions inside Heisenberg groups.
Francesco Serra Cassano: Nonparametric minimal surfaces in Heisenberg groups
In the setting of Heisenberg groups, endowed with their subRiemannian structure,
two notions for the graph of a function have
been introduced. Putting particular emphasis on
the so called intrinsic graphs, we define the
measure of a graph and prove the existence of
minimal graphs. As a first step towards the
regularity of minimal surfaces in Heisenberg groups, we show that minimal graphs must be locally bounded.
This is joint work with D. Vittone.
Marc Troyanov: Conformal mappings in Finsler geometry
After recalling what a Finsler manifold is, I will show how one can associate a Riemannian metric to any Finsler structure on a given manifold. I will then show how various problem in conformal Finsler geometry reduce to the Riemannian theory. In particular I will describe all Finsler manifold admitting a conformal nonisometriv self map.
This is common work with Vladimir Matveev.
Davide Vittone: Isodiametric sets in the Heisenberg group
In this talk we focus on isodiametric sets in the Heisenberg group, i.e. sets maximizing the volume measure among those with fixed diameter. We first show a Lipschitz regularity result for the boundary of such sets. We are able to solve the isodiametric problem in the restricted class of rotationally invariant sets, where the solution is given by the (Euclidean) convexification of CCballs. A nonuniqueness result is also shown. This is joint with G. P. Leonardi and S. Rigot.
Ben Warhurst: Rigidity classification of Carnot groups in the class of C^{2} maps via Tanaka theory
The aim of this talk is to show how the prolongation theory of Tanaka combines with the prolongation theory of Singer and Sternberg to give a precise rigidity classification of Carnot groups at the level of C^{∞} contact vector fields. We then show that this classification extends to the class of C^{2} contact maps. This is joint work with Alessandro Ottazzi.
Stefan Wenger: Lipschitz extensions to jet space Carnot groups
Lipschitz extension problems have received considerable
attention for a long time. Not much is known, however, for
CarnotCaratheodory space targets. In this talk, I will present new
Lipschitz extension and nonextension results in the case that the
target space is a jet space. Recall that jet spaces give a model for a
certain class of Carnot groups, including the (higher) Heisenberg
groups, the Engel group, and the filiform groups. Based on a
collaboration with Severine Rigot and a collaboration with Robert Young.
Roger Züst: Integration of Hölder forms and currents in snowflake spaces
For an oriented ndimensional Lipschitz manifold M we give meaning to the integral
∫_{M} f dg_{1} ^ ... ^ dg_{n} in case the functions f, g_{1},
..., g_{n} are merely Hölder continuous of a certain order by extending the construction
of the RiemannStieltjes integral to higher dimensions. More generally, we show that for α ∈
(n/n+1,1] the ndimensional locally normal currents in a locally compact metric space (X,d) represent a
subspace of the ndimensional currents in (X, d^{α}). On the other hand, for n ≥ 1 and
α ≤ n/n+1 the vector space of ndimensional currents in (X, d^{α}) is zero.
