Program and abstracts

Program

Schedule

    • 10:00 - 10:45 Welcome
    • 10:45-11:25 Morbidelli: Some properties of convex and monotone sets in two step Carnot groups
    • 11:30-12:10 Prandi: Weyl law for singular hypoelliptic operators
    • 12:15-12:55 Leonardi: Liouville-type and trace properties of divergence-measure vector fields
    • 13:00-14:10 Buffet lunch
    • 14:15-14:55 Ambrosio: Approximation in Lusin's sense of Sobolev functions by Lipschitz functions
    • 15:00-15:40 Magnani: Uniform measures in the Heisenberg group
    • 15:45-16:15 Coffee break
    • 16:15-16:55 Merlo: Geometry of uniform measures in the Heisenberg group
    • 17:00-17:40 Serra Cassano: The Bernstein problem for area-minimizing intrinsic graphs in the sub-Riemannian Heisenberg group

Abstracts

Luigi Ambrosio (Scuola Normale Superiore, Pisa)

Approximation in Lusin's sense of Sobolev functions by Lipschitz functions

In Euclidean and more generally in PI spaces, real-valued Sobolev and BV functions coincide on large sets with Lipschitz functions. The quantitative version of this property has a variety of applications (lower semicontinuity of integral functionals, flows of vector fields, theory of currents). I illustrate how this property persists even in metric measure structures that are not doubling, as finite or infinite-dimensional Gaussian spaces and more generally CD( K, ∞ ) spaces (joint work with E.Brue' and D.Trevisan). Finally, I will illustrate some applications when the map has values on a manifold, and originates from the flow of a vector field (work by E.Brue' and D.Semola).

Gian Paolo Leonardi (Dip. Mat. Univ. Modena e Reggio Emilia)

Liouville-type and trace properties of divergence-measure vector fields

It is a known fact that any bounded vector field ξ, whose distributional divergence is a measure, admits a normal trace in a suitably weak sense on any oriented, (n–1)-rectifiable set S with locally finite (n–1)-dimensional Hausdorff measure. We aim at showing that, whenever this "weak normal trace" is locally maximal at a point x of S, that is, it attains at x the value of the infinity norm of ξ in a neighbourhood of x, then the vector field ξ admits the classical trace at x, for almost all x in S. This problem is motivated by the study of capillarity for perfectly wetting fluids in "weakly-regular" cylinders. Via a blow-up argument, one can reduce the problem to a Liouville-type property for divergence-free vector fields in Rn. In the (physically relevant) case of planar vector fields we show that the property is true, while in higher dimension this is an open problem. The research is in collaboration with Giorgio Saracco.

Valentino Magnani (Dip. Mat. Univ. Pisa)

Uniform measures in the Heisenberg group

Uniform measures naturally arise as tangent measures, playing an important role in the groundbreaking theorem of Preiss on the rectifiability of measures with density. One codimensional uniform measures have been classified by Kowalski and Preiss. Recently Nimer proved the existence of new non-flat uniform measures in Euclidean space. All these results rely on the special symmetries of Euclidean metric. We present some first results on uniform measures in the Heisenberg group equipped with the Cygan-Korányi distance, establishing the classification of 1-uniform and 2-uniform measures. We also show that 3-uniform measures with vertically ruled support must be supported on a vertical plane. These results have been obtained in collaboration with J. T. Tyson e V. Chousionis.

Andrea Merlo (Scuola Normale Superiore, Pisa)

Geometry of uniform measures in the Heisenberg group

Characterisation of rectifiable measures in Euclidean spaces has been a longstanding problem for Geometric Measure Theory since the seminal works by A. Besicovitch. Such a theory came to its apex when D. Preiss in 1987 proved the equivalence between the existence of the m-dimensional density and the fact that the support of the measure is contained in countably many Lipschitz manifolds of dimension m. The argoument Preiss carried out is deeply connected with the structure of the euclidean metric and since then there have been very few attempts to tackle the density problem in other spaces. In this talk however we will show that in the Heisenberg group endowed with the Korani metric a study of the support of uniform measures (which arise naturally as blowups of measures with density) is possible, and prove that their support is contained in the zero set of second degree polynomials, in full accordance with the Euclidean theory.

Daniele Morbidelli (Univ. Bologna)

Some properties of convex and monotone sets in two step Carnot groups

I will present some recent results on convexity in Carnot groups. More precisely, I will discuss a "inner cone property" for two-step Carnot groups, generalizing a previous result of Arena, Caruso and Monti. In the second part of the talk, I will discuss an approach to the problem of classification of monotone subsets of two-step Carnot groups, with application to a model case.

Dario Prandi (L2S CentraleSupélec, Paris)

Weyl law for singular hypoelliptic operators

In this talk we present recent results on the asymptotic growth of eigenvalues of the Laplace-Beltrami operator on singular Riemannian manifolds, where all geometrical invariants appearing in classical spectral asymptotics are unbounded, and the total volume can be infinite. Under suitable assumptions, we prove that the leading term of the Weyl’s asymptotics contains information on the singularity, i.e. its Minkowski dimension and its regularized measure. We apply our results to a suitable class of almost-Riemannian structures. A key tool in the proof is a new heat trace estimate with universal remainder for Riemannian manifolds, which is of independent interest. This is a joint work with Y. Chitour and L. Rizzi.

Francesco Serra Cassano (Dip. Mat. Univ. Trento)

The Bernstein problem for area-minimizing intrinsic graphs in the sub-Riemannian Heisenberg group

We will deal with the so-called Bernstein problem for area-minimizing intrinsic graphs in the first Heisenberg group H1 ≡ ( R3, · ), understood as a Carnot group and equipped by the sub-Riemannian metric structure. More precisely, the problem reads as follows: if the intrinsic graph Γf ⊂ H1 of a function f : R2 → R, that is

Γf ⊂ H1 :={ ( 0, y, t ) · ( f( y, t ), 0, 0 ) ) : ( y, t ) ∈ R2 } ,

is (locally) area-minimizing in H1, then must Γf be a plane, in the geometry of H1 ? We will positively and negatively answer to this problem, taking the regularity of f into account.