ABSTRACTS

Sub-Riemannian Geometry and PDEs

Levico Terme (Trento), July 2-5, 2012

Luigi Ambrosio (SNS Pisa): Equivalence of weak gradients in metric measure spaces

In a joint work with Gigli and Savare', we compare several notion of weak (modulus of) gradient in metric measure spaces and prove their equivalence. This equivalence is part of the "calculus program" we developed, largely based on tools from optimal transportation theory. In particular, we prove density in energy of Lipschitz maps in Sobolev spaces independently of doubling and Poincare' assumptions on the metric measure space. The case of BV maps, together with some open questions, will also be illustrated (joint work with Di Marino).

Zoltan Balogh (Bern): The Hopf-Lax formula in Carnot groups: a control theoretical approach

I will talk about a recent joint work with Andrea Calogero and Rita Pini on the Hopf-Lax formula in Carnot groups. Our approach to this problem is by optimal control theory and allows to treat non-homogeneous state-dependent convex Hamiltonians acting on the horizontal gradient of functions defined on Carnot groups.

Ugo Boscain (Palaiseau): Small time heat kernel asymptotics at the sub-Riemannian cut locus

For a sub-Riemannian manifold provided with a smooth volume, we relate the small time asymptotics of the heat kernel at a point $y$ of the cut locus from $x$ with roughly ``how much'' $y$ is conjugate to $x$. This is done under the hypothesis that all minimizers connecting $x$ to $y$ are strongly normal, i.e.\ all pieces of the trajectory are not abnormal. Our result is a refinement of the one of Leandre  $4t\log p_t(x,y)\to -d^2(x,y)$ for $t\to 0$, in which only the leading exponential term is detected. Our results are obtained by extending an idea of Molchanov from the Riemannian to the sub-Riemannian case, and some details we get appear to be new even in the Riemannian context. 

These results permit us to obtain properties of the sub-Riemannian distance starting from those of the heat kernel and vice versa. 

For the Grushin plane endowed with the Euclidean volume we get the expansion $p_t(x,y)\sim t^{-5/4}\exp(-d^2(x,y)/4t)$ where $y$ is reached from a Riemannian point $x$ by a minimizing geodesic which is conjugate at $y$.

Luca Capogna (Minneapolis and Arkansas): Stability of the Poincare' and doubling inequalities in a  Riemannian approximation of Carnot-Carathéodory spaces and applications

In a joint paper with Giovanna Citti and Garrett Rea, we prove that given a Carnot-Carathéodory structure  $(R^n, d_0)$ associated to a system of vector fields $(X_1,...,X_m)$ satisfying  Hörmander's finite rank condition, one can choose a 

one-parameter family  of Riemannian metrics $\epsilon \to g_\epsilon$ such that the corresponding metric spaces $(R^n, d_\epsilon)$ converge in the Gromov-Hausdorff sense to $(R^n, d_0)$ and for which the constants involved in the doubling property and in the Poincar\'e inequalities can be chosen independently of $\e\ge 0$.

As application we show the long-time existence of the flow by total variation of a graph over a Carnot group of step two.

Sagun Chanillo (Rutgers): Embeddability of CR three manifolds and the CR Yamabe invariant

Our aim is to obtain CR invariant conditions that are sufficient to guarantee embedding of compact, CR three manifolds in complex spaces. The conditions are phrased in terms of the non-negativity of the CR analog of the Paneitz operator and the positivity of the Yamabe constant. We also obtain sharp lower bounds on the lowest eigenvalue of Kohn’s Laplacian. The non-negativity of the Paneitz operator is also shown to be necessary for embedding.

This is a joint work with Hung-Lin Chiu and Paul Yang.

Giovanna Citti (Bologna): Sub-Riemannian heat kernels and mean curvature flow of graphs over Carnot groups

We introduce a sub-Riemannian analogue of the Bence-Merriman-Osher algorithm [27] and show that it leads to weak solutions of the horizontal mean curvature flow of graphs over sub-Riemannian Carnot groups. The proof is based on some estimates of the integral of heat kernel on surfaces and on the notion of generalized horizontal mean curvature flow for graphs in Carnot groups.

Bruno Franchi (Bologna): Fractional sub-Laplacians in Carnot groups

In this talk we present an invariant Harnack inequality on Carnot-Carathéodory balls for fractional powers of sub-Laplacians in Carnot groups. The proof relies on an “abstract” formulation of a technique recently introduced by Caffarelli and Silvestre.

Nicola Garofalo (Purdue and Padova): Generalized curvature-dimension inequalities, Li-Yau inequalities and volume growth in negatively curved sub-Riemannian manifolds

I will discuss some recent joint works with F. Baudoin, M. Bonnefont and I. Munive. Starting from a generalization of the curvature-dimension inequality, we develop a notion of Ricci lower bound that is appropriate for large classes of sub-Riemannian manifolds. In previous works Baudoin and I had developed the case of manifolds with nonnegative ``Ricci'' tensor. In the present work we generalize those works and obtain parabolic inequalities of Li-Yau type which lead to a scale invariant Harnack inequality, Gaussian upper and lower bounds, a global volume doubling estimate for sub-Riemannian manifolds with ``Ricci'' tensor bounded from below by a negative constant'.

Enrico Le Donne (Berkeley and Orsay): Constant-normal sets in step-3 Carnot groups

In this talk we discuss subsets of Carnot groups that have constant subRiemannian normal. We reformulate the definition in geometric terms, without considering distributions. We are concerned with parametrizations of boundaries and their regularity. We give an example of constant-normal set that, when written as intrinsic upper-graph with respect to the direction of the normal, then the function de fining the set fails to be continuous. However, we give nontrivial examples of groups where constant-normal sets are upper-graphs of entire Lipschitz functions and Lipschitz-intrinsic graphs in some horizontal directions. Depending on time, we will discuss other pathological examples in higher dimension.

The talk is mostly based on a joint work with Costante Bellettini (Princeton).

Valentino Magnani (Pisa): Regularity estimates for convex functions in Carnot-Carathéodory spaces

We will present both a uniform estimate and a Lipschitz continuity estimate for convex functions along Hörmander vector fields that are locally bounded from above. Our arguments are mainly based on the geometry induced by the Hörmander vector fields and by the classical upper estimates for weak subsolutions of sub-Laplacians.

Pertti Mattila (Helsinki): Singular integrals and removability for Lipschitz harmonic functions in Heisenberg groups

The talk is based on joint work with V. Chousionis. We consider singular integrals on small (that is, measure zero and lower than full dimensional) subsets of metric groups. The main examples of the groups we have in mind are Euclidean spaces and Heisenberg groups. In Heisenberg groups we give some applications to harmonic (in the Heisenberg sense) functions of some results known earlier in Euclidean spaces.

Roberto Monti (Padova): The regularity problem of minimal surfaces in the Heisenberg group

We discuss the regularity problem  of sets in the Heisenberg group (with n>1) that are local minimizer for the Heisenberg perimeter. We present some partial results about Lipschitz and "harmonic" approximation of the boundary of minimizers. This is a joint work in progress with D. Vittone.

Séverine Rigot (Nice): Isodiametric problem in Carnot groups

In this talk we consider the isodiametric problem in Carnot groups, i.e., the problem of finding sets, called isodiametric, that maximize the Haar measure among all sets with a fixed diameter. In the Euclidean setting, it is well known that isodiametric sets are the balls. We will show that for non abelian Carnot groups, the situation is significantly different. We will for instance show that one can always find a homogeneous distance for which balls are not isodiametric. We will give various consequences of this fact, related in particular to rectifiability and density properties of the space. We will also give more refined results in the case of the Heisenberg group, this last point being a joint work with G.P. Leonardi and D. Vittone.

Manuel Ritoré (Granada): Variational problems in contact sub-Riemannian manifolds

We describe recent progress on variational problems related to the sub-Riemannian area in the class of contact sub-Riemannian manifolds. This is joint work with Matteo Galli.

Jeremy Tyson (Urbana): Projection maps and Sobolev dimension distortion in the Heisenberg group

We will discuss recent results on the metric and measure-theoretic properties of nonlinear projection mappings onto vertical subspaces of the Heisenberg group. These include dimension estimates for images of fixed subsets of the Heisenberg group under generic vertical projections, and for images of generic fibers of such a projection map under Sobolev and quasiconformal mappings.

Davide Vittone (Padova): Some new results on sub-Riemannian geodesics in Carnot groups

We present some results on sub-Riemannian geodesics in Carnot groups recently obtained in collaboration with E. Le Donne, G.P. Leonardi, and R. Monti. We show that any abnormal geodesic in a free Carnot group is contained in an algebraic variety belonging to a certain class. Applications to the problem of geodesics' regularity will be discussed.

SHORT TALKS

Davide Barilari (Palaiseau): Curvature in sub-Riemannian geometry

In this talk I will discuss a definition of curvature for general sub-Riemannian manifolds that is naturally related to the asymptotic expansion of the distance along a geodesic. It appears as a generalization of the Riemannian sectional curvature tensor.
Finally I will discuss some possible and future applications: sub-laplacian of the distance in SR geometry, measure contraction property, asymptotic of the heat kernel.
This is part of a joint project with A. Agrachev, L. Rizzi (SISSA) and P. Lee (Honk Hong).

Vladimir Goldshtein (Beer Sheva): L_{q,p}-cohomology

Erlend Grong (Bergen): PDEs as geodesic equations in infinite dimensional sub-Riemannian geometry

Ever since Arnold in 1966 showed that the Euler equation for an ideal, incompressible fluid can be thought of as geodesic equations with respect to a Riemannian structure on the group of volume preserving diffeomorphisms, there has been an interest in developing a theory of Riemannian geometry for infinite dimensional manifolds. Several known PDEs, such as the Burgers' equation and KdV, can be seen as geodesic equations in this framework. The topic of the talk will be to discuss a generalization of sub-Riemannian geometry to the infinite dimensional setting. We also discuss some application of this theory and give some examples.

Gerasim Kokarev (Muenchen): Sub-Laplacian eigenvalue bounds on CR manifolds

We prove upper bounds for sub-Laplacian eigenvalues independent of a pseudo-Hermitian structure on a CR manifold. These bounds are compatible with the Menikoff-Sjoestrand asymptotic law, and can be viewed as a CR version of Korevaar’s bounds for Laplace eigenvalues of conformal metrics.