Given a real odd-dimensional oriented submanifold $M$ of a complex manifold $X$ a natural question arises: is $M$ the boundary of a complex submanifold (or subvariety) $Y$ of $X$?
This problem, known as the boundary problem, is now more than 60 years old, but it is still alive, since the question can be asked in various different settings. Ideally the task would be to find necessary and sufficient conditions on $M$.
The first answer was due to Wermer and dates back to 1958: the case of a compact curve in $C^n$ was solved. Almost two decades later (1975 and 1977), Harvey and Lawson dealt with the case of a compact manifold of arbitrary odd-dimension in $C^n$. In order to get a solution, the regularity request on $Y$ had to be dropped, allowing also holomorphic chains as weak solution (having $dY=M$ in the sense of currents). Adding some convexity properties on $M$ allowed to prove that the solution $Y$ is indeed a variety.
More recently, in the '90s, the much more difficult problem for $M$ in some open subset of $CP^n$ was tackled by Dolbeault, Henkin and Dihn, and in the new millennium the problem in $CP^n$ was approached by Harvey, Lawson and Wermer again.
The unbounded version of the problem (i.e. when $M\subset C^n$ is not compact) was studied by Della Sala and myself, and the problem in Hilbert complex space by Mongodi and myself.
I will review known results and open problems.
Some results on codimension-one rectifiability for subsets in Carnot groups
At the beginning of the seminar, we introduce the setting of Carnot groups and, in this realm, we discuss the link between different notions of rectifiability: the general one for metric spaces by Federer and the ones proposed in the specific case of Carnot groups by Pauls and Franchi, Serapioni and Serra Cassano. We focus on codimension-one rectifiability and present some known results that highlight the relations between the various notions we present.
In the second part of the seminar we present our results. First, we show the existence of a smooth non-characteristic hypersurface in a Carnot group of topological dimension 8 that is purely Pauls unrectifiable. This shows that even very regular objects of codimension one that for sure are rectifiable according to the definition of Franchi, Serapioni and Serra Cassano are not so according to Pauls’ definition. Finally we show that such an example does not exist in the Heisenberg group H^n, with n ≥ 2. In particular we show that every smooth hypersurface in H^n, with n ≥ 2, is Pauls rectifiable, even with bi-Lipschitz maps. For n = 1 this last result was already known from previous works by Cole, Pauls and Bigolin, Vittone.
This is a joint work with Enrico Le Donne. The results we present are contained in the arXiv preprint "Pauls rectifiable and purely Pauls unrectifiable smooth hypersurfaces" (https: //arxiv.org/abs/1910.12812).
Regularity for weak solutions of quasilinear degenerate parabolic PDE in the Heisenberg group
Luca Capogna (WPI, USA)
In a joint work with Giovanna Citti (Bologna), and Nicola Garofalo (Padova) we establish gradient regularity for solutions of a class of degenerate parabolic PDE modeled on the regularized p-Laplacian (p larger or equal than 2) in the Heisenberg group or more in general on subriemannian contact manifolds . Our approach rests on a set of techniques introduced by X. Zhong in the stationary case, which we extend to the time-dependent PDE.
Invariants for sublinear bilipschitz equivalence
Gabriel Pallier (Universita' di Pisa)
Between metric spaces, sublinear bilipschitz equivalences are generalized quasiisometries for which the large-scale Lipschitz behavior is kept, while the coarse behavior is not. They appear between families of (non-pairwise quasiisometric) nilpotent or solvable connected Lie groups with close structure, in the study of asymptotic cones of Lie groups by Cornulier. In this talk, I will report on my work on revisiting certain classical quasiisometric invariants of groups to determine which of them can be turned into sublinear asymptotic invariants, in the continuation of Gromov's questions of classifying homogeneous spaces (e.g. symmetric spaces, noncompact solvmanifolds) up to quasiisometry and investigate their quasiisometric rigidity. I will present a partial classification result for the Riemannian homogeneous spaces of negative curvature. Time permitting, I will discuss the (open) problem of determining which finitely generated groups are sublinearly bilipschitz equivalent to real hyperbolic spaces, and a quantitative approach to the classification on the nilpotent side.
The Moduli space of subRiemannian Heisenberg manifolds and Mahler's compactness theorem
Kenshiro Tashiro (Kyoto University)
We introduce the Moduli space of sub-Riemannian Heisenberg manifolds and show an analogy of Mahler's compactness theorem, that is, any subset with lower bound of systoles and upper bound of measure is compact.