Geometry and Analysis

Carnot groups play a role in subRiemannian geometry analogous to that of Euclidean spaces in Riemannian geometry. Indeed, they arise as metric tangents of equiregular sub-Riemannian manifolds. Their group structure and homogeneous dilations also lead to an intrinsic notion of differentiability, namely Pansu differentiability, and hence to a natural pullback of differential forms.

For $C^1$ contact maps, the classical and Pansu pullbacks are closely related: the latter can be identified with the weight-preserving homogeneous component of the former. In this talk, I will use this relation to investigate how the Pansu pullback interacts with differential complexes on Carnot groups. I will discuss the obstructions to commutativity for the de Rham and Rumin complexes, and then introduce the spectral complexes associated with the weight decomposition of the de Rham complex, for which the Pansu pullback satisfies a commutativity property with the corresponding differentials.

Finally, I will briefly discuss weak formulations of these questions for Sobolev mappings, for which the Pansu pullback can be defined almost everywhere under suitable integrability assumptions.

Sub-Riemannian geometries can be seen as limits of Riemannian metrics. In this talk, we focus on the behaviour of the Gauss-Bonnet theorem under this Riemannian approximation scheme for surfaces embedded in three-dimensional spaces. Adopting a measure-theoretic viewpoint, we prove that the limit Gaussian curvature measure of such a surface is singular and supported on its isolated characteristic points. We identify natural geometric conditions under which this behavior occurs, namely when the surface admits characteristic points of finite order of degeneracy.

NOTES: Note the unusual room and day.

A fundamental problem in differential geometry is understanding the constraints that the topology of a manifold imposes on its geometry. In this direction, the dynamical Alekseevskii conjecture states that if the universal cover of a homogeneous space is not contractible, then every homogeneous Ricci flow solution on it has finite extinction time. Böhm proved the conjecture for compact homogeneous spaces.
In this talk, we will consider the problem for noncompact homogeneous spaces. We will show that the conjecture is true for natural families of initial homogeneous metrics satisfying certain compatibility conditions between the Lie algebra structure and the metric. We will also discuss the problem of determining the limit geometries that can be obtained after parabolic rescaling of these solutions, as well as some applications.

NOTES: Note the unusual room.
In the pseudo-Euclidean space $\mathbb{R}^{n+1,k}$, we consider the mean curvature flow of $n$-dimensional spacelike submanifolds with spacelike codimension one and arbitrary timelike codimension $k$. We show that if the initial submanifold is compact and spacelike-convex (the acceleration along every geodesic is strictly spacelike), then natural quantities measuring curvature pinching and noncollapsing are preserved under the flow. Moreover, we prove an analogue of the Huisken and Gage-Hamilton theorems in this setting, which states that the mean curvature flow deforms any such submanifold to a point in finite time, and that the solution is asymptotic to a shrinking sphere in a maximally spacelike affine subspace $\mathbb{R}^{n+1,0}\subset \mathbb{R}^{n+1,k}$.

In this talk, I will describe a strategy to obtain longtime existence results for a special type of the Lagrangian mean curvature flows (LMCF) in the Kummer K3 surface.  The argument relies on the fact that certain regions of the Kummer K3 surface are modelled on the Eguchi-Hanson space.  This allows us to use a fixed point argument to deform known solutions in the Eguchi-Hanson space into new solutions in the Kummer K3 surface. 

Sub-Riemannian structures of high codimension (greater than one) are rare on 7-manifolds. Until recently, only three such examples were known on any of the homotopy 7-spheres: two on the standard 7-sphere and one on the Gromoll–Meyer exotic sphere. In this talk I will describe new examples of 2-step, codimension-3 sub-Riemannian structures on every homotopy (exotic) 7-sphere.

Let $X$ be a six-dimensional Riemannian manifold with nonnegative sectional curvature that is a Riemannian product of a closed manifold with an Euclidean factor. We prove that every complete, finite index, non-minimal CMC hypersurface immersed in $X$ is compact. This answers affirmatively a question of do Carmo for this class of ambient Riemannian spaces, extending known lower dimensional results.
As a consequence, we complete the classification of two-sided, complete weakly stable CMC hypersurfaces immersed in the space forms of positive curvature in dimension six.
We also show that a complete, finite index CMC hypersurface immersed in the hyperbolic space $\mathbb{H}^6$ with mean curvature $|H|>7$ is compact. This gives a partial answer to a question posed by Chodosh in his survey for the ICM.

Let M be a moduli space of quadratic differentials on the sphere with poles of fixed odd orders. Such a space has a concrete realisation as a space of rational functions. In this talk, I will explain an elementary construction that gives rise to a meromorphic hyper-Kähler metric on the algebraic torus bundle that is the quotient of TM by the natural period lattice. We will meet the theory of isomonodromic deformations, the geometry of Riemann surfaces and, crucially, twistor theory. All of this is motivated by the wall-crossing behaviour of Donaldson-Thomas invariants under the variation of Bridgeland stability conditions but the technical prerequisites to understand this particular construction should be minimal. This talk is partly based on joint work with Maciej Dunajski.

NOTES: Note the unusual room!.
Abstract: Multicomplexes are variants of bicomplexes arising naturally in many geometric, topological and algebraic contexts. I will explain some recent joint work with Joana Cirici and Muriel Livernet  which explores homotopy theories related to the two spectral sequences of a truncated multicomplex.

There are potential applications to the study of homotopy types of almost and generalized complex manifolds.

NOTES: Note the unusual room!.
The Weyl energy on four-manifolds is a geometric functional related to the Chern-Gauss-Bonnet formula. Similarly to Willmore’s functional for surfaces embedded in the three-dimensional Euclidean space, it enjoys conformal invariance properties. We are interested in the interaction of the Weyl’s functionals of two manifolds under the operation of connected sum, showing conditions that decrease the energy.
Such estimates might be useful in understanding compactness properties of minimizing or critical families of metrics.
This is based on a joint project with Matt Gursky and Francesco Malizia.