The Geometry and Analysis seminar is the main seminar of the Leeds Geometry and Analysis Group. Unless otherwise stated, seminars take place on Wednesdays at 15:00. The seminar is organised by Ben Lambert and Francesca Tripaldi.
Z/2 harmonic 1-forms are generalizations of harmonic 1-forms that allow topological twisting around a subspace of codimension 2. These objects were introduced by Taubes to compactify the moduli spaces of solutions to generalized Seiberg-Witten equations, and they show up in many other gauge theoretical problems.
Donaldson showed there is a deformation theory for so-called non-degenerate Z/2-harmonic 1-forms. In this presentation metric we study the perturbations of the remaining degenerate solutions. For a natural class of degenerate examples, we prove that after a suitable perturbation of the ambient Riemannian metric, the form can be deformed to a nearby non-degenerate Z/2-harmonic 1-form.
In this talk, I will introduce spherical horoconvexity, a new notion of convexity inspired by its hyperbolic counterpart. The focus will be on its definition and geometric meaning, with geodesic spheres tangent to the equator playing the role of horospheres. I will also briefly discuss applications to curvature flows and geometric inequalities, together with a capillary analogue in the Euclidean unit ball. This is based on joint works with Julian Scheuer (Frankfurt).
Do Carmo's problem asks whether every complete, noncompact, stable constant mean curvature (CMC) hypersurface in Euclidean space must be minimal. I will first review some classical and recent progress on this question, with particular emphasis on dimension-free rigidity phenomena and on their connection with the stable Bernstein problem. I will then turn to the anisotropic counterpart of Do Carmo's question, where stability can still impose strong restrictions on the geometry at infinity. In particular, I will present a one-endedness theorem for stable constant anisotropic mean curvature hypersurfaces under a natural ellipticity condition and discuss the main ideas behind its proof.
A foundational and fascinating theme in differential geometry is to understand how geometric quantities constrain topology. In this talk, we will explain how the topological complexity of minimal surfaces in closed three-manifolds can be controlled by their variational instability and area, and how positive Ricci curvature of the ambient manifold leads to stronger bounds depending only on the Morse index. Finally, we will discuss some consequences for the topology of min–max minimal surfaces.
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.
We will talk about our recent work, a resolution of a longstanding problem on the sharp curl-Sobolev inequality for differential forms on the round sphere. We then present its geometric and variational applications that settle several open problems and conjectures in geometry and mathematical physics. This is a joint work with Mingwei Zhang.
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.
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 $\mathbb{R}^{n+1,0}\subset \mathbb{R}^{n+1,k}$.
