Paris–London Geometry Days 2023

We are organising a series of two meetings between geometry groups in London and Paris. We have scheduled two meetings, one in London with a focus on Riemannian geometry and geometric analysis, and one in Paris with a focus on Kähler geometry.

The meetings are modelled on Géométrie: échanges et perspectives. Each meeting will consist of three talks. The morning talk will be a 1.5-hour presentation including an introduction to the specific research topic. The two afternoon talks will be more classical 1-hour research talks.

The meetings are organised by Hugues Auvray, Alix Deruelle, Eleonora Di Nezza, Lorenzo Foscolo and Ilaria Mondello. We acknowledge financial support from the ANR SiGMA and the UCL Cities partnerships programme.

Participation is open to all, but we request that you register your attendance following the details below. Lunch will be offered to all registered participants. We have limited funding to support participation of early career researchers (PhD students and postdocs): contact L. Foscolo to apply.

Paris meeting: Kähler geometry

When: 30 May 2023

Where: Institut de Mathématiques de Jussieu–Paris Rive Gauche
4 Place Jussieu, Room 15-16 101. (= You should enter the campus, find tower 16, go to the first floor and take the corridor 15-16. The room number is then 101.)

Speakers: Ruadhaí Dervan (Glasgow), Hans-Joachim Hein (Münster), Enrica Mazzon (Regensburg)

Registration: by email before 15 May 2023


10:30 - coffee break

11:00 - Hans-Joachim Hein - A gluing construction for complex surfaces with hyperbolic cusps

We will describe an example of a degeneration of degree 6 algebraic surfaces in CP^3 with only ordinary triple point singularities on its central fiber. Then we will show how the unique negative Kähler-Einstein metrics on the smooth fibers, which exist by the Aubin-Yau theorem, disintegrate into three distinct geometric pieces on approach to the central fiber: (1) Kobayashi's complete Kähler-Einstein metric on the complement of the triple points, (2) long thin neck regions, and (3) Tian-Yau's complete Ricci-flat Kähler metrics in small neighborhoods of the vanishing cycles. Joint work with Xin Fu and Xumin Jiang.

12:30 - lunch

14:00 - Ruadhaí Dervan - The universal structure of moment maps in complex geometry

Much of complex geometry is motivated by linking the existence of solutions to geometric PDEs (producing "canonical metrics") to stability conditions in algebraic geometry. I will answer a more fundamental question: what is the recipe to produce geometric PDEs in complex geometry? The construction will be geometric, using a combination of universal families and tools from equivariant differential geometry. This is joint work with Michael Hallam.

15:00 - coffee break

15:30 - Enrica Mazzon - (Non-archimedean) SYZ fibrations for Calabi-Yau hypersurfaces

The SYZ conjecture is a conjectural geometric explanation of mirror symmetry. Based on this, Kontsevich and Soibelman proposed a non-archimedean approach, which led to the construction of non-archimedean SYZ fibrations by Nicaise-Xu-Yu. A recent result by Li relates explicitly the non-archimedean approach to the classical SYZ conjecture. In this talk, I will give an overview of this subject and I will focus on families of Calabi-Yau hypersurfaces in P^n. For this class, in collaboration with Jakob Hultgren, Mattias Jonsson and Nick McCleerey, we solve a non-archimedean conjecture proposed by Li and deduce that classical SYZ fibrations exist on a large open region of CY hypersurfaces.

London meeting: Riemannian geometry

When: 28 April 2023

Where: Mathematics Department, University College London. Talks are in room 505, coffee breaks and lunch in room 502

Speakers: Eva Kopfer (Bonn), Felix Schulze (Warwick), Gabriella Tarantello (Tor Vergata)


10am - coffee break

10.30 - Felix Schulze - Generic regularity for minimizing hypersurfaces in dimensions 9 and 10

Let Γ be a smooth, closed (compact and boundaryless), oriented, (n-1)-dimensional submanifold of ℝ^{n+1}. The Plateau problem asks if among all smooth, compact, oriented hypersurfaces M ⊂ ℝ^{n+1} with ∂ M = Γ, does there exist one with least area? Foundational results in geometric measure theory can be used to show that for n+1 ≤ 7 there is an smooth, compact, oriented, area-minimizing hypersurface M solving the problem. In higher dimensions smooth minimizers can fail to exist but it is nevertheless known that away from a closed set sing M ⊂ ℝ^{n+1}\ Γ of Hausdorff dimension ≤ n-7, there is a minimizer M which is a smooth hypersurface with boundary Γ. A fundamental result of Hardt-Simon shows that the singularities (necessarily isolated points) for n+1=8 can be eliminated by a slight perturbation of the boundary, Γ, thus yielding solutions to the original problem. It has been a longstanding conjecture that similar results hold in higher dimensions. We show that for n+1= 9,10 singularities can still be perturbed away as well. This is joint work with O. Chodosh and C. Mantoulidis.

12.00 - lunch

13.30 - Eva Kopfer - Bochner formulas, functional inequalities and generalized Ricci flow

Generalized Ricci flows are supersolutions of Ricci flows. We discover how to characterize generalized Ricci flow on path space using a Bochner formula for martingales, extending the results known for Ricci flows.

14.30 - coffee break

15.00 - Gabriella Tarantello - On a Donaldson functional for CMC-immersions of surfaces into Hyperbolic 3-manifolds

I discuss a parametrization for the moduli space of Constant Mean Curvature (CMC) immersions of a closed surface S (orientable and of genus at least 2) into hyperbolic 3-manifolds by pairs describing the tangent bundle of the Teichmueller space of S. For any such pair, we determine uniquely the pullback metric and the second fundamental form of the immersion by solving the Gauss-Codazzi equations. Solutions of the Gauss-Codazzi equations correspond to critical points of a suitable "Donaldson -functional" introduced by Gonsalves-Uhlenbeck (2007). In collaboration with M. Lucia and Z. Huang (2022) we proved that indeed such functional admits a global minimum as its unique critical point. In addition I shall discuss the asymptotic behavior of those minimizers and obtain a "convergence" result in terms of the Kodaira map. For example, in case of genus 2, it is possible to catch at the limit "regular" CMC 1-immersions, except in very rare situations which relate to the image, by the Kodaira map, of the six Weierstrass points of S. If time permits, I shall mention further progress for higher genus obtained in collaboration with S. Trapani.