Résumés PDE 2024

The talks will be concerned with the SYZ-conjecture for families of hypersurfaces in toric Fano manifolds. Recent work by Yang Li reduces a weak version of the SYZ-conjecture in this setting to the solvability of a real Monge-Ampère equation on the boundary of a polytope. Curiously, this Monge-Ampère equation is solvable for some families and not solvable for some families. I will explain how solvability can be described in terms of properties of the solutions to an optimal transport problem, and how another subtle aspect of the PDE, the presence of a free boundary, becomes less mysterious when viewed through the lens of optimal transport. Finally, I will highlight some of the many open problems related to this. This is based on joint work with Rolf Andreasson, Thibaut Delcroix, Mattias Jonsson, Enrica Mazzon and Nick McCleerey.

Given a complex manifold X, a totally real submanifold is a half dimensional submanifold whose tangent space contains no complex lines. Examples include Lagrangian submanifolds of Kahler manifolds, as well as small deformations of Lagrangians. In the case that X is a negatively curved Kahler-Einstein manifold or a Calabi-Yau, we demonstrate that the mean curvature flow on a totally real submanifold L will converge exponentially fast to a minimal Lagrangian submanifold, provided that the initial mean curvature vector of L, as well as the initial restriction of the Kahler form to L, are sufficiently small in the C^0 norm. This is joint work with Tristan Collins and Yu-Shen Lin.

We establish C^{1,1}-regularity and uniqueness of the first eigenfunction of the complex Hessian operator on strongly m-pseudoconvex manifolds, along with a variational formula for the first eigenvalue. From these results, we derive a number of applications, including a bifurcation-type theorem and geometric bounds for the eigenvalue. This is joint work with Jianchun Chu and Yaxiong Liu.

Distinct Calabi-Yau threefolds can be connected by degeneration and resolution. The process of degeneration and resolution may connect a Kahler Calabi-Yau threefold to a non-Kahler complex manifold. It is proposed to geometrize these non-Kahler objects by geometric structures solving the equations of heterotic string theory. This talk will survey various aspects of this problem.

In a recent work joint with Abdellah Lahdili (Université du Québec à Montréal) and Eveline Legendre (Université Lyon 1), we proposed a new approach to the existence of constant scalar curvature Kähler metrics on compact complex manifolds, by using (a version of) the Einstein-Hilbert functional. In this talk, I will explain some properties of the Einstein-Hilbert functional, particularly in connection with test configurations for a compact Kähler manifold, and then show how they can be used to give an alternative proof of the K-semistability of constant scalar curvature Kähler manifolds. I will then dedicate some time to explain the geometric motivation for considering an Einstein-Hilbert functional in this context, and possible extensions of this work.

We establish the scalar curvature and distance bounds for general finite time solutions of the Kahler-Ricci flow, extending Perelman’s work on the Fano Kahler-Ricci flow. We further prove that the Type I blow-ups of the finite time solution always sub-converge in Gromov-Hausdorff sense to an ancient solution on a family of analytic normal varieties with suitable choices of base points. As a consequence, the Type I diameter bound is proved for almost every fibre of collapsing solutions of the Kahler-Ricci flow on a Fano fibre bundle. We also apply our estimates to show that every solution of the Kahler-Ricci flow with Calabi symmetry must develop Type I singularities, including both cases of high codimensional contractions and fibre collapsing.

I will present our recent results on the openess of the coercivity for Mabuchi functional along a degenerate family of compact Kähler varieties and the existence of singular cscK metrics on ℚ-Gorenstein smoothable klt varieties when the Mabuchi functional is coercive. Our approach relies on developing a novel strong topology of pluripotential theory in families and establishing uniform estimates for cscK metrics. This is a joint work with A. Trusiani and C-M. Pan.

The Monge-Ampere functional is an important tool in the study of Monge Ampere equations. In this talk, I will discuss some interesting new Monge Ampere functionals and some related family of Monge-Ampere equations, one of which is related to the study of complete Calabi-Yau metrics. This is based on joint works with S.-T. Yau, and also with T. Collins and S.-T. Yau.

ALG gravitational instantons are complete hyper-K”ahler surfaces asymptotic to a twisted product of the complex plane and an elliptic curve. Following the classical work of Tian-Yau and Hein, etc., on Monge-Ampere methods for Ricci flat K”ahler metrics on quasi-projective varieties, we provide a geometric existence for generalized ALG Ricci flat K ̈ahler 3-folds on crepant resolutions, where the K3 fiber admits a purely non-symplectic automorphism of finite order. These metrics decay to the ALG model at any polynomial rate, and topological numbers/data can be calculated.
A local K ̈unneth formula plays a role in the Schwartz decay and an ansatz that equals a Ricci flat ALG model outside a compact set.

The proof of Schwartz’s decay relies on a non-concentration of the Newtonian potential. This can not be immediately generalized to fibration with a higher dimensional base due to the existence of a concentrating sequence of L2 normalized eigenfunctions (Zonal functions) on unit round spheres of (real) dimension 2. Moreover, in the fibred cone setting and for functions with 0−average on the fibers, we also need a weighted L2−estimate with no loss of polynomial weight.

K-stability of log Fano cone singularities was introduced by Collins-Sz\’ekelyhidi to serve as a local analog of K-stability of Fano varieties. In the Fano case, the celebrated result of Li-Xu states that to test K-stability, it suffices to test the so-called special test configurations. In this talk, I will talk about a local version of this result for log Fano cones. Our method relies on non-Archimedean pluripotential theory.

Initiated by Mabuchi, Semmes and Donaldson, homogeneous complex Monge-Ampere (HCMA) equations become a central topic in understanding uniqueness and existence of canonical metrics in K\”ahler classes. Under the setting of asymptotically locally Euclidean (ALE) K\”ahler manifolds, one of the main difficulties is the asymptotic behaviors of solutions to HCMA equations. In this talk, I will give an introduction to canonical metric problems under the setting of ALE K\”ahler manifolds and discuss the recent progress on studying asymptotic behaviors of solutions to HCMA equations.

We proved a ‘’no semistability at infinity » result for complete Calabi-Yau metrics asymptotic to cones, by eliminating the possible appearance of an intermediate K-semistable cone in  Donaldson-Sun’s 2-step degeneration theory. As a consequence, we establish a polynomial convergence rate result and a classification result for complete Calabi-Yau manifolds with Euclidean volume growth and quadratic curvature decay. This is based on joint work with Song Sun.