RESEARCH TALKS

Arunima BHATTACHARYA (University of North Carolina at Chapel Hill)

Hessian estimates for shrinkers, expanders, translators, and rotators of the Lagrangian Mean Curvature Flow

Abstract

In this talk, we will discuss interior Hessian estimates for shrinkers, expanders, translators, and rotators of the Lagrangian mean curvature flow, and further extend this result to a broader class of Lagrangian mean curvature type equations. We assume the Lagrangian phase to be hypercritical, which results in the convexity of the potential of the initial Lagrangian submanifold. Convex solutions of the second boundary value problem for certain such equations were constructed by Brendle-Warren 2010, Huang 2015, and Wang-Huang-Bao 2023.
We will also briefly introduce the fourth-order Hamiltonian stationary equation and mention some recent results on the regularity of solutions of certain fourth-order PDEs, which are critical points of variational integrals of the Hessian of a scalar function. Examples include volume functionals on Lagrangian submanifolds. This talk is partially based on joint work with Jeremy Wall.

Anne FINO (Florida International University)

Pluriclosed and Kaehler-like metrics on complex manifolds

Abstract

A Hermitian metric on a complex manifold is called pluriclosed or strong Kaehler with torsion (SKT) if the torsion of the associated Bismut connection associated is closed. In the talk I will present some general results on pluriclosed metrics in relation to the pluriclosed flow and Kaehler-like curvature conditions. Next I will discuss some recent results on compact complex manifolds admitting a Hermitian structure which is pluriclosed and CYT, but not Bismut flat.

Bin GUO (Rutgers Newark) 

Uniform estimates for complex Monge-Ampere and fully nonlinear equations

Abstract

Uniform estimates for complex Monge-Ampere equations have been extensively studied, ever since Yau’s resolution of the Calabi conjecture. Subsequent developments have led to many geometric applications to many other fields, but all relied on the pluripotential theory from complex analysis. In this talk, we will discuss a new PDE-based method of obtaining sharp uniform C^0 estimates for complex Monge-Ampere (MA) and other fully nonlinear PDEs, without the pluripotential theory. This new method extends more generally to other interesting geometric estimates for MA and Hessian equations. This is based on the joint works with D.H. Phong and F. Tong.

Hans-Joachim HEIN (WWU Münster)

Immortal solutions of the Kähler-Ricci flow

Abstract

We investigate the collapsing behavior of immortal solutions of the Kähler-Ricci flow on compact Kähler manifolds. Assuming the Abundance Conjecture, we prove that the Ricci curvature of the evolving metrics remains locally uniformly bounded away from the singular fibers of the Iitaka fibration. Joint work with Man-Chun Lee and Valentino Tosatti.

Jakob HULTGREN (Umeå University)

On the SYZ-conjecture for hypersurfaces in toric Fano manifolds

Abstract

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.

Adam JACOB (University of California Davis)

Mean curvature flow of totally real submanifolds

Abstract

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.

Nicholas McCLEERY (Perdue University)

The Eigenvalue Problem for the Complex Hessian Operator on m-Pseudoconvex Manifolds

Abstract

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.

Sébastien PICARD (University of British Columbia)

Non-Kahler Calabi-Yau Geometry

Abstract

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.

Carlo SCARPA (Université du Québec à Montréal)

The Einstein-Hilbert functional and the Donaldson-Futaki invariant

Abstract

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.

Jian SONG (Rutgers)

Finite time singularities of the Kahler-Ricci flow

Abstract

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.

Tat Dat TÔ (Sorbonne Université) 

Singular cscK metrics on smoothable varieties

Abstract

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.

Freid TONG (Harvard CMSA)

On some real Monge Ampere functionals

Abstract

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.

Yuanqi WANG (University of Kansas) 

On ALG Ricci Flat K\”ahler 3-folds with Schwartz Decay

Abstract

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