SinGular Monge-Ampère equations

Introduction

This project is driven by M-theory, String theory in theoretical physics, and the Minimal Model Problem in algebraic geometry. We study singular Kähler spaces with a focus on their special structures (of a differential geometry nature) and their interaction with various areas of analysis.

Research Focus

To be more specific, we search for special (singular) Kähler metrics with nice curvature properties, such as:

1. Kähler-Einstein (KE) metrics
2. Constant scalar curvature (cscK) metrics

The problem of the existence of these metrics can be reformulated in terms of a Monge-Ampère equation, which is a non-linear partial differential equation (PDE).

Previous Work

The KE case has been settled by:

• Aubin
• Yau (solving the Calabi conjecture)
• Chen-Donaldson-Sun (solving the Yau-Tian-Donaldson conjecture)

The cscK case has been very recently worked out by Chen-Cheng (solving a conjecture due to Tian). However, these results only hold on smooth Kähler manifolds, and one still needs to deal with singular varieties.

Pluripotential Theory

This is where Pluripotential Theory comes into play. Boucksom-Eyssidieux-Guedj-Zeriahi and the author, along with Darvas and Lu, have demonstrated that pluripotential methods are very flexible and can be adapted to work with (singular) Monge-Ampère equations.

Finding a solution to this type of equation that is smooth outside of the singular locus is equivalent to the existence of singular KE or cscK metrics.

Main Goal

At this point, a crucial ingredient is missing: the regularity of these (weak) solutions. The main goal of SiGMA is to address this challenge by using new techniques and ideas, which might also aid in tackling problems in complex analysis and algebraic geometry.

Research Group

The PI will establish a research group at her host institution focused on regularity problems of non-linear PDEs and geometric problems in singular contexts. The goal is to create a center of research excellence in this topic.