Seminars & Colloquia Calendar

Abstract: In the last decades geometric flows have been proved to be a powerful tool in the classification and uniformization problems in geometry and topology. Despite the wide range of applicability of the existing analytical methods, we are still lacking efficient tools adapted to the study of general (non-Kähler) complex manifolds. In my talk I will discuss the pluriclosed flow - a modification of the Ricci flow - which was introduced by Streets and Tian, and shares many nice features of the Ricci flow. The important open questions driving the ongoing research in complex geometry are the classification of compact non-Kähler surfaces, and the Global Spherical Shell conjecture. Our hope is that understanding the long-time behaviour and singularities of the pluriclosed flow well enough, we can use it to approach these open questions.

To apply an analytic flow to any geometric problem, we need to make the first necessary step - classify the stationary points of the flow, and, more generally, its solitons (stationary points modulo diffeomorphisms). For the pluriclosed flow, this question reduces to a non-linear elliptic PDE for an Hermitian metric on a given complex manifold. We will discuss this problem on compact/complete complex surfaces, and provide exhaustive classification under natural extra geometric assumptions. In the course of our classification we will discover a natural extension of the famous Gibbons-Hawking ansatz for hyperKähler manifolds.