September 9, 2021
TITLE: Collapsing geometry of hyperkaehler manifolds in dimension four
ABSTRACT: We will overview some recent developments on the degeneration theory of hyperkaehler 4-manifolds. As preliminaries, we will present some tools and conceptual ideas in understanding metric geometric aspect of degenerating hyperkaehler metrics. After explaining the background, this talk will focus on the following classification results in the volume collapsed setting.
First, we will classify the collapsed limits of the bounded-diameter hyperkaehler metrics on the K3 manifold. More generally, we will precisely characterize the limiting singularities of the hyperkaehler metrics with bounded quadratic curvature integral. We will also exhibit the classification of the complete non-compact hyperkaehler 4-manifolds with quadratically integrable curvature which are called gravitational instantons and appear as bubbling limits of degenerating hyperkaehler metrics.
The above ingredients constitute a relatively complete picture of the collapsing geometry of hyperkaehler metrics in dimension 4.
April 12, 2018
TITLE: Gravitational collapsing of K3 surfaces II
ABSTRACT: We will exhibit some new examples of collapsed hyperkähler metrics on a K3 surface. This is my recent joint work with Hans-Joachim Hein, Song Sun and Jeff Viaclovsky. We will construct a family of hyperkähler metrics on a K3 surface which are collapsing to a closed interval. Geometrically, each regular fiber is a Heisenberg manifold and each singular fiber is a singular circle fibration over a torus. In our example, each bubble limit is either the Taub-NUT space or a complete hyperkähler space constructed by Tian-Yau. The regularity estimates in this example in fact confirms a general picture given by the -regularity theorem we present in Lecture 2. Technically, our examples are achieved by a new gluing construction. Continuing Jeff Viaclovsky’s introduction to the construction of this example, we will go through the details of the proof. We will also discuss some variations of the main gluing construction and some possible developments.
April 9, 2018
TITLE: Quantitative nilpotent structure and regularity theorems of collapsed Einstein manifolds
ABSTRACT: This talk is on the new developments of the structure theory for collapsed Einstein manifolds. We will start with some motivating examples of collapsed Ricci-flat manifolds. Our main focus is the -regularity and structure theorems for collapsed Einstein manifolds which is included in my joint work with Aaron Naber. First, in the context of manifolds with Ricci curvature uniformly bounded from below, we show that every point on such a manifold can be associated with a nilpotent rank which has a sharp upper bound. This follows from an effective version of the Generalized Margulis Lemma. The main part of the -regularity theorem gives the following dichotomy: either the curvatures are uniformly bounded or the nilpotent rank drops.
April 9, 2018
TITLE: Introduction to Ricci curvature and the convergence theory
ABSTRACT: The first talk is an overview of the convergence and regularity theory of the manifolds with Ricci and sectional curvature bounds. Specifically, we will review some both classical and new structure theory such as the -regularity theorems, the fibration theorems, and the structure of the limit spaces. The main part is to introduce the analytic tools in studying the non-collapsing manifolds and we will see why most tools legitimately fail in the collapsed context. Another emphasis is the development of the Generalized Margulis Lemma which gives the local collapsing geometry at the level of the fundamental group.