Beijing Institute of Mathematical Sciences and Applications

In the first part, we derive an explicit formula of the Green function for the generic GJMS operator on the sphere. Our tools are the spectral theory of elliptic operators and Gegenbauer polynomials. In the second part, we study a rigidity problem of Green functions for GJMS operator of orders two and four on a hypersurface in Euclidean space. Especially, in lower dimensions, the positive mass theorem and Hartman-Nirenberg theorem are applied to obtain the strong rigidity theorem for the Green function of conformal Laplacian. This is jointly with Yalong Shi.

The Kähler-like condition is introduced by Yang-Zheng and Angella-Otal-Ugarte-Villacampa, which says the curvature of a metric connection has the same symmetry as the one of a Kähler metric. In this talk, we mainly discuss when the Bismut-Strominger connection is Kähler-like, BKL for short, and show several structural theorems on BKL manifolds and their relevant manifolds.

In this talk, we consider the moduli space of rank-two Higgs bundles with irregular singularities over the projective line. Along a curve of certain type, we show that Hitchin's hyperkähler metric is asymptotic to a simpler semi-flat metric at an exponential rate, based on the foundational works of Fredrickson, Mazzeo, Mochizuki, Swoboda, Weiss, and Witt. In our gluing construction of the harmonic metric, we introduce a new building block for weakly parabolic singularities with trivial flags. In dimension four, we explicitly compute the asymptotic limit of the semi-flat metric, which is of type ALG or ALG*. Joint work with Gao Chen.

We prove three rigidity results for Einstein manifolds with bounded covering geometry. (1) any almost flat manifold (M,g) must be flat if it is Einstein, i.e. Ric=Lg for some real number L. (2) A compact Einstein manifold with a non-vanishing and almost maximal volume entropy is hyperbolic. (3) A compact Einstein manifold admitting a uniform local rewinding almost maximal volume is isometric to a space form. This is a joint work with Cuifang Si.

Z2 harmonic 1-forms are generalize quadratic differentials on Riemann surfaces to higher dimensions, establishing deep connections with gauge theory, low-dimensional topology, and calibrated geometry. Taubes' work indicates that Z2 harmonic 1-forms are the natural boundaries for various gauge theory equations, including those for flat SL(2,C) connections. On the first half of the talk, we will introduce the background of this topic and discuss known results. On the second half of the talk, we will discuss a question introduced by Taubes-Wu, about the existence and rigidity problem of the tangent cone model of the Z2 harmonic 1-form, which will be based on joint work with J.Chen. We will explain how to apply finite group representation theory to this question.

Most of the special holonomy metrics on manifolds could be defined by specified a collection of closed differential forms satisfying some linear algebraic property. Using this observation, Ryushi Goto was able to write down a series of equations whose solutions correspond to deformations of these special holonomy metrics, and, in compact cases, to show their solvability, providing a unified treatment to previously known unobstructedness theorems of Bogomolov-Tian-Todorov (in the Sp and SU cases), and Joyce (in the G2 and Spin(7) cases). I want to show how to reinterpret Goto's equations as Maurer-Cartan equations in a certain dg-Lie algebra. The well-developed theory of Maurer-Cartan equation allows for an easy reproval of Goto's theorem (or at least its formal version), as well as suggests potential applicability of this method to other geometric structures.

Gravitational instantons are Ricci flat complete Riemannian 4-manifolds with at least quadratic curvature decay. Classical examples include the Taub-NUT and the Euclidean Kerr instanton. A classification of half-flat instantons is known but the uniqueness problem remains open in general. In this talk I will present some recent results the classification of $S^1$-symmetric instantons obtained using an identity of Israel-Robinson type and the $G$-signature theorem, together with recent results on instantons with special geometry.