Maths Seminar

Abstract: In this talk, I will describe some recent progress on the Laplacian flow for closed G2 structures.

Abstract: We establish a relation between the spectrum of  a Dirac operator on S^{5} and certain sheaf cohomologies on the complex projective 

plane. This operator comes from the deformation  of G_{2}-instantons with 1-dim singularities.  

Abstract: Ancient solutions model the singularity formation of the Ricci flow.  In two and three dimensions, we currently have complete classifications for κ-noncollapsed ancient solutions, while the higher dimensional problem remains open. This talk will survey recent developments of Ricci shrinkers, which form an important class of ancient solutions, and higher dimensional ancient solutions.

Abstract: One of fundamental questions is  how to classify open 3-manifolds with positive scalar curvature. The topology of open 3-manifolds is much complicated. For example, Geometrization conjecture is failed to be generalized to open 3-manifolds. In this talk, we give a classification for open 3-manifolds with uniformly positive scalar curvature. Precisely, we use minimal surface theory to give a prime decomposition for such manifolds. 

Abstract: Existence of constant scalar curvature Kähler (cscK) metrics on compact Kähler manifolds is a central question in complex geometry. Following the variational approach pioneered by Mabuchi in the 1980's it was recently proven (by X.X. Chen and J. Cheng) that existence of cscK metrics is equivalent to coercivity of the Mabuchi K-energy functional on the space of Kähler metrics. In this talk I will present new coercivity estimates directly related to this problem, focusing on the strongly related J-functional of Chen/Donaldson, which occurs as the “energy part” in the Chen-Tian decomposition of the K-energy, and whose Euler-Lagrange equation is Donaldson’s J-equation.

As a main result of the talk we give an explicit and optimal lower bound for the J-functional, in the sense of finding the largest possible constant in the definition of coercivity (which always exists and takes negative values in general). This has applications to stability, and sheds new light on existence criteria for cscK metrics using Tian's alpha invariant, in the spirit of Dervan and Li-Shi-Yao. As a third application we explain that there must always exist cscK metrics on compact Kähler manifolds with nef canonical bundle, thus on all smooth minimal models, and also on the blowup of any such manifold. This extends a result of Jian-Shi-Song with a proof that does not depend on the Abundance conjecture.

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Abstract: In collaboration with G. Edwards we produce (local) Calabi-Yau metrics, in two complex dimensions, with cone singularities along intersecting complex lines, for cone angles that strictly violate the Troyanov condition. We identify the tangent cone at the origin as a product of two 2-cones. In the tangent cone limit, the line with the smallest cone angle remains apart while the other lines collide into a single cone factor. 

To prove our result, we first write an approximate solution with the desired asymptotic behavior and small Ricci potential. The main difficulty is to invert the Laplacian of such approximate solution metric in suitable Holder spaces. Once this is done, we use the implicit function theorem to perturb into an actual Calabi-Yau metric.

Abstract: We will introduce our recent works on two transformations of complex structures: deformation and blow-up. We prove that the p-Kahler structure with the so-called mild ddbar-lemma is stable under small differentiable deformation. This solves a problem of Kodaira in his classic and generalizes Kodaira-Spencer's local stability theorem of Kahler structure. Using a differential geometric method, we solve a logarithmic dbar-equation on Kahler manifold to revisit Deligne's degeneracy theorem for the logarithmic Hodge to de Rham spectral sequence at E1-level and Katzarkov-Kontsevich-Pantev's unobstructedness of the deformations of a log Calabi-Yau pair. Finally, we will introduce a blow-up formula for Dolbeault cohomologies of compact complex manifolds by introducing relative Dolbeault cohomology. This talk is based on several joint works with Kefeng Liu, Xueyuan Wan, Song Yang, Xiangdong Yang, Quanting Zhao, etc.

Abstract: We define and study the harmonic heat flow for almost complex structures which are compatible with a Riemannian structure (M; g). This is a tensor-valued version of harmonic map heat flow. We prove that if the initial almost complex structure J has small energy(depending on the norm |\nabla J|), then the flow exists for all time and converges to a Kaehler structure. We also prove that there is a finite time singularity if the initial energy is sufficiently small but there is no Kaehler structure in the homotopy class. A main technical tool is a version of monotonicity formula, similar as in the theory of the harmonic map heat flow. We also construct an almost complex structure on a flat four tori with small energy such that the harmonic heat flow blows up at finite time with such an initial data.