Northwestern, Notre Dame, UIC Complex Geometry Seminar

In the study of constant scalar curvature Kähler (cscK) metrics, which are critical points of the K-energy, various versions of stability have been proposed including the coerciveness by Tian, the uniform K-stability by Yau, Tian, Donaldson, Dervan-Ross, Sjöström Dyrefelt and the uniform-slope-stability by Ross-Thomas. In this talk, I will prove that all of them are equivalent to the existence of the critical point if we replace the K-energy by J_\chi-functional. The only new part is the proof of a uniform version of Lejmi and Székelyhidi's conjecture. If time permits, I will also explain the application to cscK metrics and deformed-Hermitian-Yang-Mills (dHYM) equation.

In this talk, I will describe a new class of domains in complex Euclidean space which is defined in terms of the existence of Kaehler metrics with good geometric properties. By definition, this class is invariant under biholomorphism. It also includes many well-studied classes of domains such as strongly pseudoconvex domains, finite type domains in dimension two, convex domains, homogeneous domains, and embeddings of Teichmuller space. Analytic problems are also tractable for this class, in particular we show that compactness of the dbar-Neumann operator on (0,q)-forms is equivalent to a growth condition of the Bergman metric. This generalizes an old result of Fu-Straube for convex domains.