In an effort to find a higher dimensional analogue of the classical uniformization theorem in complex geometry, it was suggested by E. Calabi that one should study existence of "canonical metrics" on compact Kähler manifolds. These are the so called extremal metrics, characterized by the fact that they minimize a certain functional on the space of Kähler metrics. Many natural and important classes of metrics fall into this category, such as the Kähler-Einstein metrics, that play a prominent role also in physics. Kähler-Einstein metrics were proven to always exist on Calabi-Yau manifolds and on manifolds of general type. In the case of Fano manifolds however, it has been long known that they do not always exist, and it was conjectured by S-T. Yau, S. Donaldson and G. Tian that existence should be linked to stability of the underlying manifold. In the case of Kähler-Einstein metrics on Fano manifolds this conjecture was confirmed in 2012 by Chen-Donaldson-Sun and Tian independently, who showed that existence of Kähler Einstein metrics is equivalent to a stability condition in algebraic geometry called K-stability.
Generalizing the above picture further we may ask about existence of a larger class of extremal metrics, namely the constant scalar curvature metrics. In this talk I will present an analytic/pluripotential approach to stability notions that obstruct existence of such metrics on arbitrary (not necessarily projective) compact Kähler manifolds. I will explain how the methods used allow us to view classical algebraic stability notions in a new light, and gives a framework for comparing certain stability notions to each other. Finally, I will discuss how the pluripotential approach is related to recent developments regarding Tian's properness conjecture and generalized Yau-Tian-Donaldson conjectures.
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