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Title:
Energy properness and geometry of the space of Kahler metrics
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Abstract:
In the 90's, Tian introduced a notion of properness in the space of K\"ahler metrics in terms of Aubin's J-energy for Mabuchi's K-energy and formulated several conjectures on the relation between properness and K\"ahler-Einstein metrics. In joint work with Y. Rubinstein we disproved one of these conjectures, and proved the remaining ones. Our results extend to a variety of canonical metrics, in particular K\"ahler-Einstein edge metrics and K\"ahler-Ricci solitons. Lastly, we formulated a corresponding conjecture for constant scalar curvature metrics and reduce it to a regularity question on minimizers of the K-energy
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