Tian's properness conjectures
Kähler Geometry, Einstein Metrics, and Generalizations March 21, 2016 - March 25, 2016
Location: SLMath: Eisenbud Auditorium
algebraic geometry and GAGA
mathematical physics
complex differential geometry
Kahler metric
mirror symmetry
Yamabe problem
functional analysis
Finsler spaces
infinite dimensional manifolds
51D25 - Lattices of subspaces and geometric closure systems [See also 05B35]
51E14 - Finite partial geometries (general), nets, partial spreads
51Gxx - Ordered geometries (ordered incidence structures, etc.)
55P92 - Relations between equivariant and nonequivariant homotopy theory in algebraic topology
14460
In the 90's, Tian conjectured an analytic characterization of Kahler-Einstein metrics. This characterization can be viewed as a direct analogue of the celebrated Yamabe problem (for constant scalar curvature metrics in a conformal class). Tian also conjectured a Kahler-Einstein analogue of the well-known Aubin-Moser-Trudinger inequality in conformal geometry.
In joint work with T. Darvas we disprove one of these conjectures, and prove the remaining ones. Our results extend to many other types of canonical metrics in Kahler geometry aside from Kahler-Einstein metrics.
Somewhat surprisingly, the proof uses in an essential way techniques of metric space geometry applied to appropriate infinite-dimensional spaces, and in particular a Finsler metric introduced earlier by Darvas.
14460
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