Infinite dimensional geometries on the space of Kahler metrics and applications to canonical metrics
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
Finsler spaces
potential theory
gradient flows
51D25 - Lattices of subspaces and geometric closure systems [See also 05B35]
51E14 - Finite partial geometries (general), nets, partial spreads
14464
we introduce the L^p Mabuchi-Finsler structures on the space of Kahler metrics and present the resulting path length geometries. All the metric completions are geodesic metric spaces that can be characterized using elements of finite energy pluripotential theory and we focus on three particular cases of interest. Using convexity in the L^\infty geometry, one may be able to obtain C^0 estimates for variational PDE problems arising in Kahler geometry. The L^2 geometry is non-positively curved and seems to be the ideal context to study metric properties of gradient flows of various convex functionals, including the Calabi flow and the J-flow. The L^1 geometry enjoys a compactness property and has metric growth prescribed by the J functional, making it a useful tool in the study of energy properness questions, leading to the recent resolution of Tian's related conjectures
14464
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