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- Berman, Robert, 1976, et al.
(författare)
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Convexity of the K-energy on the space of Kähler metrics and uniqueness of extremal metrics
- 2017
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Ingår i: Journal of the American Mathematical Society. - : American Mathematical Society (AMS). - 0894-0347 .- 1088-6834. ; 30:4, s. 1165-1196
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Tidskriftsartikel (refereegranskat)abstract
- We establish the convexity of Mabuchi's K-energy functional along weak geodesics in the space of Kähler potentials on a compact Kähler manifold, thus confirming a conjecture of Chen, and give some applications in Kähler geometry, including a proof of the uniqueness of constant scalar curvature metrics (or more generally extremal metrics) modulo automorphisms. The key ingredient is a new local positivity property of weak solutions to the homogeneous Monge-Ampère equation on a product domain, whose proof uses plurisubharmonic variation of Bergman kernels.
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- Berman, Robert, 1976, et al.
(författare)
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Real Monge-Ampere equations and Kahler-Ricci solitons on toric log Fano varieties
- 2013
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Ingår i: Annales de la faculté des sciences de Toulouse. - : Cellule MathDoc/CEDRAM. - 0240-2963 .- 2258-7519. ; 22:4, s. 649-711
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Tidskriftsartikel (refereegranskat)abstract
- We show, using a direct variational approach, that the second boundary value problem for the Monge-Ampère equation in $\mathbb{R}^{n}$ with exponential non-linearity and target a convex body $P$ is solvable iff $0$ is the barycenter of $P.$ Combined with some toric geometry this confirms, in particular, the (generalized) Yau-Tian-Donaldson conjecture for toric log Fano varieties $(X,\Delta )$ saying that $(X,\Delta )$ admits a (singular) Kähler-Einstein metric iff it is K-stable in the algebro-geometric sense. We thus obtain a new proof and extend to the log Fano setting the seminal result of Wang-Zhou concerning the case when $X$ is smooth and $\Delta $ is trivial. Li’s toric formula for the greatest lower bound on the Ricci curvature is also generalized. More generally, we obtain Kähler-Ricci solitons on any log Fano variety and show that they appear as the large time limit of the Kähler-Ricci flow. Furthermore, using duality, we also confirm a conjecture of Donaldson concerning solutions to Abreu’s boundary value problem on the convex body $P$ in the case of a given canonical measure on the boundary of $P.$
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- Berman, Robert, 1976, et al.
(författare)
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Symmetrization of Plurisubharmonic and Convex Functions
- 2014
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Ingår i: Indiana University Mathematics Journal. - : Indiana University Mathematics Journal. - 0022-2518. ; 63:2, s. 345-365
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Tidskriftsartikel (refereegranskat)abstract
- We show that Schwarz symmetrization does not increase the Monge-Ampere energy for S-1-invariant plurisubharmonic functions in the ball. As a result, we derive a sharp Moser-Trudinger inequality for such functions. We also show that similar results do not hold for other balanced domains except for complex ellipsoids, and discuss related questions for convex functions.
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- Berndtsson, Bo, 1950
(författare)
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A Brunn–Minkowski type inequality for Fano manifolds and some uniqueness theorems in Kähler geometry
- 2015
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Ingår i: Inventiones Mathematicae. - : Springer Science and Business Media LLC. - 0020-9910 .- 1432-1297. ; 200:1, s. 149-200
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Tidskriftsartikel (refereegranskat)abstract
- For ϕ a metric on the anticanonical bundle, −KX , of a Fano manifold X we consider the volume of X ∫Xe−ϕ. In earlier papers we have proved that the logarithm of the volume is concave along geodesics in the space of positively curved metrics on −KX . Our main result here is that the concavity is strict unless the geodesic comes from the flow of a holomorphic vector field on X , even with very low regularity assumptions on the geodesic. As a consequence we get a simplified proof of the Bando–Mabuchi uniqueness theorem for Kähler–Einstein metrics. A generalization of this theorem to ‘twisted’ Kähler–Einstein metrics and some classes of manifolds that satisfy weaker hypotheses than being Fano is also given. We moreover discuss a generalization of the main result to other bundles than −KX , and finally use the same method to give a new proof of the theorem of Tian and Zhu on uniqueness of Kähler–Ricci solitons.
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