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- Berman, Robert, 1976, et al.
(author)
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Real Monge-Ampere equations and Kahler-Ricci solitons on toric log Fano varieties
- 2013
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In: Annales de la faculté des sciences de Toulouse. - : Cellule MathDoc/CEDRAM. - 0240-2963 .- 2258-7519. ; 22:4, s. 649-711
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Journal article (peer-reviewed)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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| 3. |
- Berman, Robert, 1976, et al.
(author)
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Symmetrization of Plurisubharmonic and Convex Functions
- 2014
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In: Indiana University Mathematics Journal. - : Indiana University Mathematics Journal. - 0022-2518. ; 63:2, s. 345-365
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Journal article (peer-reviewed)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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| 6. |
- Berndtsson, Bo, 1950
(author)
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Convexity on the space of Kähler metrics
- 2013
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In: Annales de la Faculté des Sciences de Toulouse. - : Cellule MathDoc/CEDRAM. - 0240-2955 .- 0240-2963 .- 2258-7519. ; 22 (6):4, s. 713-746
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Journal article (peer-reviewed)
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| 7. |
- Berndtsson, Bo, 1950
(author)
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L-2-Extension of partial derivative-closed form
- 2012
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In: Illinois Journal of Mathematics. - 0019-2082. ; 56:1, s. 21-31
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Journal article (peer-reviewed)abstract
- Generalizing and strengthening a recent result of Koziarz, we prove a version of the Ohsawa-Takegoshi-Manivel theorem for partial derivative-closed forms.
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