| 1. |
- Hultgren, Jakob, 1986, et al.
(author)
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An Optimal Transport Approach to Monge-Ampère Equations on Compact Hessian Manifolds
- 2019
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In: Journal of Geometric Analysis. - : Springer Science and Business Media LLC. - 1050-6926 .- 1559-002X. ; 29:3, s. 1953-1990
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Journal article (peer-reviewed)abstract
- In this paper we consider Monge–Ampère equations on compact Hessian manifolds, or equivalently Monge–Ampère equations on certain unbounded convex domains in Euclidean space, with a periodicity constraint given by the action of an affine group. In the case where the affine group action is volume preserving, i.e., when the manifold is special, the solvability of the corresponding Monge–Ampère equation was first established by Cheng and Yau using the continuity method. In the general case we set up a variational framework involving certain dual manifolds and a generalization of the classical Legendre transform. We give existence and uniqueness results and elaborate on connections to optimal transport and quasi-periodic tilings of convex domains.
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| 2. |
- Hultgren, Jakob, 1986, et al.
(author)
-
Coupled Kähler-Einstein Metrics
- 2019
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In: International Mathematics Research Notices. - : Oxford University Press (OUP). - 1073-7928 .- 1687-0247. ; 2019:21, s. 6765-6796
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Journal article (peer-reviewed)abstract
- We propose new types of canonical metrics on Kähler manifolds, called coupled Kähler–Einstein metrics, generalizing Kähler–Einstein metrics. We prove existence and uniqueness results in the cases when the canonical bundle is ample and when the manifold is Kähler–Einstein Fano. In the Fano case, we also prove that existence of coupled Kähler–Einstein metrics imply a certain algebraic stability condition, generalizing K-polystability.
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