| 1. |
- Andreasson, Rolf, et al.
(författare)
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Solvability of Monge-Ampère equations and tropical affine structures on reflexive polytopes
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Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- Given a reflexive polytope with a height function, we prove a necessary and sufficient condition for solvability of the associated Monge-Ampère equation. When the polytope is Delzant, solvability of this equation implies the metric SYZ conjecture for the corresponding family of Calabi-Yau hypersurfaces. We show how the location of the singularities in the tropical affine structure is determined by the PDE in the spirit of a free boundary problem and give positive and negative examples, demonstrating subtle issues with both solvability and properties of the singular set. We also improve on existing results regarding the SYZ conjecture for the Fermat family by showing regularity of the limiting potential.
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| 2. |
- Hultgren, Jakob, 1986, et al.
(författare)
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An Optimal Transport Approach to Monge-Ampère Equations on Compact Hessian Manifolds
- 2019
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Ingår i: Journal of Geometric Analysis. - : Springer Science and Business Media LLC. - 1050-6926 .- 1559-002X. ; 29:3, s. 1953-1990
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Tidskriftsartikel (refereegranskat)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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| 3. |
- Hultgren, Jakob, 1986, et al.
(författare)
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Coupled Kähler-Einstein Metrics
- 2019
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Ingår i: International Mathematics Research Notices. - : Oxford University Press (OUP). - 1073-7928 .- 1687-0247. ; 2019:21, s. 6765-6796
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Tidskriftsartikel (refereegranskat)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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| 4. |
- Hultgren, Jakob, 1986-
(författare)
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Duality of Hessian manifolds and optimal transport
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Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- This is an expository paper describing how duality theory for Hessian manifolds provides a natural setting for optimal transport. We explain how this can be used to solve Monge-Ampère equations and survey recent results along these lines with applications to the SYZ-conjecture in mirror symmetry.
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| 5. |
- Hultgren, Jakob, 1986
(författare)
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Permanental Point Processes on Real Tori
- 2016
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Licentiatavhandling (övrigt vetenskapligt/konstnärligt)abstract
- The main motivation for this thesis is to study real Monge-Ampère equations. These are fully nonlinear differential equations that arise in differential geometry. They lie at the heart of optimal transport and, as such, are related to probability theory, statistics, geometrical inequalities, fluid dynamics and diffusion equations. In this thesis we set up and study a thermodynamic formalism for a certain type of Monge-Ampère equations on real tori. We define a family of permanental point processes and show that their asymptotic behavior (when the number of particles tends infinity) is governed by Monge-Ampère equations.
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| 6. |
- Hultgren, Jakob, 1986
(författare)
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Real and complex Monge-Ampère equations, statistical mechanics and canonical metrics
- 2018
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- Recent decades has seen a strong trend in complex geometry to study canonical metrics and the way they relate to geometric analysis, algebraic geometry and probability theory. This thesis consists of four papers each contributing to this field. The first paper sets up a probabilistic framework for real Monge-Ampère equations on tori. We show that solutions to a large class of real Monge-Ampère equations arise as the many particle limit of certain permanental point processes. The framework can be seen as a real, compact analog of the probabilistic framework for Kähler-Einstein metrics on Kähler manifolds. The second paper introduces a variational approach in terms of optimal transport to real Monge-Ampère equations on compact Hessian manifolds. This is applied to prove existence and uniqueness results for various types of canonical Hessian metrics. The results can, on one hand, be seen as a first step towards a probabilistic approach to canonical metrics on Hessian manifolds and, on the other hand, as a remark on the Gross-Wilson and Kontsevich-Soibelmann conjectures in Mirror symmetry. The third paper introduces a new type of canonical metrics on Kähler manifolds, called coupled Kähler-Einstein metrics, that generalises Kähler-Einstein metrics. Existence and uniqueness theorems are given as well as a proof of one direction of a generalised Yau-Tian-Donaldson conjecture, establishing a connection between this new notion of canonical metrics and stability in algebraic geometry. The fourth paper gives a necessary and sufficient condition for existence of coupled Kähler-Einstein metrics on toric manifolds in terms of a collection of associated polytopes, proving this generalised Yau-Tian-Donaldson conjecture in the toric setting.
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| 7. |
- Hultgren, Jakob, 1986-, et al.
(författare)
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Tropical and non-Archimedean Monge-Ampère equations for a class of Calabi-Yau hypersurfaces
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Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- For a class of maximally degenerate families of Calabi-Yau hypersurfaces of complex projective space, we study associated non-Archimedean and tropical Monge--Ampère equations, taking place on the associated Berkovich space, and the essential skeleton therein, respectively. For a symmetric measure on the skeleton, we prove that the tropical equation admits a unique solution, up to an additive constant. Moreover, the solution to the non-Archimedean equation can be derived from the tropical solution, and is the restriction of a continuous semipositive toric metric on projective space. Together with the work of Yang Li, this implies the weak metric SYZ conjecture on the existence of special Lagrangian fibrations in our setting.
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| 8. |
- Hultgren, Jakob, 1986-, et al.
(författare)
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Tropical and non-Archimedean Monge–Ampère equations for a class of Calabi–Yau hypersurfaces
- 2024
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Ingår i: Advances in Mathematics. - : Elsevier. - 0001-8708 .- 1090-2082. ; 439
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Tidskriftsartikel (refereegranskat)abstract
- For a large class of maximally degenerate families of Calabi–Yau hypersurfaces of complex projective space, we study non-Archimedean and tropical Monge–Ampère equations, taking place on the associated Berkovich space, and the essential skeleton therein, respectively. For a symmetric measure on the skeleton, we prove that the tropical equation admits a unique solution, up to an additive constant. Moreover, the solution to the non-Archimedean equation can be derived from the tropical solution, and is the restriction of a continuous semipositive toric metric on projective space. Together with the work of Yang Li, this implies the weak metric SYZ conjecture on the existence of special Lagrangian fibrations in our setting.
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