Singular structures in solutions to the Monge-Ampère equation with point masses

Connor Mooney; Arghya Rakshit; Connor Mooney; Arghya Rakshit

doi:10.3934/mine.2023083

Mathematics in Engineering

2023, Volume 5, Issue 5: 1-11. doi: 10.3934/mine.2023083

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Singular structures in solutions to the Monge-Ampère equation with point masses

Connor Mooney ^,,
Arghya Rakshit

Department of Mathematics, UC Irvine, Irvine, CA 92697-3875, USA

Received: 29 April 2022 Revised: 23 October 2022 Accepted: 23 October 2022 Published: 13 April 2023

We construct new examples of Monge-Ampère metrics with polyhedral singular structures, motivated by problems related to the optimal transport of point masses and to mirror symmetry. We also analyze the stability of the singular structures under small perturbations of the data given in the problem under consideration.
- Monge-Ampère equation,
- optimal transport,
- SYZ conjecture,
- regularity and stability,
- singular solutions
Citation: Connor Mooney, Arghya Rakshit. Singular structures in solutions to the Monge-Ampère equation with point masses[J]. Mathematics in Engineering, 2023, 5(5): 1-11. doi: 10.3934/mine.2023083

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Abstract

We construct new examples of Monge-Ampère metrics with polyhedral singular structures, motivated by problems related to the optimal transport of point masses and to mirror symmetry. We also analyze the stability of the singular structures under small perturbations of the data given in the problem under consideration.

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