On the Donaldson-Scaduto conjecture
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866917440849444864 |
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| author | Esfahani, Saman Habibi Li, Yang |
| author_facet | Esfahani, Saman Habibi Li, Yang |
| contents | Motivated by $G_2$-manifolds with coassociative fibrations in the adiabatic limit, Donaldson and Scaduto conjectured the existence of associative submanifolds homeomorphic to a three-holed $3$-sphere with three asymptotically cylindrical ends in the $G_2$-manifold $X \times \mathbb{R}^3$, or equivalently similar special Lagrangians in the Calabi-Yau 3-fold $X \times \mathbb{C}$, where $X$ is an $A_2$-type ALE hyperkähler 4-manifold. We prove this conjecture by solving a real Monge-Ampère equation with a singular right-hand side, which produces a potentially singular special Lagrangian. Then, we prove the smoothness and asymptotic properties for the special Lagrangian using inputs from geometric measure theory. The method produces many other asymptotically cylindrical $U(1)$-invariant special Lagrangians in $X\times \mathbb{C}$, where $X$ arises from the Gibbons-Hawking construction. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_15432 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the Donaldson-Scaduto conjecture Esfahani, Saman Habibi Li, Yang Differential Geometry Analysis of PDEs Symplectic Geometry Motivated by $G_2$-manifolds with coassociative fibrations in the adiabatic limit, Donaldson and Scaduto conjectured the existence of associative submanifolds homeomorphic to a three-holed $3$-sphere with three asymptotically cylindrical ends in the $G_2$-manifold $X \times \mathbb{R}^3$, or equivalently similar special Lagrangians in the Calabi-Yau 3-fold $X \times \mathbb{C}$, where $X$ is an $A_2$-type ALE hyperkähler 4-manifold. We prove this conjecture by solving a real Monge-Ampère equation with a singular right-hand side, which produces a potentially singular special Lagrangian. Then, we prove the smoothness and asymptotic properties for the special Lagrangian using inputs from geometric measure theory. The method produces many other asymptotically cylindrical $U(1)$-invariant special Lagrangians in $X\times \mathbb{C}$, where $X$ arises from the Gibbons-Hawking construction. |
| title | On the Donaldson-Scaduto conjecture |
| topic | Differential Geometry Analysis of PDEs Symplectic Geometry |
| url | https://arxiv.org/abs/2401.15432 |