Searching for $J$-holomorphic curves via machine: first steps

Abstract

We assemble numerical algorithms to search for $J$-holomorphic curves in symplectic manifolds. Each algorithm employs several different numerical techniques, each technique addressing a different aspect of the geometric problem. We separately consider both classical Fourier expansion and deep neural networks in our algorithms and compare their performance. Our algorithms take as input a smooth curve in a given homology class and search for a $J$-holomorphic curve in the same homology class. We first verify we can produce explicitly known holomorphic curves in complex manifolds, for example the Weierstrass $\wp$ function on the torus and curves in $S^2\times S^2$ with the standard complex structure. Then we search for $J$-holomorphic curves in $S^2\times S^2$ with non-integrable almost complex structures: essentially we start with a known holomorphic curve in an integrable almost complex structure $J_0$, deform $J_0$ to a nearby nonintegrable almost complex structure $J_ε$, and use our methods to find the nearby $J_ε$-holomorphic curve.

Disclosure

“ectic geometry, in this case J-holomorphic curves. It would be interesting to revisit on the pure math side the usage of variational techniques to find J-holomorphic curves. Acknowledgments The code was written with the help of Claude (primarily the Opus 4.8 and Opus 5 models). We read the code line-by-line and are responsible for its correctness. The observation that ℘(1) and ℘(2) are the unique two degree 2 holomorphic maps T 2 → S 2 satisfying the marked point constr”

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Classification

Code generation, completion, or debugging

Multiplier

2

Verified

2026-08-23T05:01:48.669710Z

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