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 $J_\epsilon$, and use our methods to find the nearby $J_\epsilon$-holomorphic curve.