Existence of pseudo-holomorphic disks via non-archimedean disk potentials

Abstract

We show that if a graded monotone Lagrangian $L_0$ has a non-vanishing disk potential, then for every smooth isotopy $\{L_s\}_{s\in[0,1]}$ of Lagrangians starting from it and for every tame almost complex structure $J$, each $L_s$ bounds a $J$-holomorphic disk of Maslov index two. The main input is a non-archimedean analytic potential function, defined as an invariant up to analytic isomorphisms, generalizing the classical disk potential of a monotone Lagrangian. The techniques are inspired by recent developments in the Strominger-Yau-Zaslow mirror construction via family Floer theory and non-archimedean geometry. We also discuss applications such as recovering a simple case of Audin's conjecture.