Quantitative stratification and optimal regularity for stationary $J$-holomorphic maps

In this paper, we give an alternate proof of the $\epsilon$-regularity theorem of Wang [Comm. Cont. Math. 2003] for weakly $J$-holomorphic maps by adapting the conservation law approach of Rivière [Invent. Math. 2007]. Then we adapt the powerful quantitative stratification method of Naber–Valtorta [Ann. Math. 2017] to obtain optimal volume estimates and rectifiability for the quantitative strata of stationary $J$-holomorphic maps. In contrast with general harmonic maps, the almost complex equation gives rise to an additional symmetry improvement: Every odd-dimensional translation symmetry extends to an even-dimensional one. Consequently, consecutive even and odd strata coincide, and the $k$-th quantitative stratum satisfies a tubular-neighborhood estimate with exponent $2m-2\lfloor k\mathrm{/}2\rfloor$, where $2m$ is the real dimension of the domain. The corresponding strata are $2\lfloor k\mathrm{/}2\rfloor$-rectifiable, with almost-everywhere unique complex planes of symmetry. Finally, we prove that every stationary $J$-holomorphic map is smooth if the target { manifold} admits no non-constant $J$-holomorphic sphere, improving the corresponding theorem of Wang.