2023/06/12 by Lebl, Jiří, Mernik, Luka
#32C07 (Primary) #32E20 32D10 14P15 (Secondary) #Complex Variables (math.CV) #FOS: Mathematics
paper · doi:10.48550/arxiv.2306.07259
We prove that a singular real-analytic Levi-flat hypersurface H in \mathbb Cn being Segre-degenerate at a point p is equivalent to the existence of a so-called support curve, that is, a holomorphic curve that intersects H at exactly one point, which in turn is equivalent to the existence of support curves on at least two sides of H at p. The existence of such two-sided support provides families of analytic discs attached to H that covers a neighborhood of p. The existence of such discs has two corollaries. First, any function holomorphic on a neighborhood of a Segre-degenerate H extends to a fixed neighborhood of p. Second, the rational hull of H is a neighborhood of p, and thus no Levi-flat Segre-degenerate hypersurface in \mathbb Cn can be rationally convex.