Stably Embedded Minimal Hypersurfaces

We use Schoen-Simon-Yau's curvature estimates to prove that the subfocal tubular neighborhood of a non planar minimal hypersurface with bounded second fundamental form, stably embedded in R^{n+1}, n < 5, whose radius decays sufficiently slowly can not be embedded. In particular such hypersurfaces admit no embedded tubular neighborhoods of constant radius, whatever small the radius. However, assuming a further hypothesis on the embedding, we prove that such hypersurfaces admit an embedded tube whose radius decays sufficiently fast.