In this paper, we prove that the ODE system,
whose right-hand side is the Arnold-Beltrami-Childress (ABC) flow with parameters A = B = C = 1, has periodic orbits on $(2 \pi T)^3$ with rotation vectors parallel to (1, 0, 0), (0, 1, 0), and (0, 0, 1). An application of this result is that the well-known G-equation model for turbulent combustion with this ABC flow on $R^3$ has a linear (i.e., maximal possible) flame speed enhancement rate as the amplitude of the flow grows.
We show the existence of complete, asymptotically flat Cauchy initial data for the vacuum Einstein field equations, free of trapped surfaces, whose future development must admit a trapped surface. Moreover, the datum is exactly a constant time slice in Minkowski space-time inside and exactly a constant time slice in Kerr space-time outside.
The proof makes use of the full strength of Christodoulou’s work on the dynamical formation of black holes and Corvino-Schoen’s work on the construction of initial data sets.