Anisotropic Dynamical Horizons Arising in Gravitational Collapse

For the study of $3+1$ dimensional Einstein vacuum equations (EVEs), substantial progress has been made recently on the problem of trapped surface formation. However, very limited knowledge of existence and associated properties is acquired on the boundary of the emerged trapped region, i.e., the apparent horizon, which is composed of marginally outer trapped surfaces (MOTSs) and is of great physical importance. In this paper, concerning this emerged apparent horizon we prove a folklore conjecture relating to both cosmic censorship and black hole thermodynamics. In a framework set up by Christodoulou and under a general anisotropic condition introduced by Klainerman, Luk and Rodnianski, for $3+1$ EVEs we prove that in the process of gravitational collapse, a smooth and spacelike apparent horizon (dynamical horizon) emerges from general (both isotropic and anisotropic) initial data. This dynamical horizon censors singularities formed in gravitational collapse from non-trapped local observers near the center, and it also enables the extension of black hole thermodynamical theory along the apparent horizon to anisotropic scenarios. Our analysis builds on scale-critical hyperbolic method and non-perturbative elliptic techniques. New observations and equation structures are exploited. Geometrically, we furthermore construct explicit finger-type single and multi-valley anisotropic apparent horizons. They are the first mathematical examples of the anisotropic MOTS and the anisotropic apparent horizon formed in dynamics, which have potential applications in geometric analysis, black hole mechanics, numerical relativity and gravitational wave phenomenology.