A Proof of Weak Cosmic Censorship Conjecture for the Spherically Symmetric Einstein-Maxwell-Charged Scalar Field System

Under spherical symmetry, we show that the weak cosmic censorship holds for the gravitational collapse of the Einstein-Maxwell-charged scalar field system. Namely, for this system, with generic initial data, the formed spacetime singularities are concealed inside black-hole regions. This generalizes Christodoulou's celebrated results to the charged case. Due to the presence of charge $Q$ and the complexification of the scalar field $\phi$, multiple delicate features and miraculous monotonic properties of the Einstein-(real) scalar field system are not present. We develop a systematical approach to incorporate $Q$ and the complex-valued $\phi$ into the integrated arguments. For instance, we discover a new path, employing the reduced mass ratio, to establish the sharp trapped surface formation criterion for the charged case. Due to the complex structure and the absence of translational symmetry of $\phi$, we also carry out detailed modified scale-critical BV area estimates with renormalized quantities to deal with $Q$ and $\phi$. We present a new $C^1$ extension criterion by utilizing the Doppler exponent to elucidate the blueshift effect, analogous to the role of integrating vorticity in the Beale-Kato-Madja breakdown criterion for incompressible fluids. Furthermore, by utilizing only double-null foliations, we establish the desired first and second instability theorems for the charged scenarios and identify generic initial conditions for the non-appearance of naked singularities. Our instability argument requires intricate generalizations of the treatment for the uncharged case via analyzing the precise contribution of the charged terms and its connection to the reduced mass ratio.