Critical and extremal gravitational collapse of a spherical charged scalar field in 4+1 spacetime dimensions

We numerically investigate the gravitational collapse of a charged scalar field coupled to the Einstein and Maxwell equations in spherical symmetry, in 4+1 spacetime dimensions. At the threshold of collapse we find type-II critical phenomena very similar to what is seen in 3+1 dimensions, including self-similarity of the critical solution and power-law scaling of the black hole mass and charge. Well inside the collapse region, we also identify initial data that collapse to an extremal black hole. By varying all parameters of the initial data by up to $\sim0.5\%$, we give evidence that there is a small neighbourhood in the space of initial data where an exactly extremal charged black hole is formed in the evolution of a codimension-1 set of initial data. We believe this is the first such evidence for scalar field collapse from regular initial data.