Title: On Kerr Black Hole Formation with Complete Apparent Horizon and a New Approach Toward Penrose Inequality: Part I

Speaker: Xinliang An (National University of Singapore)

Abstract: Black hole formation is a central question in mathematical general relativity, involving nonlinear wave equations, geometric analysis, and mathematical physics. In this talk I will present a recent joint work with Taoran He. For the 3+1 dimensional Einstein vacuum equations, we extend Christodoulou’s celebrated trapped surface formation theorem to a black hole formation result. Without time-symmetric assumption, we further introduce a new approach and prove the spacetime Penrose inequality in perturbative regimes of sub-extremal Kerr black holes.

Title: On Kerr Black Hole Formation with Complete Apparent Horizon and a New Approach Toward Penrose Inequality: Part II

Speaker: Taoran He (National University of Singapore)

Abstract: This talk is a continuation of Xinliang An's talk on our recent work concerning the formation of Kerr black holes and a new method for proving the Penrose inequality. I will provide further details on both the hyperbolic and elliptic components of our proof. On the hyperbolic side, our arguments combine the scale-critical gravitational collapse result of An--Luk with the recent breakthrough of Klainerman--Szeftel on the nonlinear stability of slowly rotating Kerr black holes. This requires performing nine specific coordinate and frame transformations to adapt the stability framework to the black hole formation setting. To study the marginally outer trapped surfaces (MOTSs) along incoming null cones, we use precise Pretorius--Israel-type coordinates and construct associated incoming null hypersurfaces. We also extract detailed leading-order geometric information for their null expansions and develop new tools for solving the MOTS equation along incoming null cones. These together enable a rigorous demonstration of the global dynamics of the apparent horizon in our constructed black hole formation spacetimes, which ultimately leads to a new mathematical framework for proving the Penrose inequality in perturbative Kerr regimes. Notably, our framework allows for large angular momentum and adapts to the perturbative regimes of all subextremal  Kerr black holes.