The Existence of a Black Hole Due to Condensation of Matter
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For an asymptotically flat initial data set (three dimensional), with the mass density large on a large region, Schoen and Yau showed that there is an apparent horizon in the initial data. (SY83’) Their proof is based on a contradiction argument where assuming no apparent horizon in the region will give a global solution to Jang’s equation over the region (SY81’). Furthermore, positivity of mass density and this global solvability of Jang’s equation will give rise to positivity of a certain operator on this region. Finally, the argument is completed by showing such positivity of certain operator on the region will assert that the region can not be too large in a certain sense. Aaron Chow and the speaker used a slicing technique that was introduced in a recent paper of S. Brendle, S. Hirsch, and F. Johne (BHJ22’) to show that in a $n+1$-dimensional torical band $T^n \times [0,1]$ where $n+1\leq 7$, positivity of a similar operator will assert that the band cannot be too long. If time permits, we will also discuss the possibility of generalising Schoen-Yau existence result of Black hole to higher dimensions, with such torical band width estimate.
