球对称Jang方程的爆破解
On blow-up solutions of the Jang equation in spherical symmetry
摘要：
我们证明了Jang方程爆破解的一些相关结果。第一个：已经表明，给定最外边缘外捕获表面（MOTS）λ，存在在λ处爆炸的Jang方程的解。在这里，我们表明，此外，大类球对称的初始数据有解决方案的张方程的爆破在非最外层MOTS，即MOTS严格位于其他MOTS内，甚至严格的外部捕获表面内。然而，与最外层的MOTS，我们表明，没有\emph{always}存在爆破解决方案的内部MOTS，即使在球对称。其次，R.舍恩的证明我们包括在这里，他说，在时间对称的情况下，任何MOTS对应于张方程的爆破解决方案必须是外面积最小化，即不能包含在严格较小面积的表面。声明是假的，没有时间对称的假设，但是，我们构建了一个明确的球对称数据集提供了一个反例的一般情况下。
英文摘要：
We prove some related results concerning blow-up solutions for the Jang equation. First: it has been shown that, given an outermost marginally outer trapped surface (MOTS) Σ, there exists a solution to Jang's equation which blows up at Σ. Here we show that in addition, large classes of spherically symmetric initial data have solutions to the Jang equation which blow up at non-outermost MOTSs, i.e. MOTSs which lie strictly inside of other MOTS, and even inside of strictly outer trapped surfaces. Unlike for outermost MOTSs, however, we show that there do not \emph{always} exist blow-up solutions for inner MOTSs, even in spherical symmetry. Secondly, an unpublished result of R. Schoen, whose proof we include here, says that in the time-symmetric case, any MOTS corresponding to a blow-up solution for Jang's equation must be outer-area-minimizing, i.e. cannot be contained in a surface of strictly smaller area. The statement is false without the assumption of time-symmetry, however; we construct an explicit spherically symmetric data set providing a counterexample for the general case.
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