Riemann-Hilbert问题、Toeplitz算子与能层表面
Riemann-Hilbert problems, Toeplitz operators and ergosurfaces
AI总结：
本文利用Wiener-Hopf分解与Toeplitz算子的联系，证明有理monodromy矩阵缺失标准分解时产生的发散曲线对应旋转黑洞的能层表面。
摘要：
Riemann-Hilbert方法结合某些被称为monodromy矩阵的矩阵函数的标准Wiener-Hopf分解，使人们能够获得某些引力理论的非线性场方程的显式解。这些解编码在矩阵$M$的元素中，该矩阵依赖于Weyl坐标$\ ho$和$v$，并由该分解确定。我们利用Wiener-Hopf分解与Toeplitz算子之间的密切联系来研究这一问题，首次处理了当标准Wiener-Hopf分解不存在时会发生什么这一基础性问题。对于有理monodromy矩阵的情形，我们证明标准Wiener-Hopf分解的不存在性决定了$(\ ho,v)$平面中的曲线，在这些曲线上，$M(\ ho,v)$的某些元素趋于无穷大，但时空度规在那里仍可能表现良好。在四维时空以及在某些坐标选择下的五维时空中的不带电旋转黑洞情形中，我们表明这些曲线对应于它们的能层表面。
英文摘要：
The Riemann-Hilbert approach, in conjunction with the canonical Wiener-Hopf factorisation of certain matrix functions called monodromy matrices, enables one to obtain explicit solutions to the non-linear field equations of some gravitational theories. These solutions are encoded in the elements of a matrix $M$ depending on the Weyl coordinates $ρ$ and $v$, determined by that factorisation. We address here, for the first time, the underlying question of what happens when a canonical Wiener-Hopf factorisation does not exist, using the close connection of Wiener-Hopf factorisation with Toeplitz operators to study this question. For the case of rational monodromy matrices, we prove that the non-existence of a canonical Wiener-Hopf factorisation determines curves in the $(ρ,v)$ plane on which some elements of $M(ρ,v)$ tend to infinity, but where the space-time metric may still be well behaved. In the case of uncharged rotating black holes in four space-time dimensions and, for certain choices of coordinates, in five space-time dimensions, we show that these curves correspond to their ergosurfaces.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
