关于稳态Ricci孤子的存在性
On the existence of stationary Ricci solitons
摘要：
以前的DeTurck '技巧'已被用来渲染静态爱因斯坦方程一个良好的椭圆系统，可以解决数字几何流或直接。虽然在静态的情况下，纯引力与零或负宇宙学常数有一个简单的证明，解决修改后的“谐波”爱因斯坦方程导致原来的爱因斯坦系统的解决方案-即不是里奇孤子-在静态的情况下，这个论点不再工作。在这里，我们提供了一个新的论点，扩展静态的结果的情况下，具有“$t$ - $ϕ$“的反射对称的静态时空。从解的渐近行为定义一个“孤子电荷”，我们证明了这个量总是非正的。假设渐近条件的选择使得该电荷消失，则定态孤子不存在。
英文摘要：
Previously the DeTurck 'trick' has been used to render the stationary Einstein's equation a well posed elliptic system that may be solved numerically by geometric flow or directly. Whilst in the static case for pure gravity with zero or negative cosmological constant there is a simple proof that solving the modified "harmonic" Einstein's equation leads to a solution of the original Einstein system - i.e. not a Ricci soliton - in the stationary case this argument no longer works. Here we provide a new argument that extends the static result to the case of stationary spacetimes that possess a "$t$-$ϕ$" reflection symmetry. Defining a "soliton charge" from the asymptotic behaviour of the solution, we show that this quantity is always non-positive. Provided asymptotic conditions are chosen such that this charge vanishes, then stationary solitons cannot exist.
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