On the existence of stationary Ricci solitons

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-(empty set) 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.