On the existence of stationary Ricci solitons

TL;DR

This paper proves that under certain asymptotic conditions, stationary Ricci solitons cannot exist in spacetimes with a specific symmetry, extending previous static results to stationary cases.

Contribution

It introduces a new argument demonstrating the non-existence of stationary Ricci solitons with a $t$-$$ reflection symmetry when the soliton charge vanishes.

Findings

The soliton charge is always non-positive.

Vanishing soliton charge implies no stationary Ricci solitons exist.

Extension of static results to stationary spacetimes with symmetry.

Abstract

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 "$t$-$\phi$" 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.