On the existence of stationary Ricci solitons

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 "$t$-$\varphi$" 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.

15 pages, no figures

References in corpus (5)

• GW151226: Observation of Gravitational Waves from a 22-Solar-Mass Binary Black Hole Coalescence
• A Higher Dimensional Stationary Rotating Black Hole Must be Axisymmetric
• Symmetries of Higher Dimensional Black Holes
• On the uniqueness of smooth, stationary black holes in vacuum
• AdS flowing black funnels: Stationary AdS black holes with non-Killing horizons and heat transport in the dual CFT

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