content_image/344232.jpg content_image/344233.jpg content_image/344231.jpg content_image/344230.jpg content_image/344229.jpg content_image/344228.jpg
The r$_{σ}$ part can be absorbed into a redefinition of the σ coordinate and we essentially have r = r ( λ ). These simplify the equations to give (up to 2 nd order), 

Since the equations are now in a variable separable form, for consistency at the leading order we must have f ′′ $_{σ}$+ f ′ 2 σ = const$_{1}$ and ( f ′′ σ + f ′ 2 σ )+3 f ′ 2 σ = const$_{2}$ ; i.e. f ′ σ = k = const. and f$_{σ}$ = kσ . Note that this condition comes only from the leading order terms. The equations in λ become r 2 e 2 f λ ˙ ( e 2 $^{f$_{λ}$}$) = ˙ ( r $^{2}$), which gives e 2 f$_{λ}$ = r 2 = Ω( λ ) ≥ 0. 

<img src='content_image/344232.jpg'>

<img src='content_image/344233.jpg'>

Thus we have a solution with the metric Eq.(26) which is conformal to the Anti-de Sitter metric, as stated in Prop.IV.1. The Ricci and Kretschmann scalars, R = g $^{ij}$R$_{ij}$ and K = R$_{ijkl}$R ijkl respectively, for such a metric are given by – 

Thus the curvature diverges and the manifolds become singular at Ω = 0. 

<img src='content_image/344231.jpg'>

<img src='content_image/344230.jpg'>

Using the separable form for the functions in Eq.(5), we get (at third order), 

<img src='content_image/344229.jpg'>

where F ( n ) = d $^{n}$F dσ $_{n}$. But, from the separability at leading order, we already have f ′ = k and thus both the terms reduce to e − 2 $^{f}$r $^{6}$β (3) µµ = r $^{4}$β (3) σσ = − 8 k $^{6}$. Similarly all higher derivative terms vanish at order 4 leaving e − 2 $^{f}$r $^{8}$β (4) µµ = r $^{6}$β (4) σσ = 2 (3 + 5 ζ (3)) k $^{8}$. 

This leads to the following ODE for Ω – 

<img src='content_image/344228.jpg'>

We can readily see that k = 0 is a fixed point of the flow. This corresponds to a 5 

