content_image/369879.jpg content_image/369875.jpg content_image/369874.jpg
k = 0: 

<img src='content_image/369879.jpg'>

As we discussed in the last section, small black brane with a radius r$_{+}$ exist when α > α$_{ext}$ . Figures 7 and 8 show that these small black branes are stable provided α > α$_{crit}$ . On the other side, large black branes are stable as one may see in Figs. 9 and 10. 

k = − 1: 

Figures 7 and 8 show that small black branes exist only for medium values of α ( α$_{1ext}$ < α < α$_{2ext}$ ), but they are unstable. On the other side, large black branes are stable as one may see in Figs. 9 and 10. 

In order to confirm the stability analysis of the black branes, one can also use the gener- alized free energy [38] 

<img src='content_image/369875.jpg'>

which applies to any value of r$_{+}$ with a fixed temperature T . This off-shell free energy reduces to the on-shell free energy at T = T$_{+}$ : 

<img src='content_image/369874.jpg'>

which is the Legendre transform of M with respect to S . As an example of using the stability analysis by off-shell free energy, we plot F off versus r$_{+}$ for k = 0, α = 1 . 2 and α = 4. These are plotted in Figs. 11 and 12, which show that the black brane is unstable for α = 1 . 2, while it is stable for α = 4. These figures confirm the stability analysis of Fig. 8. One may also confirm other stability analysis given in this section. 

