I am looking for example of steady Ricci soliton with indefinite or nonpositive Ricci curvature. 
Any help will be appreciated. 
Thanks!



Example of steady Ricci soliton whith indefinite or nonpositive Ricci curvature
I have found doing some calculation that the metric:



> 
> $$g(t)=\frac{dx^2+dy^2}{e^{-4t}-x^2-y^2}$$
> 
> 
> 


satisfies $\frac{dg(t)}{dt}=-2Ric(t)$


Where 



> 
> $$\frac{dg(t)}{dt}=\frac{4e^{-4t}(dx^2+dy^2)}{(e^{-4t}-x^2-y^2)^2}\ {\rm and}
> \ Ric(t)=\frac{-2e^{-4t}( dx^2+dy^2)}{(e^{-4t}-x^2-y^2)^2}$$
> 
> 
> 


this should be a solution to the Ricci flow on $R^2$.


Furthermore, with some other calculation, I found that this metric $g(t)$ is the only solution within the family: $g\_{\lambda}(t)=\frac{dx^2+dy^2}{e^{-4\lambda t}-x^{2 \lambda}-y^{2 \lambda}}$, in fact, the only solution to the Ricci flow is only for $\lambda=1$.


Said this, my analysis has been to see the behavior in time for $g(t)$ and $Ric(t)$, when $t$ tends to $\infty$ and $- \infty$, but I want that 
the metric $g(t)$  remain positive.


a) Then for $t \rightarrow \infty$, to ensure that $g(t)$ is positive, I fixed $x=0$ and $y=0$ and I get $g(t)$ tends to $\infty$ and $Ric(t)$ tends to $-2$


b) While for $t \rightarrow - \infty$ I get $g(t)$ tends to $0$ and $Ric(t)$ tends again to $-2$.


For $t=0$,  


I get $g\_0= \frac{dx^2+dy^2}{1-x^2-y^2}$ and with a simple calculations, I found that the 1-parameter family of diffeomorfism is $\phi\_t=(e^{2t}x, e^{2t}y)$.


Considering $r^2=(x^2+y^2)$, the Scalar curvature, for $t=0$, is
$R=\frac{-4}{1-r^2}$.


Now If I make an analysis of singularities (always for $(1-r^2)>0$), I found that if $r$ tends to $1-$, $g\_0$ tends to $\infty$ and $R$ tends $- \infty$.


This soliton should be a steady soliton, it has negative curvature and it is not bunded from below.


