50 1. RICCI SOLITONS9.3. Examples. The Buscher dual of a rotationally symmetric soliton
solution on a surface
I> g = dr^2 + w (r )^2 d0^2is the soliton
For example,
dr^2 + tanh^2 r d0^2 dr^2 + coth^2 r d0^2
dr^2 + tan^2 r d0^2 dr^2 + cot^2 r d0^2
dr^2 + r^2 d0^2 dr^2 + r-^2 de^2
dr^2 + d0^2 dr^2 + d0^2
Let (Nm, h) be an Einstein metric with Re= c:g, c: E R Consider the
doubly-warped product metric on ffi.^2 x Nm:
(^0) g ~ ds (^2) + F (s) (^2) d0 (^2) + G (s) (^2) h,
where s is the radial coordinate, e is the coordinate on the circle, and F
and G are positive functions. The gradient Ricci soliton equation I> Rab +
2 °\7 a^0 \7bf = 0 becomes (see equation (1) in [219]) the following system for
the triple (F, G, !):
(1.83)(1.84)(1.85)The Buscher dual metric is
We know that Ug is a steady gradient Ricci soliton if I> g is, and so we leave it
to the reader to verify that indeed the triple (-j;i, ~' f -log F) is a solution
to (1.83)-(1.85) if (F, G, f) is.
10. Summary of results and open problems on Ricci solitons
In this section we collect some known results and open problems about
the properties and classification of gradient Ricci solitons. Besides the dis-
cussion earlier in this chapter, one may consult [111] for some of the proofs.