New examples of complete Ricci solitons

The Ricci soliton condition reduces to a set of ODEs when one assumes that the metric is a doubly-warped product of a ray with a sphere and an Einstein manifold. If the Einstein manifold has positive Ricci curvature, we show there is a one-parameter family of solutions which give complete noncompact Ricci solitons. INTRODUCTION A Ricci soliton is a solution to the Ricci flow ag/Ot = -2Ric(g) such that the metric changes only by diffeomorphisms as time goes on; since the diffeomorphisms of the underlying manifold are symmetries of the evolution equation, it would be more accurate to call this a similarity solution for the Ricci flow. Soliton solutions are important to the study of the Ricci flow because they represent extremal cases for the Harnack estimate [H2] and may be limiting cases for the Ricci flow near singularities (cf. [A]). A Ricci soliton is generated by an initial metric g and a vector field V such that Svg = 2Ric(g); then V generates the diffeomorphisms. A gradient soliton is one where V is the gradient of some function h with respect to g; the corresponding condition is that the Hessian V2 h coincide with the Ricci tensor. Up to now, the known examples of complete Ricci solitons were the radially symmetric 'cigar' metric on R2 [Hi], the radially symmetric soliton on R3 discovered by Bryant (which easily generalizes to Rn), and the U(n)symmetric soliton on Cn disovered by Cao [C]. Let (Mn, da2) be a compact Einstein manifold with Einstein constant e > 0 . Let dO2 denote the standard metric of constant curvature + 1 on Sk, k > 1 . On Rk+1 x M with radial coordinate t > 0 consider the doubly-warped product metric (*) ds2 = dt2 + f(t)2d02 + g(t)2da2. For the metric to be smooth near t = 0 we require that f extend smoothly to Received by the editors December 2, 1992. 1991 Mathematics Subject Classification. Primary 53C25; Secondary 34C99.