On Locally Conformally Flat Gradient Steady Ricci Solitons

In this paper, we prove that a complete noncompact non-flat conformally flat gradient steady Ricci soliton is, up to scaling, the Bryant soliton. 1. The result A complete Riemannian metric gij on a smooth manifold M n is called a gradient steady Ricci soliton if there exists a smooth function F on M such that the Ricci tensor Rij of the metric gij is given by the Hessian of F : Rij = ∇i∇jF. (1.1) The function F is called a potential function of the gradient steady soliton. Gradient steady solitons play an important role in Hamilton’s Ricci flow as they correspond to translating solutions, and often arise as Type II singularity models. Thus one is interested to classify them and understand their geometry. It turns out that compact gradient steady solitons must be Ricci flat (cf. Hamilton [11], Ivey [13]). In dimension n = 2, Hamilton [10] discovered the first example of a complete noncompact gradient steady soliton on R, called the cigar soliton, where the metric is given by ds = dx + dy 1 + x2 + y2 , with potential function F = log(1 + x + y). The cigar soliton has positive curvature and is asymptotic to a cylinder of finite circumference at infinity. Furthermore, Hamilton [10] showed that the only complete steady soliton on a two-dimensional manifold with bounded (scalar) curvature R which assumes its maximum at an origin is, up to scaling, the cigar soliton. For n ≥ 3, Bryant [2] proved that there exists, up to scaling, a unique complete rotationally symmetric gradient Ricci soliton on R, see, e.g., Chow et al. [6] for details. The Bryant soliton has positive sectional curvature, linear curvature decay and volume growth on the order of r. We remark that the first author [3] constructed complete gradient Kähler-Ricci soliton on C, for m ≥ 2, with positive sectional curvature which is invariant under U(m) symmetry. A well-known conjecture is that, in dimension n = 3, the Bryant soliton is the only complete noncompact (κ-noncollapsed) gradient steady soliton with positive sectional curvature. For n ≥ 4, it is also natural to ask if the Bryant soliton is the only complete noncompact, positively curved, locally conformally flat gradient The first author was partially supported by NSF; the second author was supported by a Dean’s Fellowship of the School of Arts and Sciences at Lehigh University. 1Perelman ([14], 11.9) claimed the result but didn’t give any detail, nor sketch, of a proof. 1 2 HUAI-DONG CAO AND QIANG CHEN steady soliton. In this paper, we classify n-dimensional (n ≥ 3) complete noncompact locally conformally flat gradient steady solitons, and give an affirmative answer to the latter question. Our main result is: Theorem 1.1. Let (M, gij, F ) (n ≥ 3) be a complete noncompact locally conformally flat gradient steady Ricci soliton. Then, (M, gij , F ) is either flat or isometric to the Bryant soliton. Our proof of Theorem 1.1, showing that (M, gij , F ) must be rotationally symmetric, was motivated in part by the works of physicists Israel [12] and Robinson [15] concerning the uniqueness of the Schwarzschild black hole among all static, asymptotically flat vacuum space-times. Amazingly, in their setting, the Einstein field equations take the form