On Complete Gradient Shrinking Ricci Solitons

In this paper we derive a precise estimate on the growth of potential functions of complete noncompact shrinking solitons. Based on this, we prove that a complete noncompact gradient shrinking Ricci soliton has at most Euclidean volume growth. The latter result can be viewed as an analog of the well-known theorem of Bishop that a complete noncompact Riemannian manifold with nonnegative Ricci curvature has at most Euclidean volume growth. 1. The results A complete Riemannian metric gij on a smooth manifold M n is called a gradient shrinking Ricci soliton if there exists a smooth function f on M such that the Ricci tensor Rij of the metric gij is given by Rij +∇i∇jf = ρgij for some positive constant ρ > 0. The function f is called a potential function of the shrinking soliton. Note that by scaling gij one can normalize ρ = 1 2 so that Rij +∇i∇jf = 1 2 gij . (1.1) Gradient shrinking Ricci solitons play an important role in Hamilton’s Ricci flow as they correspond to self-similar solutions, and often arise as Type I singularity models. In this paper, we investigate the asymptotic behavior of potential functions and volume growth rates of complete noncompact gradient shrinking solitons. Our main results are: Theorem 1.1. Let (M, gij , f) be a complete noncompact gradient shrinking Ricci soliton satisfying (1.1). Then, the potential function f satisfies the estimates 1 4 (r(x) − c1) ≤ f(x) ≤ 1 4 (r(x) + c2) . Here r(x) = d(x0, x) is the distance function from some fixed point x0 ∈ M , c1 and c2 are positive constants depending only on n and the geometry of gij on the unit ball Bx0(1). Remark 1.1. In view of the Gaussian shrinker, namely the flat Euclidean space (R, g0) with the potential function |x|2/4, the leading term 1 4r2(x) for the lower and upper bounds on f in Theorem 1.1 is optimal. We also point out that it has been known, by the work of Ni-Wallach [12] and Cao-Chen-Zhu [3], that any 3dimensional complete noncompact non-flat shrinking gradient soliton is necessarily the round cylinder S × R or one of its Z2 quotients. The first author was partially supported by NSF grants DMS-0354621 and DMS-0506084; the second author was partially supported by CNPq and FAPERJ, Brazil. 1 2 HUAI-DONG CAO AND DETANG ZHOU Remark 1.2. When the Ricci curvature of (M, gij , f) is assumed to be bounded, Theorem 1.1 was shown by Perelman [13]. Also, under the assumption of Rc ≥ 0, a lower estimate of the form f(x) ≥ 1 8 r(x) − c′1 was shown by Ni [11]. Moreover, the upper bound in Theorem 1.1 was essentially observed in [3], while a rough quadratic lower bound, as pointed out by Carrillo-Ni [5], could follow from the argument of Fang-Man-Zhang in [7]. Theorem 1.2. Let (M, gij , f) be a complete noncompact gradient shrinking Ricci soliton. Then, there exists some positive constant C1 > 0 such that Vol(Bx0(r)) ≤ C1r for r > 0 sufficiently large. Remark 1.3. In an earlier version of the paper, we had an extra assumption R(x) ≤ αr(x) +A(r(x) + 1), (1.2) with 0 ≤ α < 1 4 and A > 0, on the scalar curvature R. However, as observed by Ovidiu Munteanu, assumption (1.2) actually is not needed in our proof because there holds estimate (3.4) on the average of the scalar curvature in general. Note that, as stated in Lemma 2.3, R(x) ≤ 14 (r(x) + c) holds for any complete noncompact gradient shrinking soliton. It remains interesting to find out whether R is bounded from above by a constant. Remark 1.4. Feldman-Ilmanen-Knopf [8] constructed a complete noncompact gradient Kähler shrinker on the tautological line bundle O(−1) of the complex projective space CP (n ≥ 2) which has Euclidean volume growth, quadratic curvature decay, and with Ricci curvature changing signs. This example shows that the volume growth rate in Theorem 1.2 is optimal. Note that Carrillo-Ni [5] showed that any non-flat gradient shrinking soliton with nonnegative Ricci curvature Rc ≥ 0 must have zero asymptotic volume ratio, i.e., limr→∞ Vol(Bx0(r))/r n = 0. Combining Theorem 1.1 and Theorem 1.2, we also have the following consequence, which was obtained previously in [10] and [15] respectively. Corollary 1.1. Let (M, gij , f) be a complete noncompact gradient shrinking Ricci soliton. Then we have