On Uniqueness of Tangent Cones for Einstein Manifolds

We show that for any Ricci-flat manifold with Euclidean volume growth the tangent cone at infinity is unique if one tangent cone has a smooth cross-section. Similarly, for any noncollapsing limit of Einstein manifolds with uniformly bounded Einstein constants, we show that local tangent cones are unique if one tangent cone has a smooth cross-section. 0. Introduction By Gromov’s compactness theorem, [GLP], [G], if M is an n-dimensional manifold with nonnegative Ricci curvature, then any sequence of rescalings (M, r i g), where ri → ∞, has a subsequence that converges in the Gromov-Hausdorff topology to a length space. Any such limit is said to be a tangent cone at infinity of M . Compactness follows from that (0.1) r Vol(Br(x)) is monotone nonincreasing in the radius r of the ball Br(x) for any fixed x ∈ M by the Bishop-Gromov volume comparison. As r tends to 0, this quantity on a smooth manifold converges to the volume of the unit ball in R and, as r tends to infinity, it converges to a nonnegative number VM . If VM > 0, then M is said to have Euclidean volume growth and, by [ChC1], any tangent cone at infinity is a metric cone. An important well-known question is whether the cross-section of the tangent cone at infinity of a Ricci-flat manifold with VM > 0 depends on the convergent sequence of blowdowns or is unique and independent of the sequence. Our main theorem is the following: Theorem 0.2 (Uniqueness at ∞). Let M be a Ricci-flat manifold with Euclidean volume growth. If one tangent cone at infinity has a smooth cross-section, then the tangent cone at infinity is unique. In fact, we prove an effective version of uniqueness that is considerably stronger. Theorem 0.2 settles in the affirmative a very strong form of conjecture 1.12 in [CN1]. The results of this paper were announced in [C2] and again in [CM3]. Theorem 0.2 describes the asymptotic structure of Einstein manifolds with Euclidean volume growth and vanishing Ricci curvature. These arise in a number of different fields, including string theory, general relativity, and complex and algebraic geometry, amongst The authors were partially supported by NSF Grants DMS 11040934, DMS 0906233, and NSF FRG grants DMS 0854774 and DMS 0853501. 1A metric cone C(X) with cross-sectionX is a warped product metric dr+r d2X on the space (0,∞)×X . For tangent cones at infinity of manifolds with Ric ≥ 0 and VM > 0, by [ChC1] any cross-secton is a length space with diameter ≤ π. 2In fact, we prove that the scale invariant distance to the tangent cone converges to zero like (log r) for some β > 0, where r is the distance to a fixed point. 1 2 TOBIAS HOLCK COLDING AND WILLIAM P. MINICOZZI II others, and there is a extensive literature of examples; see, e.g., [BGS], [DS], [K1], [K2], [MS1], [MS2], [MSY1], [MSY2], [TY1] and [TY2]. Most examples fall into several different classes, including ALE spaces (like the Eguchi-Hanson metric and, more generally, noncollapsing gravitational instantons, etc.), Kähler-Einstein metrics constructed by blowing up divisors, or cones over Sasaki-Einstein manifolds. Our arguments will also show that local tangent cones of limits of noncollapsing Einstein metrics are unique: Theorem 0.3 (Local uniqueness). Let (Mi, xi) be a sequence of pointed n-dimensional Einstein metrics with uniformly bounded Einstein constants and Vol(B1(xi)) ≥ v > 0. If (M∞, x∞) is a Gromov-Hausdorff limit of (Mi, xi) and one tangent cone at y ∈ M∞ has a smooth cross-section, then the tangent cone at y is unique. Similar to the case of tangent cones at infinity, the above statement follows from a stronger effective version of uniqueness of local tangent cones. It is well-known that uniqueness may fail without the two-sided bound on the Ricci curvature. Namely, there exist a large number of examples of manifolds with nonnegative Ricci curvature and Euclidean volume growth and nonunique tangent cones at infinity; see [P2], [ChC1], [CN2]. In fact, by [CN2], it is known that any smooth family of metrics on a fixed closed manifold can occur as cross-sections of tangent cones at infinity of a single manifold with nonnegative Ricci curvature and Euclidean volume growth provided the following two necessary assumptions are satisfied for any element in the family: (1) The Ricci curvature is ≥ than that of the round unit (n− 1)-dimensional sphere. (2) The volume is equal to a fixed constant. Since the space of cross-sections of tangent cones at infinity of a given manifold with nonnegative Ricci curvature and Euclidean volume growth is connected and closed under the Gromov-Hausdorff topology, it follows that if a smooth family of closed manifolds occurs as cross-sections, then so does any metric space in the closure. There is a rich history of uniqueness results for geometric problems and equations. In perhaps its simplest form, the issue of uniqueness or not comes up already in a 1904 paper entitled “On a continuous curve without tangents constructible from elementary geometry” by the Swedish mathematician Helge von Koch. In that paper, Koch described what is now known as the Koch curve or Koch snowflake. It is one of the earliest fractal curves to be described and, as suggested by the title, shows that there are continuous curves that do not have a tangent in any point. On the other hand, when a set or a curve has a well-defined tangent or well-defined blow-up at every point, then much regularity is known to follow. Tangents at every point, or uniqueness of blow-ups, is a ‘hard’ analytical fact that most often is connected with a PDE, as opposed to say Rademacher’s theorem, where tangents are shown to exist almost everywhere for any Lipschitz functions. Uniqueness is a key question for the regularity of Geometric PDE’s; for instance, as explained in [W]: “Whether nonuniqueness of tangent cones ever happens remains perhaps the most fundamental open question about singularities of minimal varieties”. Two of the most prominent early works on uniqueness of tangent cones are Leon Simon’s hugely influential 3Strictly speaking, for the construction in [CN2], one must assume strict inequality for the Ricci curvature. UNIQUENESS OF TANGENT CONES 3 paper [S1] from 1983, where he proves uniqueness for tangent cones of minimal surfaces with smooth cross-section. The other is Allard-Almgren’s 1981 [AA] paper where uniqueness of tangent cones with smooth cross-section is proven under an additional integrability assumption on the cross-section; see also [S2] and [H] for more references about uniqueness. Earlier work on uniqueness for Ricci-flat metrics includes Cheeger-Tian’s 1994 paper [ChT], where uniqueness is shown if all tangent cones have smooth cross-sections and all are integrable. In each of these geometric problems, existence of tangent cones comes from monotonicity, while the approaches to uniqueness rely on showing that the monotone quantity approaches its limit at a definite rate. However, estimating the rate of convergence seems to require either integrability and/or a great deal of regularity (such as analyticity). For instance, for minimal surfaces or harmonic maps, the classical monotone quantities are highly regular and are well-suited to this type of argument. This is not at all the case in the current setting where the Bishop-Gromov is of very low regularity and ill suited: the distance function is Lipschitz, but is not even C, let alone analytic. This is a major point (cf. page 496 of [ChT]). In contrast, the functional A (that we describe below) is defined on the level sets of an analytic function (the Green’s function) and does depend analytically and, furthermore, its derivative has the right properties. In a sense, the scale invariant volume is already a regularization of the quantity that, if one could, one would most of all like to work with. Namely, one would like to work directly with the scale invariant Gromov-Hausdorff distance between the manifold and the cone that best approximates it on the given scale and try to prove directly some kind of decay (in the scale) for this quantity. However, not only is it not clear that it is monotone, but as a purely metric quantity it is even less regular than the scale invariant volume. Throughout, C will denote a constant which will be allowed to change from line to line. When the dependence is important, we will be more explicit. M will always be an open n-dimensional Ricci-flat manifold with Euclidean volume growth where n ≥ 3. Moreover, dGH(X, Y ) will denote the Gromov-Hausdorff distance between metric spaces X and Y . 0.1. Proving uniqueness. Next we will try to explain the key points in the proof of uniqueness; a much more detailed discussion can be found in Section 1. Let p ∈ M be a fixed point in a Ricci flat manifold with Euclidean volume growth. We would like to show that the tangent cone at infinity is unique; that is, does not depend on the sequence of blow-downs. To show this, let Θr be the scale invariant Gromov-Hausdorff distance between the annulus B4r(p) \Br(p) and the corresponding annulus centered at the vertex of the cone that best approximates the annulus. (By scale invariant distance, we mean the distance between the annuli after the metrics are rescaled so that the annuli have unit size; see (1.51).) The first key point is to find a positive quantity A = A(r) that is a function of the distance to p, is monotone A ↓ and so for some positive constant C −A(r) ≥ C Θr r . (0.4) 4In addition to integrability of all cross-sections, [ChT] assumed that the sectional curvatures decay at least quadratically at infinity. This can be seen (by [C1]) to be equivalent to that all tangent cones at infinity have smooth cross-sections. 4 TOBIAS HOLCK COLDING AND WILLIAM P. MINICOZZI II (The quantity A with this property was found in [C2]. Perelman’s monotone W functional is also potentially a candidate, but it comes from integrating over the entire space which introduces so many other serious difficulties that it cannot be used.) In fact, we shall use that for Q roughly equal to −r A(r), Q is monotone nonincreasing and [Q(r/2)−Q(8r)] ≥ C Θr . (0.5) We claim that uniqueness of tangent cones is implied by showing that A converges to its limit at infinity at a sufficiently fast rate or, equivalently, that Q decays sufficiently fast to zero. Namely, by the triangle inequality, uniqueness is implied by proving that