We discuss an elementary consequence of the works of (1) Brett Kotschwar and
Lu Wang and (2) Ovidiu Munteanu and Jiaping Wang

Chow, Bennett

Lu, Peng

English

arXiv

Numérisation de Documents Anciens Mathématiques

C. R. Acad. Sci. Paris, Ser. I 353 (2015) 1007–1009Contents lists available at ScienceDirectC. R. Acad. Sci. Paris, Ser. Iwww.sciencedirect.comDifferential geometryUniqueness of asymptotic cones of complete noncompact shrinking gradient Ricci solitons with Ricci curvature decayUnicité des cônes asymptotiques des solitons gradients de Ricci contractants complets non compacts avec courbure de Ricci décroissanteBennett Chow a, Peng Lu b,1a Department of Mathematics, University of California San Diego, La Jolla, CA 92093, United Statesb Department of Mathematics, University of Oregon, Eugene, OR 97403, United Statesa r t i c l e i n f o a b s t r a c tArticle history:Received 15 July 2015Accepted after revision 14 September 2015Available online 23 October 2015Presented by Étienne GhysWe show that any complete noncompact shrinking gradient Ricci soliton with (1) | Rc | → 0at infinity or (2) R → 0 at infinity, |Rm| bounded, and κ-noncollapsed has a unique asymptotic cone.© 2015 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.r é s u m éNous montrons que tout soliton gradient de Ricci contractant complet non compact vérifiant la propriété (1) | Rc | → 0 à l’infini ou (2) R → 0 à l’infini, avec | Rm | bornée et κ-non-effrondée, possède un cône asymptotique unique.© 2015 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.Let (Mn, ̄g, f̄ ) be a complete noncompact shrinking gradient Ricci soliton (GRS). The study of its geometry near infinity is of crucial importance. A basic question is the volume growth of ḡ . By Cao and Zhou [1], with an observation of Munteanu using Chen’s result in [5] that R ≥ 0, the volume growth is at most Euclidean. Moreover, in [1] a strong lower bound for the potential function f̄ was obtained. This may be interpreted as generally indicating rigidity for shrinking GRS. Further evidence reflecting the aforementioned rigidity are the classification and nonexistence results starting with Hamilton [8,9]and Perelman [18] and then Cao, Chen and Zhu [2], Ni and Wallach [16], Naber [15], and Petersen and Wylie [19]. The most general and recent result in this direction is by Munteanu and Wang [14]. Regarding curvature estimates, notable are the works [12] and [13] of Munteanu and Wang.Invariants reflecting the geometry near infinity of a complete noncompact Riemannian manifold (N n, h) are its asymp-totic cones, where an asymptotic cone is defined as the pointed Gromov–Hausdorff limit of (N , λ−1i dh, p)for some sequence λi → ∞ and p ∈ N (this limit is independent of p). By the compactness theorem of Gromov [7], if Rch ≥ 0, then there exists such a metric space limit. This leads to the questions of the uniqueness and regularity of such limits. E-mail addresses: benchow@math.ucsd.edu (B. Chow), penglu@uoregon.edu (P. Lu).1 P.L. is partially supported by a grant from the Simons Foundation.http://dx.doi.org/10.1016/j.crma.2015.09.0091631-073X/© 2015 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.1008 B. Chow, P. Lu / C. R. Acad. Sci. Paris, Ser. I 353 (2015) 1007–1009That is, are the limits independent of the sequence λi → ∞? Is an asymptotic cone necessarily a regular asymptotic cone (i.e., the cross section is a smooth (n − 1)-dimensional manifold)? Even in the presence of positive Ricci curvature, the limit may be nonunique by the examples of Perelman [17]. For the study of complete noncompact Ricci flat manifolds, there are the deep works of Cheeger and Colding (e.g., [3]), Cheeger and Tian [4], and Colding and Minicozzi (e.g., [6]), where in the last paper it is proved that if there exists a regular asymptotic cone, then the asymptotic cone is unique.Regarding the geometry near infinity of GRS, Kotschwar and Wang [11] proved that for a simply-connected complete noncompact shrinking GRS (Mn1, g1, f1), if a topological end is geometrically asymptotic near infinity to a unique regular asymptotic cone, then any other simply-connected shrinking GRS (Mn2, g2, f2) with a topological end geometrically asymp-totic to the same cone must be isometric to (Mn1, g1). This holographic principle was based in part on their earlier separate works on backward uniqueness of the Ricci flow (see Kotschwar [10]) and the holographic principle for shrinking self-similar solutions to the mean curvature flow (see Wang [20]).Let (Mn, ̄g, f̄ ) be a complete noncompact shrinking GRS with f̄ normalized, so that Rc ḡ +∇2ḡ f̄ = 12 ḡ and R ḡ +|∇ f̄ |2 = f̄ . We assume that | Rc ḡ |(x) → 0 as x → ∞. Munteanu and Wang [13] proved that, fixing p ∈M, there exists a constant C < ∞such that | Rmḡ |(x) ≤ C f̄ (x)−1 ≤ C(dḡ(x, p) + 1)−2for x ∈M. This implies that asymptotic cones are regular. In this paper, based mostly on §2 of [11], we consider the uniqueness issue.Proposition 1. Let (Mn, g(t), f (t)), t < 1, be the associated time-dependent canonical form of (M, ̄g, f̄ ). Then there exists a compact set K ⊂ M such that, as t → 1− , g(t) converges pointwise in C∞ on compact subsets of M − K to a smooth Riemannian metric g1, where (M − K , g1) is isometric to the complement of a compact set in a regular cone.Proof. Firstly, observe that the quadratic curvature decay implies that |∇ f̄ |2 ≥ f̄ − a2f̄for some positive constant a. By the definition of the canonical form, there exist diffeomorphisms ϕt : M → M, defined by ∂∂t ϕt (x) = 11−t(∇ḡ f̄)(ϕt (x)), ϕ0 = id, such that g(t) = (1 − t)ϕ∗t ḡ is a solution to the Ricci flow and f (x, t)  f̄ (ϕt (x)) > 0 satisfies Rcg(t) +∇2g(t) f (t) −12(1−t) g(t) = 0 and ∂ f∂t (x, t) = 11−t |∇ḡ f̄ |2 (ϕt (x)). Hence∂ f∂t(x, t) ≥ 11 − t(f (x, t) − a2f (x, t)). (1)Suppose that x ∈M satisfies f̄ (x) ≥ a√1−ε2 , ε > 0. By ff 2−a2∂ f∂t ≥ 11−t , we have f (x, t)2 − a2 ≥ (1 − t)−2(f̄ (x)2 − a2). There-foref (x, t) ≥ (1 − t)−1 ( f̄ (x)2 − a2)1/2 ≥ ε (1 − t)−1 f̄ (x) for t ∈ [0,1). (2)We have|Rmg(t) |g(t)(x) = (1 − t)−1∣∣Rmḡ∣∣ḡ (ϕt (x)) ≤ C(1 − t) f (x, t) ≤C√1 − ε2aε.By this uniform bound for curvature, by ∫ 10∣∣ ∂∂t g (x, t)∣∣g(x,t) dt ≤ C√1−ε2εa , and by Shi’s local derivative of curvature estimates, there exists a smooth metric g1 on { f̄ > a} such that g(t) converges to g1 in C∞ on { f̄ ≥ a + ε}, for every ε > 0.Now ∂ f∂t (x, t) ≤ 11−t f (x, t) implies that h(x, t)  (1 − t) f (x, t) ≤ f̄ (x). By 0 ≤ R ḡ (ϕt (x)) ≤ C(1−t)ε f̄ (x) and∂h∂t(x, t) = − f (x, t) + |∇ḡ f̄ |2 (ϕt (x)) = −R ḡ (ϕt (x)) (3)for x ∈ { f̄ ≥ a√1−ε2 } and t ∈ [0, 1), we see that h(t) converges in C0 on { f̄ > a} as t → 1 to a function h1. By (1 − t) Rcg(t) +∇2g(t)h(t) − 12 g(t) = 0 and standard elliptic theory, the convergence is in C∞ . Taking the limit of this equa-tion as t → 1, we obtain ∇2g1 h1 −12g1 = 0. Since (1 − t)2 R g(t) + |∇h(t)|2g(t) = h(t), we have |∇h1|2g1 = h1. Moreover, ε f̄ (x) ≤ h1(x) ≤ f̄ (x). Since ∣∣∣ ∂h∂t∣∣∣ ≤ C(1−t)ε f̄, we have |h(x, t) − h1 (x)| ≤ C(1−t)2ε f̄ (x) on { f̄ ≥ a√1−ε2 } × [0, 1).Define  = {h1 > a} ⊂ M. Taking ε = 1√2 , we obtain { f̄ >√2a} ⊂  ⊂ { f̄ > a}. The function ρ1  2√h1 on  satis-fies ∇2g1(ρ21) = 2g1, |∇ρ1|2g1 = 1, ∇ g1 (ρ21 ) is a vector field generating a 1-parameter family {ϕ1t }t∈[0,∞) of homotheties of (, g1) into itself, the integral curves to ∇ g1ρ1 are geodesics, and there is a diffeomorphism between  and the product of (2√a, ∞) and a compact manifold 	n−1 such that g1 = dρ21 + ρ21 g̃1, where g̃1 is a C∞ metric on 	. This implies that (, g1) extends to a regular asymptotic cone. Theorem 2. Any two asymptotic cones of a complete noncompact shrinking gradient Ricci soliton (Mn, ̄g, f̄ ) with | Rc |(x) → 0 as x → ∞ are isometric.B. Chow, P. Lu / C. R. Acad. Sci. Paris, Ser. I 353 (2015) 1007–1009 1009Proof. Let O be a minimum point of f̄ , so that ϕt(O ) = O . Suppose that an Euclidean metric cone C is the pointed Gromov–Hausdorff limit of (M, λ−1i dḡ, O)for some sequence λi → ∞. Since g(1 −λ−2i ) = λ−2i ϕ∗1−λ−2i ḡ converges pointwise in C∞ on compact subsets of  to g1, we have that (ϕ1−λ−2i (), λ−2i ḡ) converges in the C∞ Cheeger–Gromov sense using the diffeomorphisms ϕ1−λ−2i . Sincedλ−2i ḡ(ϕ1−λ−2i (x), O ) = d(ϕ−11−λ−2i )∗ g(1−λ−2i )(ϕ1−λ−2i (x),ϕ1−λ−2i (O ))= dg(1−λ−2i )(x, O ) ≤ Cdḡ(x, O ),the Cheeger–Gromov convergence matches with the pointed Gromov–Hausdorff convergence. We obtain that (, g1) is isometric to the complement of a compact set in C. So C is independent of the choice of λi . Finally, we observe a sufficient condition for a shrinking GRS to satisfy |Rc| → 0.Proposition 3. If a κ-noncollapsed complete noncompact shrinking GRS (Mn, ̄g, f̄ ) satisfies |Rm| ≤ C and R(x) → 0 as x → ∞, then |Rc| (x) → 0 as x → ∞.Proof. Suppose |Rm| ≤ C and R(x) → 0 as x → ∞, but |Rc| (x)  0 as x → ∞. Choose a sequence of points xi → ∞ with |Rc| (xi) ≥ c > 0. Let (M, g(t), f (t)) be the canonical form of (M, ̄g, f̄ ). Then (M, g(t), xi) will subconverge to a complete ancient solution (Mn∞, g∞(t), x∞) with bounded curvature and |Rc| (x∞, 0) ≥ c and R(x∞, 0) = 0. This is a contradiction since by the strong maximum principle applied to the equation ∂ R∂t = R + 2| Rc |2, a complete ancient solution to Ricci flow with R = 0 at some point must be Ricci flat. References[1] H.-D. Cao, D.-T. Zhou, On complete gradient shrinking Ricci solitons, J. Differ. Geom. 85 (2010) 175–186.[2] H.-D. Cao, B.-L. Chen, X.-P. Zhu, Recent developments on Hamilton’s Ricci flow, in: Geometric Flows, in: Surveys in Differential Geometry, vol. XII, Int. Press, Somerville, MA, USA, 2008, pp. 47–112.[3] J. Cheeger, T.H. Colding, Lower bounds on Ricci curvature and the almost rigidity of warped products, Ann. of Math. (2) 144 (1996) 189–237.[4] J. Cheeger, G. Tian, On the cone structure at infinity of Ricci flat manifolds with Euclidean volume growth and quadratic curvature decay, Invent. Math. 118 (1994) 493–571.[5] B.-L. Chen, Strong uniqueness of the Ricci flow, J. Differ. Geom. 82 (2009) 363–382.[6] T.H. Colding, W.P. Minicozzi II, On uniqueness of tangent cones for Einstein manifolds, Invent. Math. 196 (2014) 515–588.[7] M. Gromov, Metric Structures for Riemannian and Non-Riemannian Spaces, with appendices by M. Katz, P. Pansu and S. Semmes, translated from the French by Sean Michael Bates, Progress in Mathematics, vol. 152, Birkhäuser Boston, Inc., Boston, MA, USA, 1999.[8] R.S. Hamilton, The Ricci flow on surfaces, in: Mathematics and General Relativity, Santa Cruz, CA, 1986, in: Contemp. Math., vol. 71, 1988, pp. 237–262.[9] R.S. Hamilton, The formation of singularities in the Ricci flow, in: Surveys in Differential Geometry, vol. II, Cambridge, MA, 1993, International Press, Cambridge, MA, USA, 1995, pp. 7–136.[10] B. Kotschwar, Backwards uniqueness for the Ricci flow, Int. Math. Res. Not. 21 (2010) 4064–4097.[11] B. Kotschwar, L. Wang, Rigidity of asymptotically conical shrinking gradient Ricci solitons, J. Differ. Geom. 100 (2015) 55–108.[12] O. Munteanu, J. Wang, Geometry of shrinking Ricci solitons, arXiv:1410.3813.[13] O. Munteanu, J. Wang, Conical structure for shrinking Ricci solitons, arXiv:1412.4414.[14] O. Munteanu, J. Wang, Positively curved shrinking Ricci solitons are compact, arXiv:1504.07898.[15] A. Naber, Noncompact shrinking four solitons with nonnegative curvature, J. Reine Angew. Math. 645 (2010) 125–153.[16] L. Ni, N. Wallach, On a classification of gradient shrinking solitons, Math. Res. Lett. 15 (2008) 941–955.[17] G. Perelman, A complete Riemannian manifold of positive Ricci curvature with Euclidean volume growth and nonunique asymptotic cone, in: Compar-ison Geometry, Berkeley, CA, 1993–1994, in: Math. Sci. Res. Inst. Publ., vol. 30, Cambridge University Press, Cambridge, UK, 1997, pp. 165–166.[18] G. Perelman, The entropy formula for the Ricci flow and its geometric applications, arXiv:math.DG/0211159.[19] P. Petersen, W. Wylie, On the classification of gradient Ricci solitons, Geom. Topol. 14 (2010) 2277–2300.[20] L. Wang, Uniqueness of self-similar shrinkers with asymptotically conical ends, J. Amer. Math. Soc. 27 (2014) 613–638.

Uniqueness of asymptotic cones of complete noncompact shrinking gradient Ricci solitons with Ricci curvature decay

Bennett Chow

Peng Lu

Comptes Rendus Mathématique

http://www.numdam.org/item/10.1016/j.crma.2015.09.009.pdf

Uniqueness of asymptotic cones of complete noncompact shrinking gradient Ricci solitons with Ricci curvature decay

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