## On the structure of spaces with Ricci curvature bounded below. I (1997)

Venue: | J. DIFFERENTIAL GEOM |

Citations: | 88 - 7 self |

### BibTeX

@ARTICLE{Cheeger97onthe,

author = {Jeff Cheeger and Tobias H. Colding},

title = {On the structure of spaces with Ricci curvature bounded below. I},

journal = { J. DIFFERENTIAL GEOM},

year = {1997},

volume = {45},

pages = {37--74}

}

### Years of Citing Articles

### OpenURL

### Abstract

### Citations

764 |
Geometric Measure Theory
- Federer
- 1969
(Show Context)
Citation Context ...etic treatment of Ricci curvature Generalizations of the notion of "smooth function" have long played a very important role in analysis and in questions of a geometric analytic nature; see e=-=.g. [10], [29], [51], [5-=-7]. After the pioneering work of Alexandrov and Gromov ([2], [37]) analogous notions of "generalized Riemannian manifold" have begun to play an increasingly significant role in Riemannian ge... |

368 | Minimal Surfaces and functions of Bounded Variation - Giusti - 1984 |

365 |
Theorie Des Distributions
- Schwartz
- 1997
(Show Context)
Citation Context ...reatment of Ricci curvature Generalizations of the notion of "smooth function" have long played a very important role in analysis and in questions of a geometric analytic nature; see e.g. [1=-=0], [29], [51], [57]. Af-=-ter the pioneering work of Alexandrov and Gromov ([2], [37]) analogous notions of "generalized Riemannian manifold" have begun to play an increasingly significant role in Riemannian geometry... |

175 |
Lectures on geometric measure theory
- Simon
- 1983
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Citation Context ...ff ! 1, then there is a subcollection, (2:26) B 0 = fB r ff 0 (z ff 0 )g; of mutually disjoint balls, such that (2:27) [ ff 0 B 6r ff 0 (z ff 0 ) oe [ ff B r ff (z ff ); see Chapter 1, Theorem 3.3 of =-=[56]-=-. This statement, which we call the Covering Theorem, has the following standard consequence. Let / be a oe-finite measure on Z, such that for all R ? 0, there exists c(R), such that for all z 2 Z, (2... |

170 |
Multiple Integrals in the Calculus of Variations
- Morrey
- 1966
(Show Context)
Citation Context ...ffi (Y n ) ae ffi R " (Y n ); see Theorem A.1.5. For subsets of R n , an analog of the condition which defines R " is known as "Reifenberg's condition", as was pointed out to us by=-= Bruce Kleiner; see [43]-=-, [50]. In Section 6 we show that S(Y n ) ae S n\Gamma2 (Y n ). Thus, in the noncollapsed case, the singular set has codimension at least 2. Obvious 2-dimensional examples show that this result is opt... |

156 |
Ebin – Comparison theorems in Riemannian geometry
- Cheeger, G
- 1975
(Show Context)
Citation Context ... S 7 +s2 g S 3 ; wheres= (ffi) and (ffi) ! 0 as ffi ! 0. Since the Ricci curvature of g S 7 +s2 g S 3 is uniformly positive, the above mentioned fact is a direct consequence of O'Neill's formula; see =-=[18]-=-. More precisely, as ffi ! 0, the sectional curvatures of planes which are contained in the fibre, S 3 , are equal to ffi \Gamma2 , the curvatures of horizontal planes approach 4 and the curvatures of... |

128 |
Aleksandrov spaces with curvatures bounded below
- Burago, Gromov, et al.
- 1992
(Show Context)
Citation Context ...ecture 0.8. If jRic M n i js(n \Gamma 1), then S(Y n ) ae S n\Gamma4 (Y n ) and H n\Gamma4 (S n\Gamma4 (Y n )) ! 1. Finally, we point out that the results of this paper should be compared to those of =-=[9]-=-, [48], which treat analogous questions in the context of a lower bound on sectional curvature, i.e., for Alexandrov spaces. Recall that in [24]--[26], one finds the first theorems on Ricci curvature ... |

89 |
MGromov,Collapsing Riemannian manifolds while keeping their curvature bounded
- Cheeger
- 1990
(Show Context)
Citation Context ...j 2 ;j 1 . The modifications are effected by the proceedure of [11], where a situation closely related to the one considered here is treated. Thus, we will refer to [11] for certain details (see also =-=[21]). If-=- x i;k 1 2 Q i;1 ; x i;k 2 2 Q i;2 ; are such that I i;k 2 ;k 1 (B 4\Delta2 \Gammai (0 i;k 1 )) " B 4\Delta2 \Gammai (0 i;k 2 ) is nonempty, we put e I i;k 2 ;k 1 = I i;k 2 ;k 1 : Note that for x... |

89 | Differential equations on Riemannian manifolds and their geometric applications - Cheng, Yau - 1975 |

57 |
Some function-theoretic properties of complete Riemannian manifold and their applications to geometry
- Yau
- 1976
(Show Context)
Citation Context ...n; [8], [10], [37]. Calabi emphasized that Laplacian comparison holds in a useful generalized sense, even at points where the distance function fails to be smooth i.e. on the cut locus; compare [20], =-=[62]-=-. In terms of mean curvature, this principle can also be formulated as follows. Namely, rate of change of the logarithm of the area of the intersection of any (thin) angular sector of minimal geodesic... |

51 |
Finiteness theorems for Riemannian manifolds
- Cheeger
- 1970
(Show Context)
Citation Context ... n ) ffl;j \Gamma1 (M n i ). Hence, by the discussion preceding the proof, there is a definite bound on the C 1;ff -harmonic radius at q i . By using arguments which by now are standard (compare e.g. =-=[11]-=-, [7]) and Theorem 6.1, which asserts that S ae S n\Gamma2 , the proof can be concluded. In view of Theorem 7.2 and 7.3, an obvious application of Gromov's compactness theorem gives Theorem 7.5. Given... |

49 |
Analysis of and on uniformly rectifiable sets
- David, Semmes
- 1994
(Show Context)
Citation Context ...nection with the results of [26] and Fred Almgren for some helpful conversations concerning it. We are also grateful to Stephen Semmes, for discussions and for pointing out his paper, [52], and book, =-=[28], which de-=-al with situations closely related to Reifenberg's. Let (R k ) ";r denote the set of points such that for some u ? r, (4.6) holds for all s 2 (0; u] and R k \Theta X = R k . Thus, (R k ) " =... |

34 |
Collapsing of Riemannian manifolds and eigenvalues of the laplace operator, Invent
- Fukaya
- 1987
(Show Context)
Citation Context ...annian measures on the manifolds, M n j . Here, the renormalization is such that renormalized volume of the unit ball, B 1 (p i ), is equal to 1. These renormalized limit measures were constructed in =-=[30]-=-; see also Section 1 and compare [36]. In the noncollapsed case, it turns out that any such measure, , is just a multiple of the Hausdorff measure, H n ; see Theorem 5.9. However, in the collapsed cas... |

34 | A generalized sphere theorem - Grove, Shiohama - 1977 |

30 | The fundamental groups of almost non-negatively curved manifolds - Fukaya, Yamaguchi - 1992 |

26 |
An extension of E. Hopf’s maximum principle with an application to Riemannian geometry
- Calabi
- 1958
(Show Context)
Citation Context ... synthetic treatment of Ricci curvature Generalizations of the notion of "smooth function" have long played a very important role in analysis and in questions of a geometric analytic nature;=-= see e.g. [10], [29], [5-=-1], [57]. After the pioneering work of Alexandrov and Gromov ([2], [37]) analogous notions of "generalized Riemannian manifold" have begun to play an increasingly significant role in Riemann... |

23 |
C α -compactness for manifolds with Ricci curvature and injectivity radius bounded below
- Anderson, Cheeger
- 1992
(Show Context)
Citation Context ...fl;j \Gamma1 (M n i ). Hence, by the discussion preceding the proof, there is a definite bound on the C 1;ff -harmonic radius at q i . By using arguments which by now are standard (compare e.g. [11], =-=[7]-=-) and Theorem 6.1, which asserts that S ae S n\Gamma2 , the proof can be concluded. In view of Theorem 7.2 and 7.3, an obvious application of Gromov's compactness theorem gives Theorem 7.5. Given k 0 ... |

21 |
A theorem on triangles in a metric space and some applications
- Alexandrov
- 1951
(Show Context)
Citation Context ...ooth function" have long played a very important role in analysis and in questions of a geometric analytic nature; see e.g. [10], [29], [51], [57]. After the pioneering work of Alexandrov and Gro=-=mov ([2], [37]) an-=-alogous notions of "generalized Riemannian manifold" have begun to play an increasingly significant role in Riemannian geometry. In this appendix, we will discuss some related issues in conn... |

20 |
Introduction to Measure and Integration
- Munroe
- 1953
(Show Context)
Citation Context ...z i ; r i )jB r i (z i ); r isffi g: ricci curvature 417 By standard measure theory,sis a metric outer measure and the corresponding measure, also denoted by , is a Radon measure; see Theorem 13.7 of =-=[44]-=-. Theorem 1.10. There is a unique Radon measure, , such that for all, z; R; (1:11) (BR (z)) = V 1 (z; R): In particularssatisfies the inequality, (1:12) (B r 1 (z)) (B r 2 (z)) V n;\Gamma1 (r 1 ) V n;... |

20 |
Solution of the Plateau Problem for m-dimensional surfaces of varying topological type
- Reifenberg
- 1960
(Show Context)
Citation Context ... n ) ae ffi R " (Y n ); see Theorem A.1.5. For subsets of R n , an analog of the condition which defines R " is known as "Reifenberg's condition", as was pointed out to us by Bruce=-= Kleiner; see [43], [50]-=-. In Section 6 we show that S(Y n ) ae S n\Gamma2 (Y n ). Thus, in the noncollapsed case, the singular set has codimension at least 2. Obvious 2-dimensional examples show that this result is optimal. ... |

20 |
Collapsing and pinching under lower curvature bound
- Yamaguchi
- 1991
(Show Context)
Citation Context ...ts and every such segment extends to a line in the sense of the splitting theorem. Since the splitting theorem holds for Alexandrov spaces (see [39] and for the case of Gromov-Hausdorff limit spaces, =-=[61]-=-) it follows that X n is not an Alexandrov space unless it is isometric to Euclidean space, R n . (Of course this can also be checked directly). None-the-less, it is easy to verify (A.2.2) (for H =0).... |

19 |
On complete manifolds with nonnegative Ricci curvature
- Abresch, Gromoll
- 1990
(Show Context)
Citation Context ...mit measure, can be regarded as having positive Ricci curvature in a generalized sense, although not in the classical sense. Example 8.71 (Smooth limit spaces). Let h be a smooth positive function on =-=[0; 1]-=-, such that (8:72) fi fi fi fi fi h 0 h fi fi fi fi fi ! ffi r ; 452 jeff cheeger & tobias h. colding (8:73) fi fi fi fi fi h 00 h fi fi fi fi fi ! ffi r 2 : Here, ffi is so small that the doubly warp... |

19 |
Shape of manifolds with positive Ricci curvature
- Colding
- 1996
(Show Context)
Citation Context ...ts proved in [26]. Recall that in [26], using the solution of a conjecture of Anderson-Cheeger proved there, combined with a conjecture of Anderson-Cheeger and Perelman analogous to the one proved in =-=[24], the-=- following was shown. If RicM ns\Gamma"(n) and some ball in M n is Gromov-Hausdorff close to the corresponding ball in R n , then every sub-ball (whose center is not very close to the boundary) i... |

19 | Bounding homotopy type by geometry - Grove, Peterson - 1988 |

19 |
Geometric finiteness theorems via controlled topology
- GROVE, PETERSEN, et al.
- 1990
(Show Context)
Citation Context ...esemble results which play a basic role in Alexandrov space theory. Moreover, some of the results of Sections 5, 6 of this paper should be contrasted with the theorem of Grove-Petersen [38] (see also =-=[40]-=-, [41]) giving a lower bound on the relative contractibility radius at all points of M n , under the assumptions diam(M n )sd; Vol (M n )sv ? 0; KMs\Gamma1. Here KM denotes sectional curvature. Well k... |

18 |
On the cone structure at infinity of Ricci flat manifolds with Euclidean volume growth and quadratic curvature decay, Invent
- Cheeger, G
- 1994
(Show Context)
Citation Context ...published) who constructed a metric on R 4 with positive Ricci curvature, Euclidean volume growth and quadratic curvature decay, for which the tangent cone at infinity is not unique; compare [15] and =-=[22]-=-. In Perelman's example, he views R 4 n f0g as R+ \Theta S 3 , on which he constructs a metric of triply warped product type, based on the Hopf fibration, S 1 ! S 3s! S 2 . Our examples are based on d... |

15 | Geometric conditions and existence of bilipschitz parametrisations - Toro - 1995 |

14 |
Chord-arc surfaces with small constant II
- Semmes
- 1991
(Show Context)
Citation Context ...ic M n i i ? 0, we have n i ! 1, then for the limit space, all good properties (such as the splitting theorem) can fail to hold. In Appendix 1, we reformulate the theorem of Reifenberg [50] (see also =-=[52]-=-) in an intrinsic setting. In combination with the results of [24]--[26] (in particular with the conjectures of Anderson--Cheeger and Perelman proved there) this implies a sharpening of Perelman's low... |

10 |
Existence and non-existence of homogeneous Einstein metrics
- Wang, Ziller
- 1986
(Show Context)
Citation Context ...e locally constant dimension, do not arise as limit spaces; Finally in [12], we specialize the results of the present paper to the case in which the manifolds, M n i ; are homogeneous spaces; compare =-=[59]-=-, [60]. We close this introduction with some additional remarks and conjectures. Conjecture 0.7. The interior of Y n n S n\Gamma4 (Y n ) is a topological manifold. In a subsequent joint paper with Gan... |

7 |
Synthetic geometry in Riemannian manifolds
- Gromov
- 1978
(Show Context)
Citation Context ...ergent sequences,sM n i ! Y m , can lead to different limit measures; see Example 1.24. Thus, a renormalized limit measure encodes information on the collapsing sequence from which it arises; compare =-=[34]-=-. Even in the collapsed case, the renormalized limit measures and Hausdorff measure are closely related; see [13] for further discussion. Any renormalized limit measure, , has the crucial property tha... |

6 |
curvature and volume convergence
- Ricci
- 1997
(Show Context)
Citation Context ...RicM n is\Gamma(n \Gamma 1) and Y n is a manifold, also plays a direct role in the present discussion. The continuity of the volume in the above case was conjectured by Anderson-Cheeger and proved in =-=[26]-=-. The remainder of this paper is divided into 8 sections and two appendices. 1. Renormalized limit measures. 2. Arbitrary limit spaces. 3. dim Ysn \Gamma 1 in the collapsed case. 4. Polar limit spaces... |

5 |
Metrics of Positive Ricci curvature with large diameter
- Anderson
- 1990
(Show Context)
Citation Context ...uct type, based on the Hopf fibration, S 1 ! S 3s! S 2 . Our examples are based on doubly warped product constructions, and we will begin by briefly reviewing some properties of such metrics; compare =-=[3]-=-. Let I be an interval and let Z be a manifold. Consider a family of Riemannian metrics, g Z (r) on Z, parameterized by r 2 I . Assume that for each p 2 Z, the metrics g Z (r)jZ p can all be simultane... |

5 |
A relation between volume, mean curvature and diameter
- Bishop
- 1963
(Show Context)
Citation Context ...d below", which in the smooth case, is mean curvature comparison, or equivalently (in the smooth case) Laplacian comparison, one needs a version of (0.5) which is localized with respect to direct=-=ion; [8]-=-, [10], [37]. Calabi emphasized that Laplacian comparison holds in a useful generalized sense, even at points where the distance function fails to be smooth i.e. on the cut locus; compare [20], [62]. ... |

4 |
Structures Metriques Pour Les Varieties Riemanniennes
- Gromov
- 1981
(Show Context)
Citation Context ...onding statements for manifolds and limit spaces follows directly from the continuity of the geometric quantities in question under GromovHausdorff limits, together with Gromov's compactness theorem, =-=[37]-=-; Theorems 2.45, 5.12 (see also Remark 5.13), 7.5, 7.6, are examples of Received April 5, 1996, and, in revised form, March 20, 1997. The first author was supported in part by NSF Grant DMS 9303999, a... |

4 | Di¤erential equations on Riemannian manifolds and their geometric applications - Cheng, Yau - 1975 |

3 | Constraints on singularities under Ricci curvature bounds - Cheeger, Colding, et al. - 1997 |

3 | On the structure of complete manifolds of nonnegative sectional curvature - Cheeger, Gromoll - 1972 |

3 | Manifolds with Positive Ricci Curvature - Large - 1996 |

3 |
Geometric niteness theorems via controlled topology
- Grove, Petersen, et al.
- 1990
(Show Context)
Citation Context ...esemble results which play a basic role in Alexandrov space theory. Moreover, some of the results of Sections 5, 6 of this paper should be contrasted with the theorem of Grove-Petersen [38] (see also =-=[40]-=-, [41]) giving a lower bound on the relative contractibility radius at all points of M n , under the assumptions diam(M n ) d� Vol (M n ) v>0� KM ;1. Here KM denotes sectional curvature. Well known ex... |

2 |
geodesics and gravitational instantons
- Short
- 1990
(Show Context)
Citation Context ...diam(M n )sd; Vol (M n )sv ? 0; KMs\Gamma1. Here KM denotes sectional curvature. Well known examples show that this fails to hold if the bound KMs\Gamma1 is weakened to RicM ns\Gamma(n \Gamma 1); see =-=[5]-=-. However, according to Theorem 5.12 and Remark 5.13, the complement of a set of codimension 2 can be written as a union of sets, on each of which, every point has a neighborhood of a definite size di... |

2 |
bounds on Ricci curvature and the almost rigidity of warped products
- Lower
- 1996
(Show Context)
Citation Context ... Remark 4.9. Our results, most of which were announced in [14], are applications of the "almost rigidity" theorems for manifolds of almost nonnegative Ricci curvature, announced in [14] and =-=proved in [15]. In parti-=-cular, we use the generalized splitting, "volume cone implies metric cone" and (implicitly) integral Toponogov theorems, together with tangent cone analysis of the sort employed in geometric... |

2 |
Spaces on and beyond the boundary of existence
- Petersen, Wilhelm, et al.
- 1995
(Show Context)
Citation Context ...his class is not defined synthetically, even though it contains members more general than smooth n-dimensional Riemannian manifolds. In fact, for H ? 0, it is known that sec(n; H) 6ae Alex(n; H); see =-=[49]-=-. However, it is not known whether there exists N(X) ! 1; c(X)H ? \Gamma1; such that if X 2 Alex(n; H), then X 2 sec(N(X); c(X)H). If this were known, we would say that the class sec(N(\Delta); c(\Del... |

2 |
On the cone structure at in nity of Ricci at manifolds with Euclidean volume growth and quadratic curvature decay, Invent
- Cheeger, Tian
- 1994
(Show Context)
Citation Context ... (unpublished) who constructed a metric on R 4 with positive Ricci curvature, Euclidean volume growth and quadratic curvature decay, for which the tangent coneatinnity is not unique� compare [15] and =-=[22]-=-. In Perelman's example, he views R 4 nf0g as R + S 3 , on which he constructs a metric of triply warped product type, based on the Hopf bration, S1 ! S3 ! S2 . Our examples are based on doubly warped... |

1 |
of metrics with bounded curvature and applications to critical metrics of Riemannian functionals
- Degenerations
- 1993
(Show Context)
Citation Context ...rd argument based on characteristic numbers and curvature identities implies that this uniform bound actually holds, with ps2; see [17]. On the other hand, the following conjecture is well known; see =-=[6]-=-, Conjecture 2.3. Conjecture 0.8. If jRic M n i js(n \Gamma 1), then S(Y n ) ae S n\Gamma4 (Y n ) and H n\Gamma4 (S n\Gamma4 (Y n )) ! 1. Finally, we point out that the results of this paper should be... |

1 |
non-linear spectra and width, Lecture Notes
- Dimension
- 1988
(Show Context)
Citation Context ..., from which it follows that Lip r 1 ;r 2 (x; f) is a continuous function of (x; r 1 ; r 2 ). Hence, the function, (A:2:4) Lip(x; f) := lim r 2 !0 lim r 1 !0 Lip r 1 ;r 2 (x; f); is measurable. As in =-=[35]-=-, we define a generalized Dirichlet functional by (A:2:5) Q(f; f) = Z X (Lip(x; f)) 2 ; whenever the integral is finite. If (X) ! 1, then this holds for all Lipschitz functions, f . Note that at least... |

1 |
and geometric meaning of curvature, Rondi
- Sign
- 1991
(Show Context)
Citation Context ... j . Here, the renormalization is such that renormalized volume of the unit ball, B 1 (p i ), is equal to 1. These renormalized limit measures were constructed in [30]; see also Section 1 and compare =-=[36]-=-. In the noncollapsed case, it turns out that any such measure, , is just a multiple of the Hausdorff measure, H n ; see Theorem 5.9. However, in the collapsed case, different Gromov-Hausdorff converg... |

1 | private communication). 480 jeff cheeger & tobias h. colding - Perelman |

1 |
of positive Ricci curvature with almost maximal volume
- Manifolds
- 1994
(Show Context)
Citation Context ...th the conjectures of Anderson--Cheeger and Perelman proved there) this implies a sharpening of Perelman's lower bound on the relative contractibility radius in the presence of almost maximal volume; =-=[46]-=-. As a consequence, we obtain sharpenings of most of the results of [24]--[26] and additional new results; see in particular, Theorem A.1.11. As a specific example, it follows that there exists ffi(n)... |

1 |
Volume comparison and its applications in Riemannian-Finsler geometry
- Shen
(Show Context)
Citation Context ...nonical examples do satisfy generalized Laplacian comparison (as well as (A.2.2)) but not the basic rigidity and integral Toponogov theorems. These examples were pointed out to us by Z. Shen; compare =-=[54]-=-, [55]. Let X n denote a normed vector space of dimension n. As usual, we regard X n as a complete metric space by setting v 1 ; v 2 = jv 1 \Gamma v 2 j. Let ricci curvature 475sdenote the associated ... |

1 |
un th'eor`em d'analyse fonctionalle
- Sobolev, Sur
- 1938
(Show Context)
Citation Context ...nt of Ricci curvature Generalizations of the notion of "smooth function" have long played a very important role in analysis and in questions of a geometric analytic nature; see e.g. [10], [2=-=9], [51], [57]. After th-=-e pioneering work of Alexandrov and Gromov ([2], [37]) analogous notions of "generalized Riemannian manifold" have begun to play an increasingly significant role in Riemannian geometry. In t... |