introduction to ricci curvature and the convergence theory

The ε-Regularity with Integral Curvature BoundsThe ε-Regularity without Integral Curvature Bounds

Collapsed Manifolds with Bounded CurvatureCollapsed Spaces with Ricci Curvature Bounds

Introduction to Ricci Curvature and theConvergence Theory

Ruobing Zhang (Stony Brook University)

Structure of Collapsed Special Holonomy SpacesDuke University,April 9- 13, 2018

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Table of contents

1 The ε-Regularity with Integral Curvature Bounds

2 The ε-Regularity without Integral Curvature Bounds

3 Collapsed Manifolds with Bounded Curvature

4 Collapsed Spaces with Ricci Curvature Bounds

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Elliptic Theory

(Elliptic Inequality)Rm = Rm ∗Rm +∇2 Ric

Ricg ≡ λ · g, λ ∈ R(1)

=⇒ ∆|Rm | ≥ −C(n) · |Rm |2. (2)

(Sobolev) Let u ∈ C∞0 (Ω),(ˆΩu

2nn−2

)n−22n ≤ CS

(ˆΩ|∇u|2

) 12. (3)

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Elliptic Theory

(Moser Iteration) In dimension 4, there are δ(CS) > 0 andQ(CS) > 0 let u satisfy

∆u ≥ −u2, (4)

and ˆB2(x)

|u|2 ≤ δ, (5)

then

supB1(x)

|u| ≤ Q ·(ˆ

B2(x)|u|2) 1

2. (6)

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Elliptic Theory

(Croke) Let (Mn, g) satisfy Ricg ≥ −(n− 1) andVol(B1(p)) ≥ v > 0, then in B2(p) we have

CS ≤ C0(n, v) <∞. (7)

(Classical ε-Regularity) Let (M4, g, p) be an Einsteinmanifold with |Ricg | ≤ 3. Assume Vol(B1(p)) ≥ v > 0, thenthere are constants ε(v) > 0 and C(v) <∞ such that

ˆB2(p)

|Rm |2 < ε ⇒ supB1(p)

|Rm | ≤ C(v). (8)

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Chern-Gauss-Bonnet and Integral Curvature Bounds

Let (M4, g) be a closed 4-manifold, thenChern-Gauss-Bonnet theorem states that

χ(M4) =

ˆM4

Pχ, (9)

wherePχ ≡

1

8π2(|Rm |2 − 4|Ric |2 +R2). (10)

If (M4, g) is Einstein, then

Pχ =1

8π2|Rm |2. (11)

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The ε-Regularity Theorems and Integral CurvatureBounds

Theorem (M. Anderson)

Given n ≥ 2, there are dimensional constants ε(n) > 0 andC(n) > 0 such that the following holds. Let (Mn, g, p) be anEinstein manifold with |Ricg | ≤ n− 1, then

B2(p)

|Rm |n2 < ε =⇒ sup

B1(p)|Rm | ≤ 1. (12)

This type of ε-regularity mostly applies in thenon-collapsing case.There is a much stronger ε-regularity when n = 4 due toCheeger-Tian.

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B2(p)

|Rm |n2 < ε =⇒ sup

B1(p)|Rm | ≤ 1. (12)

This type of ε-regularity mostly applies in thenon-collapsing case.

There is a much stronger ε-regularity when n = 4 due toCheeger-Tian.

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B2(p)

|Rm |n2 < ε =⇒ sup

B1(p)|Rm | ≤ 1. (12)

This type of ε-regularity mostly applies in thenon-collapsing case.There is a much stronger ε-regularity when n = 4 due toCheeger-Tian.

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Theorem (Cheeger-Tian, 2005)There exist absolute constants ε > 0, C <∞ such that thefollowing holds. Let (M4, g) be an Einstein 4-manifold withRicg ≡ λ · g and |λ| ≤ 3. Then

ˆB2(p)

|Rm |2g dvolg < ε =⇒ supB1(p)

|Rm |g ≤ C. (13)

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C1-Harmonic Radius

Definition (C1-harmonic coordinates)

Let u = (u1, . . . , un) : Br(p)→ Rn with u(p) = 0 and u adiffeomorphism onto its image. We call u a C1-harmoniccoordinates system with ‖u‖r ≤ 1 if the following propertieshold:

For each 1 ≤ k ≤ n, uk is harmonic.If gij = g(∇ui,∇uj) is the metric in coordinates, then

|gij − δij |C0(Br(p)) + r|∂gij |C0(Br(p)) < 10−6, (14)

where the scale-invariant norms are taken in the euclideancoordinates.

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C1-Harmonic Radius

Definition (C1-Harmonic Radius)

For x ∈Mn we define the harmonic radius rh(x) by

rh(x) ≡ supr > 0| ∃ C1 − harmonic coordinates u : Br(x)→ Rn

with‖u‖r ≤ 1. (15)

Definition (Curvature Radius)

For x ∈Mn we define the harmonic radius r|Rm |(x) by

rh(x) ≡ supr > 0

∣∣∣ supBr(x)

r2|Rm | ≤ 1. (16)

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C1-Harmonic Radius

In harmonic coordinates, we have the following expressionof Ricci tensor,

Ricij =1

2gkl

∂2gij∂xk∂xl

+Q(∂grs∂xm

). (17)

With |Ricg | ≤ n− 1, then by the standard elliptic regularity,within the C1-harmonic radius, the metric has uniformlybounded W 2,p-norm for any p > 1.If (Mn, g) is Einstein, then rh(x) ≥ r0 implies that |Rm | isuniformly bounded around x.

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C1-Harmonic Radius


Ricij =1

2gkl

∂2gij∂xk∂xl

+Q(∂grs∂xm

). (17)

With |Ricg | ≤ n− 1, then by the standard elliptic regularity,within the C1-harmonic radius, the metric has uniformlybounded W 2,p-norm for any p > 1.

If (Mn, g) is Einstein, then rh(x) ≥ r0 implies that |Rm | isuniformly bounded around x.

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C1-Harmonic Radius


Ricij =1

2gkl

∂2gij∂xk∂xl

+Q(∂grs∂xm

). (17)

With |Ricg | ≤ n− 1, then by the standard elliptic regularity,within the C1-harmonic radius, the metric has uniformlybounded W 2,p-norm for any p > 1.If (Mn, g) is Einstein, then rh(x) ≥ r0 implies that |Rm | isuniformly bounded around x.

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C1-Harmonic Radius

Theorem (M. Anderson, 1990)

Let (Mn, g, p) be a Riemannian manifold with |Ricg | ≤ n− 1.Assume

InjRad(x) ≥ i0 > 0 (18)

for every x ∈ B2(p), then there exists r0(n, i0) > 0 such that forall x ∈ B1(p),

rh(x) ≥ r0 > 0. (19)

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C1-Harmonic Radius


Let (Mn, g, p) be a Riemannian manifold with |Ricg | ≤ n− 1and Volg(B1(p)) ≥ v > 0, then there are uniform constantsε0(n, v) > 0 and r0(n, v) > 0 such that if

ˆB2(p)

|Rm |n2 ≤ ε0, (20)

then for all x ∈ B1(p),

rh(x) ≥ r0 > 0. (21)

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C1-Harmonic Radius

Theorem (Cheeger-Tian, 2005)There exist absolute constants ε > 0 and r0 > 0 such that thefollowing holds. Let (M4, g) be a 4-manifold with |Ricg | ≤ 3 and

ˆB2(p)

|Rm |2 < ε, (22)

then for every x ∈ B1(p),

rh(x) ≥ r0 > 0, (23)

where x ∈ ˜B10r0(x).

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ε-Regularity Theorems absent a Priori Integral Bounds


There exists ε(n) > 0 such that if a Riemannian manifold(Mn, g, p) satisfies |Ricg | ≤ (n− 1)ε2 and

Vol(B3/2(p))

Vol(B3/2(0n))> 1− ε, (24)

thenrh(p) ≥ 1. (25)

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ε-Regularity Theorems absent a Priori Integral Bounds

Theorem (Cheeger-Colding, 1997)

Given n ≥ 2, there exists ε(n) > 0 such that the following holds.Let (Mn, g, p) be a Riemannian manifold with |Ricg | ≤ (n− 1)ε2

anddGH(B2(p), B2(0n)) < ε, 0n ∈ Rn, (26)

then rh(p) ≥ 1. In particular, if (Mn, g) is Einstein, then‖Rmg ‖ ≤ 1.

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Quantitative Symmetry and ε-Regularity

DefinitionLet (X, d, p) be a metric space,

we say X is k-symmetric at p if there exists a compactmetric space Y such that X ≡ Rk × C(Y ),we say X is (k, ε, r)-symmetric at p if there exists acompact metric space Y such that

dGH(Brε−1(p), Brε−1(0k, y∗)) < rε, (0k, y) ∈ Rk × C(Y ),(27)

where C(Y ) is a metric cone with a cone tip y∗.

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we say X is k-symmetric at p if there exists a compactmetric space Y such that X ≡ Rk × C(Y ),

we say X is (k, ε, r)-symmetric at p if there exists acompact metric space Y such that



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we say X is k-symmetric at p if there exists a compactmetric space Y such that X ≡ Rk × C(Y ),we say X is (k, ε, r)-symmetric at p if there exists acompact metric space Y such that



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Theorem (Cheeger-Colding’s Metric Cone Theorem, 1996)

Let (Mnj , gj , pj) be a sequence of non-collapsing manifolds with

Ricgj ≥ −(n− 1) such that

(Mnj , gj , pj)

GH−−→ (X, d, p), (28)

then for every x ∈ X, each tangent cone at x is a metric cone.

Theorem (Cheeger-Colding’s Metric Cone Theorem, 1996)

Let (X, d, p) be a non-collapsed limit space under lower Ricci,then every tangent cone over p is k-symmetric for some k ≥ 0.

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Let (Xn, d) be a Ricci limit space.

Let 1 ≤ k ≤ n, we define

Sk(X) ≡x ∈ X

∣∣∣no tangent cone at x is (k+1)−symmetric

(29)and

S(X) ≡ Sn−1(X), R(X) ≡ X \ S(X). (30)

By definition,

S0(X) ⊂ S1(X) ⊂ . . . ⊂ Sn−1(X) = S(X). (31)

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Theorem (Cheeger-Colding’s Stratification Theorem, 1997)

Let (Xn, d, p) be a non-collapsing Ricci-limit space, then

dimH(Sk) ≤ k (32)

andS0 ⊂ S1 ⊂ . . . ⊂ Sn−2 = S. (33)

In particular,dimH(S) ≤ n− 2. (34)

The half Euclidean space Rn+ cannot be a non-collapsingRicci limit space or a tangent cone in Xn.

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Theorem (Cheeger-Naber, 2014)

Let (Mnj , gj , pj)

GH−−→ (Xn∞, d∞, p∞) satisfy |Ricgj | ≤ n− 1, then

the singular set satisfies

S(Xn∞) = Sn−4(Xn

∞). (35)

In particular, dimH(S) ≤ n− 4.

If a tagent cone TpXn∞ ≡ Rn−3 × C(Y ), then Y ≡ S3 and

TpXn∞ ≡ Rn.

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Given n ≥ 2, v > 0, there exists ε(n, v) > 0 such that thefollowing holds. Let (Mn, g, p) satisfy |Ricg | ≤ (n− 1)ε2,Vol(B1(p)) ≥ v > 0 and Mn is (n− 3, ε, 2)-symmetric at p, thenrh(p) ≥ 1.

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Theorem (Cheeger-Naber 2014, Naber-Jiang 2016)

Let (Mnj , gj , pj) be Einstein manifolds with |Ric | ≤ n− 1 and

Vol(B1(pj)) ≥ v such that

(Mnj , gj , pj)

pGH−−−→ (Xn, d∞, p∞), (36)

then the following holds:For every q < 2,

Vol(Tr(Br)) ≤ C(n, v, q)r2q, (37)

where Br ≡ x ∈Mn|r|Rm |(x) ≤ r.(Naber-Jiang) q can be improved to 2.

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Cheeger-Colding’s metric cone theorem works only fornon-collapsed limits, so the symmetry assumption wouldbe very unnatural in the collapsed setting.

In fact, even in the non-collapsed setting, the coneassumption is unnecessary. Such an improvement isuseful in the study of regularity in the collapsed setting.Roughly, in the context of bounded Ricci curvature,non-collapsed limit with Rn−3-splitting in effect impliessmoothness.The above improvement mainly follows from a quantitativedifferentiation argument which is the quantitative version ofCheeger-Colding’s metric cone structure theorem.

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Cheeger-Colding’s metric cone theorem works only fornon-collapsed limits, so the symmetry assumption wouldbe very unnatural in the collapsed setting.In fact, even in the non-collapsed setting, the coneassumption is unnecessary. Such an improvement isuseful in the study of regularity in the collapsed setting.Roughly, in the context of bounded Ricci curvature,non-collapsed limit with Rn−3-splitting in effect impliessmoothness.

The above improvement mainly follows from a quantitativedifferentiation argument which is the quantitative version ofCheeger-Colding’s metric cone structure theorem.

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Cheeger-Colding’s metric cone theorem works only fornon-collapsed limits, so the symmetry assumption wouldbe very unnatural in the collapsed setting.In fact, even in the non-collapsed setting, the coneassumption is unnecessary. Such an improvement isuseful in the study of regularity in the collapsed setting.Roughly, in the context of bounded Ricci curvature,non-collapsed limit with Rn−3-splitting in effect impliessmoothness.The above improvement mainly follows from a quantitativedifferentiation argument which is the quantitative version ofCheeger-Colding’s metric cone structure theorem.

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Given n ≥ 2, v > 0, there exists δ(n, v) > 0, r0(n, v) > 0 s.t. if(Mn, g, p) satisfies |Ricg | ≤ (n− 1)δ2, Vol(B1(p)) ≥ v > 0 and

dGH(B2(p), B2(0n−3, y)) < δ, (0n−3, y) ∈ Rn−3 × Y, (38)

where (Y, y) is a metric space, then rh(p) ≥ r0 > 0.

Some key points in the proof:

DefinitionLet (X, d, p) be a metric space. For α ∈ N, let rα ≡ 2−α > 0.Let δ > 0, rα is called a good scale if X is (0, δ, rα)-symmetric.Otherwise, rα is a bad scale.

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Theorem (Quantitative Metric Cone Structure Theorem)

Let (Mn, g, p) be a Riemannian manifold with Ricg ≥ −(n− 1)and Vol(B1(p)) ≥ v > 0, then for every δ > 0, there existsN(δ, n, v) > 0 such that every x ∈Mn has at most N badscales.

The above theorem immediately implies that for everyδ > 0 and x ∈Mn, there exists 2−N−1 < r < 2−N such thatx is (0, δ, r)-symmetric.We can choose δ > 0 sufficiently small such that thequantitative Rn−3-splitting assumption gives that x is(n− 3, ε1, r)-symmetric, where ε1 > 0 is the constant inCheeger-Naber’s ε-regularity theorem (symmetric version).Then the proof is complete.

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The above theorem immediately implies that for everyδ > 0 and x ∈Mn, there exists 2−N−1 < r < 2−N such thatx is (0, δ, r)-symmetric.

We can choose δ > 0 sufficiently small such that thequantitative Rn−3-splitting assumption gives that x is(n− 3, ε1, r)-symmetric, where ε1 > 0 is the constant inCheeger-Naber’s ε-regularity theorem (symmetric version).Then the proof is complete.

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The above theorem immediately implies that for everyδ > 0 and x ∈Mn, there exists 2−N−1 < r < 2−N such thatx is (0, δ, r)-symmetric.We can choose δ > 0 sufficiently small such that thequantitative Rn−3-splitting assumption gives that x is(n− 3, ε1, r)-symmetric, where ε1 > 0 is the constant inCheeger-Naber’s ε-regularity theorem (symmetric version).Then the proof is complete.

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Almost Flat Manifolds

Theorem (Gromov, 1978)

There exists ε(n) > 0 and w(n) <∞ such that if (Mn, g) is aclosed manifold satisfying

‖ secg ‖C0(Mn) · diam2g(M

n) < ε, (39)

then Mn is finitely covered by a nilmanifold Nn/Γ of order≤ w(n), where Nn is a simply-connected nilpotent Lie groupand Γ ≤ Nn.

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Almost Flat Manifolds

Gromov’s theorem was improved by E. Ruh.

Theorem (Ruh, 1982)

There exists ε(n) > 0 and w(n) <∞ such that if (Mn, g) is aclosed manifold satisfying

‖ secg ‖C0(Mn) · diam2g(M

n) < ε, (40)

then Mn is an infra-nilmanifold. That is, the universal cover Nn

is a simply-connected nilpotent Lie group and

Λ ≡ π1(Mn) ≤ N o Aut(N). (41)

Moreover, [Λ : Λ ∩Nn] ≤ w(n).

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Collapse with Bounded Curvature

Theorem (K. Fukaya, Smooth Limit)

Let (Mnj , gj) satisfy | secgj | ≤ Λ0 and diamgj (M

nj ) ≤ D0 such

that(Mn

j , gj)GH−−→ (Mk

∞, g∞), (42)

where (Mk∞, g∞) is a smooth Riemannian manifold.

Then forany sufficiently large j ≥ J0(n,Λ0, D0), there is a fiber bundlemap fj : Mn

j →Mk∞ with the following properties:

1 For every x ∈Mk∞, diamgj (f

−1j (x))→ 0.

2 For every x ∈Mk∞, ‖ IIf−1

j (x) ‖ ≤ C(n,Λ0, D0).

3 (Gromov and Ruh) Each fiber is diffeomorphic to an(n− k)-dimensional infra-nilmanifold.

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Theorem (K. Fukaya, Smooth Limit)

Let (Mnj , gj) satisfy | secgj | ≤ Λ0 and diamgj (M

nj ) ≤ D0 such

that(Mn

j , gj)GH−−→ (Mk

∞, g∞), (42)

where (Mk∞, g∞) is a smooth Riemannian manifold. Then for

any sufficiently large j ≥ J0(n,Λ0, D0), there is a fiber bundlemap fj : Mn

j →Mk∞ with the following properties:

1 For every x ∈Mk∞, diamgj (f

−1j (x))→ 0.

2 For every x ∈Mk∞, ‖ IIf−1

j (x) ‖ ≤ C(n,Λ0, D0).

3 (Gromov and Ruh) Each fiber is diffeomorphic to an(n− k)-dimensional infra-nilmanifold.

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Theorem (K. Fukaya, General Limit)

Let (Mnj , gj) be a sequence of closed manifolds with

| secgj | ≤ Λ, diamgj (Mnj ) ≤ D (43)

and (Mnj , gj)

GH−−→ (Xk∞, d∞). Then there is a diagram

(F (Mnj ), O(n))

eqGH //

prj

(Y∞, O(n))

pr∞

(Mnj , gj)

GH // (Xk∞, d∞)

(44)

such that Y∞ is a smooth manifold with a C1,α-metric.

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Theorem (Fukaya, General Limit)Moreover, for each sufficiently large j, there is anO(n)-equivariant fiber bundle map

Γ\N → F (Mnj )

Fj−→ Y∞ (45)

with nilpotent fibers, which induces a (singlar) infranil fibration

N ′ →Mnj

Fj−→ Xk∞. (46)

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Theorem (J. Cheeger, 1969)

Given n ≥ 2, v > 0 and D > 0, there exists C(n,D, v) > 0 suchthat the class of closed manifolds (Mn, g) satisfying

| secg | ≤ 1, diamg(Mn) ≤ D, Volg(M

n) ≥ v > 0, (47)

contains finite diffeomorphism types of number bounded byC(n,D, v).

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Theorem (Fukaya)

Let (Mn, g) be a complete Riemannian manifold with| secg | ≤ 1. There exists δ(n) > 0 such that for every x ∈Mn,there is some open neighborhood Bδ(x) ⊂ Ux ⊂ B10δ(x) withthe fiber bundle structure

Dk −→ Ux −→ Nn−k, (48)

where Nn−k is an infranilmanifold.

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Definition (Pure Nilpotent Structure)

A pure nilpotent structure is given by the above O(n)-invariantfibration structure Γ\N → F (Mn)→ Y .

Definition (Mixed Nilpotent Structure)

A mixed nilpotent structure (Oα,Nα) is an atlas on Mn suchthat

each (Oα,Nα) is a pure nilpotent structure(compatibility) If Oα ∩ Oβ 6= ∅, then restricting to Oα ∩ Oβ,(Oα,Nα) is a substructure (Oβ,Nβ) or vice versa.

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Theorem (Cheeger-Fukaya-Gromov, 1992)

There exists v0(n) > 0 such that if (Mn, g) is complete with

| secg | ≤ 1, Volg(B1(x)) < v0, ∀x ∈Mn, (49)

then there is a mixed N -structure of positive rank on Mn andfor every ε > 0 there exists an N -invariant gε nearby g such that

1 e−εg < gε < eεg,2 |∇g −∇gε | < ε,3 |∇kgε Rmgε | ≤ Ck(n, ε), ∀k ∈ N.

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Theorem (Q. Cai - X. Rong, 2009)

If (Mn, g) admits an N -structure of positive rank, then there area family of invariant metrics gε satisfying

| secgε | ≤ 1, InjRadgε(x) ≤ ε, ∀x ∈Mn. (50)

In particular, MinVol(Mn) = 0 and all characteristics of Mn

vanish.

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The Margulis Lemma

Theorem (Margulis)Let G be a connected Lie group and let G0 be its identitycomponent, then there is some open neighborhood

e ∈ Ze ≤ G0 (51)

such that if Γ ≤ G is discrete, then 〈Γ ∩ Ze〉 is nilpotent.

Ze is called the Zassenhaus neighborhood.

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The Margulis Lemma

Theorem (Margulis)

Let (Mn, g) be a complete manifold with −1 ≤ secg ≤ 0, thenthere exists δ(n) > 0 and w(n) > 0 such that for every p ∈Mn,the group Γδ(p) contains a nilpotent subgroup of index ≤ w(n).

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The Margulis Lemma

Theorem (Heintze-Margulis)

Let (Mn, g) be a complete manifold with −1 ≤ secg < 0 andInjRad→ 0, then there exists δ(n) > 0 and p ∈Mn such that

InjRad(p) ≥ δ > 0. (52)

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The Generalized Margulis Lemma

Theorem (Cao-Cheeger-Rong, 2004)

There exists δ(n) > 0 such that the following holds. Let (Mn, g)be a closed manifold with secg ≤ 0 and at some point Ricg < 0.Then for any metric h with | sech | ≤ 1, there is some p ∈Mn

such thatInjRadh(p) ≥ δ > 0. (53)

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The Generalized Margulis Lemma

Theorem (Cheeger-Tian, 2005)

Let (M4, g, p) be a complete Einstein 4-manifold with

Ricg = ±3g (54)

and ˆM4

|Rm |2 ≤ Λ0. (55)

Given any ε > 0, there exists δ(Λ0, ε) > 0 such that

Volx ∈M4| InjRad(x) ≥ δVol(M4)

≥ 1− ε. (56)

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Collapse with Locally Bounded Curvature

DefinitionWe say U ⊂ (Mn, g) is v0-collapsed with locally boundedcurvature if

Vol(Br|Rm |(p)) ≤ v0 · (r|Rm |(p))n (57)

for all p ∈ U .

Theorem (Cheeger-Tian, 2005)

Let (Mn, g) be Einstein, then there exists v0(n) > 0 such that ifU ⊂Mn is v0-collapsed with locally bounded curvature, then Uadmits an N -structure of positive rank.

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Example: Codimension-1 Collapse

Let (T3, g0) be a 3-dimensional flat torus with diam(T3) = 1.Then we perform a surgery by M. Anderson:

Remove a small ball Bε(p) ⊂ T3 with Bε(p) ∼= D3.Take the metric product N4 ≡ S1

ε × (T3 \Bε(p)) with∂N4 = S1

ε × S2ε .

Let (R2 × S2ε , gS) be the Schwarzschild space with

RicgS ≡ 0. Choose Oε ⊂ R2 × S2ε with ∂Oε = S1

ε × S2ε .

Attach Oε on N4, after smoothing, then the resultingmanifold (M4, gε) is closed with |Ricgε | ≤ 3 and| secgε | → +∞ as ε→ 0.

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Remove a small ball Bε(p) ⊂ T3 with Bε(p) ∼= D3.

Take the metric product N4 ≡ S1ε × (T3 \Bε(p)) with

∂N4 = S1ε × S2

ε .



ε × S2ε .


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Remove a small ball Bε(p) ⊂ T3 with Bε(p) ∼= D3.Take the metric product N4 ≡ S1

ε × (T3 \Bε(p)) with∂N4 = S1

ε × S2ε .



ε × S2ε .


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Example: Codimension-1 Collapse of a K3 Surface

L. Foscolo constructed a family of hyperkahler metrics gε on K3

(K3, gε)GH−−→ T3/Z2 (58)

with a punctured subset

T∗ ≡ T3 \ q1, . . . , q8, p1, τ(p1), . . . , pn, τ(pn) (59)

where n satisfies some “balancing condition” such that:

There is a non-trivial S1-fibration over T∗.Curvatures blow up around the punctures but uniformlybounded away from the punctures.Each bubble is an ALF gravitational instanton.

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(K3, gε)GH−−→ T3/Z2 (58)


T∗ ≡ T3 \ q1, . . . , q8, p1, τ(p1), . . . , pn, τ(pn) (59)


There is a non-trivial S1-fibration over T∗.

Curvatures blow up around the punctures but uniformlybounded away from the punctures.Each bubble is an ALF gravitational instanton.

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(K3, gε)GH−−→ T3/Z2 (58)


T∗ ≡ T3 \ q1, . . . , q8, p1, τ(p1), . . . , pn, τ(pn) (59)


There is a non-trivial S1-fibration over T∗.Curvatures blow up around the punctures but uniformlybounded away from the punctures.

Each bubble is an ALF gravitational instanton.

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(K3, gε)GH−−→ T3/Z2 (58)


T∗ ≡ T3 \ q1, . . . , q8, p1, τ(p1), . . . , pn, τ(pn) (59)


There is a non-trivial S1-fibration over T∗.Curvatures blow up around the punctures but uniformlybounded away from the punctures.Each bubble is an ALF gravitational instanton.

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Due to Gross-Wilson, there are a family of hyperkahler metricsgε on a K3 surface with (K3, gε)

GH−−→ (S2, d∞):

Singular Fibration Structure: There is a holomorphicfibration, fε : K3 −→ S2 with a finite set S = qα24

α=1 suchthat f−1

ε (q)homeo∼= T2, q ∈ S2 \ S,

f−1ε (q)

homeo∼= I1, q ∈ S.(60)

Geometric Structure:

the tangent cone Tq(S2) is isometric to R2, ∀q ∈ S2.‖Rmgε ‖ blows up along each I1, and ‖Rmgε ‖ are boundedalong T2.Each bubble is isometric to the Taub-NUT metric.

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GH−−→ (S2, d∞):


α=1 suchthat f−1

ε (q)homeo∼= T2, q ∈ S2 \ S,

f−1ε (q)

homeo∼= I1, q ∈ S.(60)


the tangent cone Tq(S2) is isometric to R2, ∀q ∈ S2.

‖Rmgε ‖ blows up along each I1, and ‖Rmgε ‖ are boundedalong T2.Each bubble is isometric to the Taub-NUT metric.

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GH−−→ (S2, d∞):


α=1 suchthat f−1

ε (q)homeo∼= T2, q ∈ S2 \ S,

f−1ε (q)

homeo∼= I1, q ∈ S.(60)


the tangent cone Tq(S2) is isometric to R2, ∀q ∈ S2.‖Rmgε ‖ blows up along each I1, and ‖Rmgε ‖ are boundedalong T2.

Each bubble is isometric to the Taub-NUT metric.

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GH−−→ (S2, d∞):


α=1 suchthat f−1

ε (q)homeo∼= T2, q ∈ S2 \ S,

f−1ε (q)

homeo∼= I1, q ∈ S.(60)


the tangent cone Tq(S2) is isometric to R2, ∀q ∈ S2.‖Rmgε ‖ blows up along each I1, and ‖Rmgε ‖ are boundedalong T2.Each bubble is isometric to the Taub-NUT metric.

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Example: Codimension-3 Collapse of a K3-Surface

G. Chen-X. Chen constructed a family of hyperkahler metrics gεon a K3 surface which collapse to a closed interval,

(K3, gε)GH−−→ ([0, 1], dt2) (61)

such thatthere is a T3-fibration over the interval (10−4, 1− 10−4),curvatures blow up around the two ends and curvaturesare uniformly bounded in the interior.Each bubble X is an ALH gravitational instanton with

ˆX|Rm |2 = 96π2. (62)

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(K3, gε)GH−−→ ([0, 1], dt2) (61)

such that

there is a T3-fibration over the interval (10−4, 1− 10−4),curvatures blow up around the two ends and curvaturesare uniformly bounded in the interior.Each bubble X is an ALH gravitational instanton with

ˆX|Rm |2 = 96π2. (62)

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(K3, gε)GH−−→ ([0, 1], dt2) (61)

such thatthere is a T3-fibration over the interval (10−4, 1− 10−4),

curvatures blow up around the two ends and curvaturesare uniformly bounded in the interior.Each bubble X is an ALH gravitational instanton with

ˆX|Rm |2 = 96π2. (62)

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(K3, gε)GH−−→ ([0, 1], dt2) (61)

such thatthere is a T3-fibration over the interval (10−4, 1− 10−4),curvatures blow up around the two ends and curvaturesare uniformly bounded in the interior.

Each bubble X is an ALH gravitational instanton withˆX|Rm |2 = 96π2. (62)

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(K3, gε)GH−−→ ([0, 1], dt2) (61)

such thatthere is a T3-fibration over the interval (10−4, 1− 10−4),curvatures blow up around the two ends and curvaturesare uniformly bounded in the interior.Each bubble X is an ALH gravitational instanton with

ˆX|Rm |2 = 96π2. (62)

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Quantitative Splitting Theorem

Without assuming bounded curvature, a priori there is nofibration structure or N -structure.Let Ricg ≥ −(n− 1).

1 Cheeger and Colding discovered a replacement of thefibration map which controls the collapsing geometry “in theL2 sense”.

2 In the case of | secg | ≤ 1, the N -structure contains all thecollapsing information. In general, the collapsed informationat the level of the fundamental group is controlled by theGeneralized Margulis Lemma (Kapovitch-Wilking).

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Without assuming bounded curvature, a priori there is nofibration structure or N -structure.

Let Ricg ≥ −(n− 1).1 Cheeger and Colding discovered a replacement of the

fibration map which controls the collapsing geometry “in theL2 sense”.


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Without assuming bounded curvature, a priori there is nofibration structure or N -structure.Let Ricg ≥ −(n− 1).

1 Cheeger and Colding discovered a replacement of thefibration map which controls the collapsing geometry “in theL2 sense”.


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Let (Mnj , gj , pj) be a sequence of manifolds with

Ricgj ≥ −(n− 1)δ2j such that

(Mnj , gj , pj)

GH−−→ (X∞, d∞, p∞). (63)

If X∞ admits a line, then X∞ ≡ Rk × Y∞ and Y∞ does notadmit any line.

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Definition (Cheeger-Colding’s ε-splitting map)

An ε-splitting map Φ ≡ (u(1), . . . , u(k)) : Br(p)→ Rk is aharmonic map (i.e. ∆u(α) = 0) such that

k∑α,β=1

Br(p)

|〈∇u(α),∇u(β)〉 − δαβ|+ Br(p)

|∇2u(α)|2 < ε. (64)

The above gradient and Hessian estimates amount to the“Toponogov Theorem” in the L2 sense.There is some Ψ(ε|n, r) > 0 such that

‖dt(p)− dt(p)‖L2 + ‖∠t − ∠t‖L2 < Ψ. (65)

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∀ ε > 0, n ≥ 2, r > 0, ∃ δ(n, ε, r) > 0 such that1 if Ricg ≥ −(n− 1)δ2 and there is an ε-splitting map

Φ ≡ (u1, . . . , uk) : B4r(p)→ Rk, then

dGH(Br(p), Br(0k, x)) < εr, Br(0

k, x) ⊂ Rk ×X, (66)

for some complete length space (X, d).2 if

Ricg ≥ −(n− 1)δ2

dGH

(Bδ−1(p), Bδ−1(0k, x)

)< δ,

(67)

then there is an ε-splitting map Φ : B4r(p)→ Rk.

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Theorem (Cheeger-Colding 1996, Cheeger-Colding-Tian 2002)

The ε-splitting map Φ ≡ (u(1), . . . , u(k)) : B4R(p)→ Rk satisfies:1 Vol(BR(0k) \ u(BR(p))) < Ψ(ε|n,R).2 Let ω` ≡ du1 ∧ . . . ∧ du` for 1 ≤ ` ≤ k,

BR(p)

|Ric(∇u(α),∇u(α))|+ BR(p)

∣∣∣|ω`| − 1∣∣∣ < Ψ(ε|n,R).

(68)

Cheeger-Colding-Tian proved the fibers are almost totallygeodesic in the L2-sense.

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THANK YOU!

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introduction to ricci curvature and the convergence theory

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