Universit`
a degli Studi di Pisa
FACOLT `A DI MATEMATICA Corso di Laurea in MatematicaTesi di laurea magistrale
Limits of Riemannian manifolds with Ricci curvature bounded
from below
Candidato
Gioacchino Antonelli
Relatori
Prof. Luigi Ambrosio Dott. Luciano Mari
Controrelatore
Contents
Introduction 1
1 Tools and Notation 5
1.1 Bochner formula . . . 6
1.2 Segment inequality . . . 6
1.3 Poincar´e inequality on manifolds. . . 7
1.4 Cheng-Yau Gradient estimate . . . 8
1.5 Comparison functions . . . 8
1.6 Maximum principle . . . 12
1.7 A function with uniformly bounded Laplacian on Riemannian manifolds with Ricci bounded below . . . 12
1.8 Gromov-Hausdorff distance and convergence . . . 13
1.9 Hausdorff measures . . . 21
2 The Almost Splitting Theorem 23 2.1 Statement of the Theorem . . . 23
2.2 Outline of the proof . . . 24
2.3 The harmonic replacement of the shifted distance function . . . 26
2.4 The gradient and the Hessian of β+ . . . . 31
2.5 Approximate Pythagorean Theorem . . . 34
2.6 Conclusion of the proof . . . 43
3 Preliminaries on Ricci-limit spaces 51 3.1 Preliminary definitions and observations . . . 51
3.2 Statement of some structure results with recent developments and main theorem . . . 57
4 GH-close and Volume-close balls 61 4.1 Statement: GH-close ⇒ Volume-close . . . 61
4.2 Outline of the proof . . . 61
4.3 Statement: Volume-close ⇒ GH-close . . . 68
5 The Singular Set of the Limit Space: Non-Collapsed case 69 5.1 First step: S = Sn−1 . . . 69
5.2 Second step: every tangent cone is a metric cone. . . 70
5.3 Third step: dimHSk ≤ k . . . 72
INTRODUCTION
The aim of this thesis is the study of some structure properties of the so-called Ricci-limit spaces. In this introduction we will give a brief historical account of the subject and then we will pass to analyze the content of every chapter.
The main ingredient for the definition of a Ricci-limit space is the the notion of pointed Gromov-Hausdorff convergence, introduced by Gromov in a paper which dates back to 1981 [25]: there, in particular, this tool was used to prove that a group with polinomial growth admits a nilpotent subgroup of finite index.
Immediately after this definition was given, it was understood that this tool could have been used in many other different situation, e.g. to take the limit of pointed complete Riemannian manifolds. Indeed, from the compactness criterion already presented in [25] and the Bishop-Gromov inequality, it follows that a sequence of complete pointed Riemannian n-dimensional manifolds (Mi, mi) which satisfy the uniform lower bound
RicMi ≥ −(n − 1) (0.0.1)
admits a subsequence which converges, in the pointed Gromov-Hausdorff sense, to some pointed metric space (Y, y). These limit points are the so-called Ricci-limit spaces.
It was soon understood that the limit is not necessarily a Riemannian manifold. Nevertheless, it is predictable that the limit could have good geometric properties: the aim of studying such properties led Cheeger and Colding to publish a series of papers [10, 11, 12, 13] dealing with such spaces. In this study they made a great use of tools and estimates coming from Riemannian Geometry.
One of the most important topics studied by them is the analysis of tangent spaces. Indeed, by using the pointed Gromov-Hausdorff convergence, it is possible to define a notion of tangent space to a boundedly compact metric space X at one of its points x. The first thing one can expect is that almost every point of a Ricci-limit space should have a tangent space which is isometric to some Euclidean space. This appears to be true if one takes as measure the limit (which appears not to be unique), in the appropriate sense, of the renormalized volume measures on the complete Riemannian manifolds [11]. The next question naturally arises: what is the Hausdorff dimension of the singular points S, i.e. the points where the Ricci-limit space does not have a unique tangent space which is isometric to some Euclidean space? In [11] the answer to the question is given in a particular case: if we assume the so-called non-collapsed condition, i.e. the existence of some v > 0 such that
vol(B1(mi)) ≥ v ∀i ≥ 1, (0.0.2)
then
In the non-collapsed case it appears also that every limit measure constructed on the Ricci-limit space is, up to constants, the Hausdorff n-dimensional measure Hn [11].
The most important tools to show (0.0.3) are the almost splitting theorem and the fact that every tangent space, in the non-collapsed case, is a metric cone. The first result, whose proof is in [10], is a generalization of a well-known theorem in Riemannian Ge-ometry, i.e. the splitting theorem, shown in [14, 20]. According to the splitting theorem, a complete Riemannian n-dimensional manifold M with RicM ≥ 0 that contains a line, splits isometrically as R × N, where N is a complete Riemannian (n − 1)-dimensional manifold with RicN ≥ 0; the almost splitting theorem implies an analogous on Ricci-limit spaces Y whose bounds on Ricci curvature converge to zero: in particular if such a space contains a line, then it splits isometrically as R × X, with X a length space. The second result, i.e. the fact that in the non-collapsed case every tangent space is a metric cone, whose proof is provided in [10,11], gives a rigidity to the structure of tangent spaces which makes possible to perform a dimension reduction argument: in particular this rigidity is important to show that
dimHSk ≤ k (0.0.4)
where Sk is the set of singular points, in the Ricci-limit space, for which no tangent cone splits off a factor Rk+1 isometrically.
Historically, a deeper understanding of what happens in the collapsed case came later with the work of Colding and Naber [17] in which it is shown that the dimension of the tangent space is the same almost everywhere according to every limit measure. This was already known from [11] in the non-collapsed case: indeed, in this case, the dimension is Hn-almost everywhere equal to n.
In very recent times the study of this subject evolved a lot: the aim of obtaining more synthetic proofs of the results known for Ricci-limit spaces and, for example, of better understanding compactness properties under pointed Gromov-Hausdorff convergence, led to the idea of developing a synthetic theory about spaces with Ricci curvature bounded from below which works for metric measure spaces. An overview of this recent field of study is in [2]. Three seminal papers, in this sense, are [31], [41] and [42] in which the authors independently gave the definition of the condition CD(K, N ), which is to say the synthetic version of Ric ≥ K, dim ≤ N , where K ∈ R and 1 ≤ N ≤ ∞. In the last years, in particular, a huge literature is expanding around the class of RCD(K, N ) and RCD?(K, N ) spaces [2].
The introductory chapter 1 contains the tools which will be very useful throughout the thesis.
The first part is more geometric: the Bochner formula, the Poincar´e inequality on manifolds, the Cheng-Yau gradient estimate and the Volume comparison, also known as the Bishop-Gromov inequality, are recalled. A key inequality is also stated, the famous segment inequality shown by Cheeger and Colding in [10], which is a fundamental tool in the proof of the almost splitting theorem. In this part we define also the so-called comparison functions which are used to construct functions whose Laplacian is greater (or smaller) than 1 on a complete Riemannian n-dimensional manifold M with RicM ≥ −(n − 1)k. The existence of such functions, combined with the maximum-minimum principle in the weak sense due to Calabi [8], which is recalled in this first part, is very useful in some estimates.
conver-Contents
gence, with particular attention to the realization of the convergence. We state also the famous compactness criterion due to Gromov and show some lemmas about the commu-tation of Gromov-Hausdorff distance and products and the transmission of the inequality dGH(BL(p), BL(0)) ≤ δ on small scales.
In the third part we recall some definitions and basic results about Hausdorff measures. Chapter 2 is a detailed proof of the splitting theorem for limit spaces, shown by following the line presented in [9] and filling the details. The general idea of the proof, which is rather long, is presented in section 2.2. The first step is to reduce the statement to a purely Riemannian one which passes to the limit: this result is a sort of almost splitting theorem. The key step to show this result is to prove an approximate Pythagorean theorem which in turn is sufficient to conclude that some ball in the Riemannian manifold is GH-close to suitable ball in some product metric space.
In order to show this approximate Pythagorean theorem it is useful an estimate which goes back to Abresch and Gromoll [1] together with all the tools presented in the first part of chapter 1. Here the underlying idea is to obtain an estimate regarding lengths passing through estimates regarding the gradient and the Hessian of the harmonic replacement of a suitable modification of a distance function.
In section 2.6, after having recovered the original result by passing to the limit the Riemannian one, the iteration of such splitting procedure is discussed. Indeed, while it is readily seen that the classical splitting theorem can be iterated, it is not clear whether it is the case for the splitting theorem for limit spaces. In this section we show in an elementary way that, if we know a priori that the line is, under the isometry, a fiber R × {x}, which is true when the line in the limit space is a limit line, then we can iterate the procedure. The general case can be fixed by making use of the recent result due to Gigli [22], which is a generalization of the splitting theorem to the context of RCD(0, N ) spaces.
In chapter 3 we introduce some preliminary definitions and observations about Ricci-limit spaces.
In the first part we describe how it is possible to construct tangents to these spaces and show the existence of at least one tangent space at each point. After having noticed that each iterated tangent space is itself a Ricci-limit space, we notice that, by using an argument which relies on Bishop-Gromov inequality, the dimension of a Ricci-limit space, as well as the dimension of any iterated tangent space, cannot exceed n. We also present the construction of the limit measures on a Ricci-limit space and notice that these measures satisfies an inequality similar to the Bishop-Gromov one. In particular we obtain that we can equip each iterated tangent space with a 2n-doubling measure.
In the second part we briefly state some of the results which are presented in [11] with recent developments in the non-smooth context. This is the case, for example, of the generalization of Colding’s volume convergence [16] to the case of Ricci-limit spaces, contained in [11], and the so-called dimensional gap, according to which in the collapsed case the Hausdorff dimension of the limit decreases at least by 1 [11]. Recent developments about these results are contained in [19,37, 6, 15, 30].
In chapter 4 we briefly discuss the following question: is it true that in a complete Riemannian n-dimensional manifold with almost-nonnegative Ricci, a ball of radius R is GH-close to a ball of radius R in Rn if and only if it is volume-close? The answer is yes and we only give an outline of the proof of the fact that being GH-close implies being
volume-close.
The key idea is explained at the beginning of section 4.2 and it consists in improving the estimates obtained in chapter 2 in order to construct an harmonic chart which is almost an isometry on the ball of the n-dimensional Riemannian manifold.
In the final chapter 5 we analyze the steps of the proof of (0.0.3). We divide it in four steps, some of which are only stated: in particular the fact that, in the non-collapsed case, S = Sn−1 and Sn−1\ Sn−2 = ∅. We give a brief discussion of some geometric properties regarding metric cones which in turn are useful to give a detailed proof of the fact that
dimHSk≤ k. (0.0.5)
The proof of this fact strongly relies on the splitting theorem for limit spaces and on the fact that, in the non-collapsed case, every tangent space is a metric cone: of this last fact we give also a precise statement.
Ringraziamenti: Desidero ringraziare vivamente i miei due relatori prof. Luigi Am-brosio e prof. Luciano Mari. Al prof. Luigi AmAm-brosio vanno i miei ringraziamenti per i moltissimi consigli che mi ha dato in questi anni, sia per il lavoro della tesi magistrale che per quello della tesi triennale, e per l’enorme disponibilit`a con cui condivide il suo sapere matematico. Al prof. Luciano Mari vanno i miei ringraziamenti per la grandissima pas-sione, instancabile dedizione e disponibilit`a con cui ha seguito questo lavoro, e anche per il clima sereno e informale in cui ci siamo trovati a lavorare. Senza di loro, sicuramente, questo lavoro non sarebbe potuto esistere.
Un grazie a tutti coloro che mi hanno sostenuto in questo percorso universitario, specialmente ai miei amici, quelli delle giornate pisane e delle serate terlizzesi. Senza di loro sarebbe sicuramente stato tutto pi`u difficile: sono loro che riescono sempre a risollevarmi il morale e a fornirmi continuamente nuovi spunti di riflessione. Un grazie alla mia famiglia perch´e `e sempre presente e infine a chi sa che, con un solo sorriso, riesce a farmi sembrare meno pesante tutto.
CHAPTER 1
TOOLS AND NOTATION
In this initial chapter we will list some results which will be useful later on. In the first part we will deal with facts concerning Riemannian geometry, in the second we will explore properties of Gromov-Hausdorff convergence and in the third we will remind some basic facts about Hausdorff measures.
Below there is the list of some of the symbols we will use throughout the thesis.
(M, h·, ·i) A generic (connected) smooth manifold M of dimension n with complete Riemannian metric h·, ·i
BR(p) Ball (open or closed) of radius R and center p in (M, h·, ·i)
Hn n-dimensional Hausdorff measure
Hn
s n-dimensional spherical Hausdorff measure
Hess Hessian on a Riemannian manifold Ric Ricci tensor on a Riemannian manifold d vol Volume form on a Riemannian manifold
vol(A) Volume of the set A on a Riemannian manifold
vol(∂A) Hn−1 measure of the boundary of A on a Riemannian manifold
Mkn Model space of dimension n and sectional curvature k snk A function depending on k defined in(1.5.2)
∆ Laplacian in the model space Mn k
Vk(r) Volume of a ball of radius r in the model space Mkn
vk(r) Hn−1 measure of the boundary of a ball of radius r
in the model space Mn k
dX Distance on X
BdRX(x) Ball (open or closed) of radius R and center x in some metric space (X, dX)
ϕk(ρ, l) Decreasing comparison function defined in(1.5.5)
dGH Gromov-Hausdorff distance or pointed
Gromov-Hausdorff distance defined in(1.8.1) and (1.8.8) dH Hausdorff distance between (compact) sets
CovX(ε) Minimum number of ε-balls it takes to cover X
CapX(ε) Maximum number of 2ε disjoint balls in X BR(p) Closed ball of center p ∈ Rn and radius R
BR((a, x)) Closed ball of radius R and center (a, x) in some product Rk× X with the Hilbertian metric
Ψ(ε1, . . . , εk|c1, . . . , cn) A positive quantity which depends on positive ε1, . . . , εk and
c1, . . . , cN and which goes to zero as ε1+ · · · + εk→ 0
while c1, . . . , cN are fixed
SBR(p) Unit tangent vector bundle of BR(p)
(C(X), vX, d
c) Metric cone (with vertex vX and metric dc)
1.1. Bochner formula
The following identity, known as the Bochner formula will be very useful. It can be found, for example, in [9, page 19].
Formula 1.1.1. [Bochner formula] Let (M, h·, ·i) be a Riemannian manifold. Let f : M → R be a smooth function. Then
1
2∆|∇f |
2 = |Hess f |2+ h∇f, ∇∆f i + Ric(∇f, ∇f ). (1.1.1) For more discussions about this formula, one can also read [39, chapter 9].
1.2. Segment inequality
The following proposition, known as segment inequality, has been proved firstly in the paper [10]. It is useful for the proof of the almost splitting theorem, which is the reason why Cheeger and Colding proved this result, but it is also interesting per se. For ex-ample Haj lasz and Koskela in [27] investigated how this segment inequality with other hypotheses, which are satisfied in the Riemannian case with Ricci bounded from below for example, give a Sobolev-type inequality. One can find a rigorous discussion of this tool also in [39, pages 285-293].
Proposition 1.2.1 (Segment inequality). Let (M, h·, ·i) be a complete Riemannian man-ifold of dimension n, let k ≥0 be a real number and suppose RicM ≥ −(n − 1)k.
Let f : M → [ 0, +∞ ) be an integrable function and let A, B, W Borel subsets of M such that
• A ⊂ W , B ⊂ W ;
• There exists D ≥ 0 such that d(x, y) ≤ D for each x ∈ A and y ∈ B;
• For each x ∈ A and y ∈ B you have selected a segment cx,y : [ 0, 1 ] → M which connects x and y and which is entirely contained in W .
Then there exists a constant C = C(n, k, D) such that. ˆ
A×B ˆ 1
0
f(cx,y(t)) dt d volx d voly ≤ C(vol A + vol B) ˆ
W
fd vol . (1.2.1) One can iterate this segment inequality and obtain the so called iterated segment inequality. Once we have fixed segments cx,y, we can use the following notation
Ff(x, y)=. ˆ 1
0
f(cx,y(t)) dt. (1.2.2)
Then, having fixed z ∈ M and having chosen a segment from z and (almost every) w ∈ W, we can apply (1.2.1) with Ff(z, ·) instead of f , provided that the same hypotheses of Proposition 1.2.1 (Segment inequality) are satisfied. In this way, being cz,cx,y(t)(s) :
[ 0, 1 ] → M the segment form z and cx,y(t) with t fixed, one has ˆ A×B ˆ 1 0 ˆ 1 0
f(cz,cx,y(t)(s)) ds dt d volx d voly ≤ C(vol A + vol B)
ˆ W
Ff(z, x) d volx. (1.2.3) For an explicit value of the constant C, one can see [39, page 285].
1.3. Poincar´e inequality on manifolds
Remark 1.2.2. In this remark we mean with measure the volume measure on the manifold. It is known that, given p ∈ M , the cut locus of p is a closed set of measure zero. For the defintion and the basic properties of the cut locus one may read [39, pages 215-220] while the paper in which it is shown that dimHcut(p) ≤ n − 1 is [28]. A good reference for properties of cut(p) is also [36]. It is worth noticing here that some properties of the distance function from a point that are useful in the development of Cheeger-Colding theory are also contained in [5, Chapter 2].
From the properties of the cut locus, it is true that the subset E of pairs (x, y) ∈ A×B for which y /∈ cut(x) has full measure in A × B and, for such pairs of points, there exists only one segment connecting x to y. Then, to be more precise and avoid measurability issues, we may substitute the third hypothesis of Proposition 1.2.1(Segment inequality) with the request that every segment joining x ∈ A to y ∈ B is entirely contained in W and we may consider the integral in (1.2.1) on E rather than on A × B. For a rigorous statement in this direction, one can look also [5, Proposition 5.4].
We also have to be careful, in the same way, for the iterated segment inequality, also defining Ff(z, ·) on W \ cut(z) which has full measure in W .
1.3. Poincar´
e inequality on manifolds
From theProposition 1.2.1 (Segment inequality), with some work, one can deduce some Sobolev-type inequality. Let us fix the notation.
Given u : M → [ 0, +∞ ) a smooth function and A ⊂ M , we will denote uA
. =
A
ud vol . (1.3.1)
Given k ∈ [ 0, +∞ ), we will also denote kukk . = M |u|k 1k , kukk,A . = A |u|k 1k . (1.3.2)
By making use of classical results about the maximal function, and using a very clever iteration argument, one may show the following, whose proof is very well explained in [39]:
Proposition 1.3.1. Let us assume (M, h·, ·i) is a compact Riemannian manifold of di-mension n and there exists k ≥ 0 such that RicM ≥ −(n − 1)k. Let us assume that M has diameter R.
Given u : M → R a smooth function, there exists C = C(n, k, R) such that.
ku − uMkn−1n ≤ CRk∇uk1. (1.3.3)
Moreover C is uniformly bounded from above as R →0.
By making some additional work and using some ideas which can be found in [27], one can obtain the following local version of the previous proposition, which is stated in [39].
Theorem 1.3.2. Let us assume (M, h·, ·i) is a complete Riemannian manifold of dimen-sion n, and there exists k ≥0 such that RicM ≥ −(n − 1)k.
Given u : M → R a smooth function and D > 0 a real number, we have that there exists a constant C = C(n, k, D) such that, for all R ≤ D,.
ku − uBR(x)kn−1n ,BR(x) ≤ CRk∇uk1,BR(x). (1.3.4)
1.4. Cheng-Yau Gradient estimate
The following estimate is very important in our discussion. It is known as Cheng-Yau gradient estimate. It can be found, for example, in [40, pages 17-23], but also in [32, pages 151-155]. In these references the estimate is given for the case of an harmonic positive function. A version of the Cheng-Yau gradient estimate for the heat equation is in [5, Chapter 9] and related topics can be found also in [34]. We will give more general statement, though not quantitative, which can be found in [9, pages 39-41].
Proposition 1.4.1 (Cheng-Yau gradient estimate). Let (M, h·, ·i) be a complete Rieman-nian manifold of dimension n, let k ≥0 be a real number and suppose RicM ≥ −(n − 1)k. Let p ∈ M , R2 > R1 > 0 be real numbers and u : BR2(p) → ( 0, +∞ ) a smooth
function defined on the open ball BR2(p) of center p and radius R2. Let K : R → R be a
C∞ function and let u satisfy
∆u = K(u) on BR2(p). (1.4.1)
If the function F(x) = e. −xK(ex) satisfies F0(x) ≤ 0, then there exists a constant C = C(n, R1. , R2, k) for which one has the following estimate on the ball BR1(p)
|∇u|2
u2 ≤ max 2u −1
K(u), C + 2u−1K(u) − 2K0(u) . (1.4.2) It has to be noticed that Proposition 1.4.1 (Cheng-Yau gradient estimate) can be used for equations ∆u = k, where k > 0 is a real number, or also ∆u = λu where λ is a real number. Also it is important to remind that, as R1 approaches to R2, the constant C(n, R1, R2, k) goes to infinity.
1.5. Comparison functions
We will denote, from now on, the standard model space of dimension n and sectional curvature k with Mn
k. In geodesic polar coordinates the metric on this space is given by h·, ·i = dr2 + snk(r)2h·, ·i Sn−1 (1.5.1) where snk(r) is 1 √ ksin √ kr if k > 0, r if k = 0, 1 √ −k sinh √ −kr if k < 0. (1.5.2)
From now on we will denote by vk(r) the Hn−1 measure of ∂Br(¯p), with ¯p ∈ Mn k, and with Vk(r) the volume of Br(¯p). It can be shown that
1.5. Comparison functions
where σn−1 is the area of the unit sphere in Rn, and Vk(r) = ˆ r 0 vk(s) ds = σn−1 ˆ r 0 snn−1k (s) ds. (1.5.4) Following the notation of [1, page 360], we can introduce some comparison functions, which is to say special functions with constant Laplacian in the model space. We will denote ϕk(ρ, l)=. ˆ ˆ ρ≤t≤τ ≤l snk(τ ) snk(t) n−1 dτ dt = ˆ l ρ Vk(l) − Vk(t) vk(t) dt. (1.5.5) This function is well defined for 0 < ρ ≤ l, provided kl < π2. This means, for example, that if k ≤ 0 this function is defined for all 0 < ρ ≤ l < +∞. Once l is fixed, the function is decreasing and convex in ρ.
If we fix p ∈ Mn
k it can be showed, in the case kl < π2, that considering the function h(q)= ϕk(d(p, q), l) defined on the punctured closed ball Bl(p) \ {p}, one has, by making. computations,
(
∆h(q) = 1 on the punctured ball Bl(p) \ {p},
h(q) = 0, ∇h(q) = 0 on ∂Bl(p), (1.5.6)
where, with ∆, we mean the Laplacian in the model space Mn k.
It can be also produced a comparison function with constant Laplacian 1 which is increasing. We will denote
θk(ρ)=. ˆ ˆ 0≤t≤τ ≤ρ snk(τ ) snk(t) n−1 dτ dt = ˆ ρ 0 Vk(t) vk(t) dt. (1.5.7) This function is well defined for kρ < π2. It is also increasing in ρ.
If we fix p ∈ Mn
k and set g(q) = θk(d(p, q)) it can be shown that
∆g(q) = 1 on the ball Bρ(p) (1.5.8)
provided that kρ < π2.
These functions are very important because they can give us functions which has Laplacian greater or lesser than 1 in Riemannian manifolds with Ricci bounded from below, just by using Laplacian comparison. For the sake of completeness, let us remind below what is the Laplacian comparison, following [9, page 23].
Theorem 1.5.1 (Laplacian comparison). Let (M, h·, ·i) be a complete Riemannian man-ifold of dimension n, let k be a real number and suppose RicM ≥ (n − 1)k.
Let p ∈ M and let r(q)= d(p, q). Let ∆ be the Laplacian in the model space M. n k and f : M → R be a smooth function. Then, either in the barrier sense or in the distribution sense, it holds that
(
∆f (r) ≤ ∆f (r) if f0 ≥ 0,
∆f (r) ≥ ∆f (r) if f0 ≤ 0. (1.5.9)
Details on what barrier sense means can be found, for example, in [9, pages 30-31]. One can also read [36]. When we say distribution sense we mean that, testing(1.5.9)with every smooth positive compactly supported function, we have the inequality. Another good reference for this theorem is [5, Chapter 3].
We will now state a lemma which will be very useful later on and which is a conse-quence of Theorem 1.5.1 (Laplacian comparison).
Lemma 1.5.2. Let (M, h·, ·i) be a complete Riemannian manifold of dimension n, let k ≥0 be a real number and suppose RicM ≥ −(n − 1)k.
Let R > 0 and p, q ∈ M such that d(p, q) > R. Let r(x) = d(p, x). Then, either in. the barrier sense or in the distribution sense,
∆r ≤ (n − 1) 1 d(p, q) − R + √ k on the ball BR(q). (1.5.10) Proof. It is a direct consequence of Theorem 1.5.1 (Laplacian comparison) and triangle inequality if k = 0 and a consequence of Theorem 1.5.1 (Laplacian comparison) and the estimate √krcoth√kr ≤1 +√kr if k > 0.
The Theorem 1.5.1 (Laplacian comparison) and the discussion at the start of Sec-tion 1.5(Comparison functions) give the following:
Proposition 1.5.3. Let (M, h·, ·i) be a complete Riemannian manifold of dimension n, let k ≥0 be a real number and suppose RicM ≥ −(n − 1)k.
For fixed l > 0, let ϕ−k(ρ, l) as in (1.5.5). Then, fixed p ∈ M , being h(q) =. ϕ−k(d(p, q), l), one has that, either in the barrier sense or in the distribution sense, ∆h(q) ≥ 1 on the ball Bl(p). (1.5.11) Analogously let θ−k(ρ) as in (1.5.7). Then, fixed p ∈ M , being g(q)
.
= θ−k(d(p, q)), one has that, either in the barrier sense or in the distribution sense,
∆g(q) ≤ 1 on M. (1.5.12)
These functions are useful, for example, to prove some quantitative maximum prin-ciples on manifolds with Ricci bounded from below. These quantitative results are, for example, in [9, Chapter 8].
A part of the following estimate will be very useful in the following chapter, during the proof of the almost splitting theorem. Also it gives a general way to estimate the total mass of the Laplacian of the distance function, so it is worth mentioning it.
Lemma 1.5.4. Let (M, h·, ·i) be a complete Riemannian manifold of dimension n. Let us assume RicM ≥ −(n − 1)k for some k ≥ 0.
Let p ∈ M be a point and consider r(x) = d(p, x). One can define the signed finite. Radon measure ∆r on some fixed ball BR(q) with q ∈ M by means of the action on ϕ ∈ Cc∞(BR(q))), by declaring
h∆r, ϕi = −. ˆ
BR(q)
h∇r, ∇ϕi d vol . (1.5.13)
Consider |∆r| and (∆r)+ the total mass and the positive part of such a signed Radon measure which are defined, for example, in [3, page 3]. Then, in the sense of measures on BR(q), ∆r ≤ (∆r)+ ≤ (n − 1)sn 0 k(r) snk(r)d vol, |∆r| ≤ 2(n − 1)sn 0 k(r) snk(r)d vol −∆r. (1.5.14)
1.5. Comparison functions
Proof. In [36] it has been remarked that there exists a set U ⊆ cut(p) with Hn−1(cut(p) \ U) = 0 and such that, for every q ∈ U , there exist exactly two segments from p to q. If we denote with ∇r+ and ∇r− the unit tangent vectors to these segments in q, the distributional laplacian ∆r on the manifold is the signed Radon measure
∆r = ˜∆r d vol −|∇r+− ∇r−|Hn−1|cut(p). (1.5.15) Here we mean that ˜∆r is the standard laplacian of r which exists, in the classical sense, outside cut(p). For the details one may read [36]. Then (1.5.15) also holds in BR(q) where we also know that this measure is finite.
From (1.5.15), it follows
(∆r)+≤ ( ˜∆r)+d vol . (1.5.16)
Theorem 1.5.1 (Laplacian comparison) holds also in the standard sense outside the cut locus: see, for example, [40, Corollary 1.1]. Then
˜
∆r ≤ (n − 1)sn 0 k(r)
snk(r) on M \ cut(p) (1.5.17)
and, as the right-hand side is positive, also ( ˜∆r)+≤ (n − 1)sn
0 k(r)
snk(r) on M \ cut(p). (1.5.18) The last equation, with (1.5.16), gives exactly the first line of (1.5.14). To prove the second line just notiche that
|∆r| = 2(∆r)+− ((∆r)+− (∆r)−
) = 2(∆r)+− ∆r. (1.5.19)
Finally we will briefly remind the statement of the so called volume comparison the-orem. The theorem is a consequence of Theorem 1.5.1 (Laplacian comparison). The statement and the proof can be found, for example, in [39, page 279]. To reach the fol-lowing more complete statement one can read a combination of [33, Corollary 2.3] and [40, Theorem 1.2 and 1.3]. It can be also useful to read the reference [5, Chapter 4]. Theorem 1.5.5 (Volume comparison). Let (M, h·, ·i) be a complete Riemannian manifold of dimension n, let k be a real number and suppose RicM ≥ (n − 1)k.
Let p ∈ M , let Vk(r) the volume of the ball of radius r in the model space Mn k and vk(r) the Hn−1 measure of the boundary of the ball of radius r in the model space Mn
k. Then vol(∂Br(p)) vk(r) is non-increasing in r and vol(∂Br(p)) vk(r) ↑r→0+ 1, vol(Br(p)) Vk(r) is non-increasing in r and vol(Br(p)) Vk(r) ↑r→0+ 1. (1.5.20)
Remark 1.5.6. A more general version of the monotonicity, which can be found in [40, Corollary 2.3] or which follows from the previous result by applying a simple monotonicity lemma contained in [40, page 10], is also true: given 0 ≤ r1 < r2 ≤ r3 < r4 we have
vol(Br4(p)) − vol(Br3(p))
Vk(r4) − Vk(r3) ≤
vol(Br2(p)) − vol(Br1(p))
Vk(r2) − Vk(r1) (1.5.21) and passing to the limit in (1.5.21) after having chosen r2 = r3, we obtain also that for almost every 0 < r1 < r2, vol(∂Br2(p)) vk(r2) ≤ vol(Br2(p)) − vol(Br1(p)) Vk(r2) − Vk(r1) ≤ vol(∂Br1(p)) vk(r1) . (1.5.22) It is also worth noticing that, fromTheorem 1.5.5(Volume comparison), it holds also that for every r > 0
vol(∂Br(p)) vk(r) ≤
vol(Br(p))
Vk(r) . (1.5.23)
Indeed, by using the co-area formula, which holds also in the setting of smooth Rie-mannian manifolds [38, Theorem 2.6], and the monotonicity in the first line of (1.5.20), one obtains vol(Br(p)) = ˆ r 0 vol(∂Bs(p)) ds = ˆ r 0 vol(∂Bs(p)) snn−1k (s) sn n−1 k (s) ds ≥ ≥ vol(∂Br(p)) snn−1 k (r) ˆ r 0 snn−1 k (s) ds = vol(∂Br(p)) vk(r) Vk(r) (1.5.24) because vk(r) = σn−1snn−1
k (r), where σn−1 is the area of the unit sphere in Rn. Then dividing one reaches the conclusion in (1.5.23).
1.6. Maximum principle
We remind the maximum principle on manifolds. This result can be found in [39, pages 280-283] or in [9, pages 53-54]. The minimum principle is analogous.
It has to be said that the maximum principle in this weak sense was firstly proved by E. Calabi in [8].
Theorem 1.6.1 (Maximum principle). If f : (M, h·, ·i) → R is continuous and subhar-monic in the barrier sense ( i.e. ∆f ≥ 0 in the barrier sense), then f is constant in a neighborhood of every local maximum. In particular, if f has a global maximum, then it is constant.
1.7. A function with uniformly bounded Laplacian on
Rieman-nian manifolds with Ricci bounded below
Using the results which can be found in [9, Chapter 8], i.e some sort of quantitative maximum principles and the Cheng-Yau gradient estimate, we can construct a function which will be very useful in the proof of the almost splitting theorem. The construction is in [9, pages 45-46] and also in the original paper by Cheeger and Colding [10]. For a detailed proof it is also suggested the reading of [5, Chapter 11].
1.8. Gromov-Hausdorff distance and convergence
Proposition 1.7.1. Let (M, h·, ·i) be a complete Riemannian manifold of dimension n, let k ≥0 be a real number and suppose RicM ≥ −(n − 1)k.
Fix p ∈ M and R2 > R1 > 0. There exists a constant C .
= C(n, R1, R2, k) and a smooth function ϕ: M → [ 0, 1 ] such that
ϕ|BR1(p) ≡ 1, ϕ|BR2(p)c ≡ 0, 0 ≤ ϕ ≤ 1, |∆ϕ| ≤ C(n, R1, R2, k). (1.7.1)
1.8. Gromov-Hausdorff distance and convergence
In this section we explore some properties about Gromov-Hausdorff distance and conver-gence. The main reference is Petersen’s book [39, pages 395-495], but an overview may also be found in Cheeger’s notes [9, pages 27-28]. Also, one can see Gromov’s book [26, Chapter 3] or may tray benefit from the pedagogic approach of Burago in [7, Chapters 7-8]. One good reference is also the beautiful introduction given by Villani in [43, Chapter 27]. It is important to notice that each ball is intended to be a closed ball from now on. Definition 1.8.1 (Gromov-Hausorff distance). Let (X, dX) and (Y, dY) be two metric spaces. We call a metric admissibile on X t Y , the disjoint union of X and Y , if it is a metric on X t Y whose restriction on X (respectively Y ) is dX (respectively dY). Then the Gromov-Hausdorff distance (possibly infinite) is defined as
dGH (X, dX), (Y, dY) .
= inf{dH(X, Y ) : d is an admissibile metric on X t Y }, (1.8.1) where dH is the usual Hausdorff distance between sets (see, for example, [39, page 396]). It is useful to remind here a theorem which will be used later on. It can be found, for example, in [7, Theorem 7.3.8].
Theorem 1.8.2 (Blaschke’s selection theorem). Let (X, dX) be a compact metric space and let K(X) be the set of compact subsets of X equipped with the Hausdorff distance dX
H. Then K(X), dX
H is a compact metric space.
It is known that, if M is the class of all compact metric spaces, then (M, dGH) is a pseudo-metric space and it becomes a metric space if we consider the equivalence classes under isometry ˜M. Also M, d˜ GH
is a complete separable metric space. In order to give the idea of how it is proved that, if X, Y ∈ M, then
dGH(X, Y ) = 0 ⇒ X ∼= Y, (1.8.2)
we will introduce the notion of GH-approximation and a useful proposition, which will also useful later on.
Definition 1.8.3 (GH-approximation). Consider f : X → Y a map between two metric spaces. We say that f is an ε-GH approximation if
sup (x1,x2)∈X×X
|dY(f (x1), f (x2)) − dX(x1, x2)| ≤ ε, ∀y ∈ Y ∃x ∈ X : dY(y, f (x)) ≤ ε
(1.8.3)
Remark 1.8.4. For maps defined as in Definition 1.8.3 (GH-approximation) one can also found the terminologies rough isometry, ε-rough isometry, ε-isometry.
The following lemma is contained in [43, Equation (27.3) and following lines], and also the next proposition is in [43, Lemma 2.74].
Lemma 1.8.5. Suppose we have (X, dX) and (Y, dY) two metric spaces. Then it holds 2
3dGH(X, Y ) ≤ inf {ε > 0 : there exists an ε-GH approximation f : X → Y } ≤ 2dGH(X, Y ). (1.8.4) Moreover if there exists f : X → Y an ε-GH approximation, then there exists also g : Y → X a 4ε-GH approximation.
Remark 1.8.6. If in the previous lemma we are dealing with (X, dX) and (Y, dY) compact metric spaces and we have f : X → Y an ε-GH isometry, we can take {y1, . . . , yk} a finite ε-dense subset in Y and consider the set {x1, . . . , xk} where the xi’s are the elements corresponding to the yi’s according to the second of (1.8.3). Eventually adding some element in order to obtain A= {x1. , . . . , xh} an ε-dense subset of X and enlarging the set of y’s to B = {y1, . . . , yh} by adding the images of the new elements, we thus obtain two. finite ε-dense subsets of X and Y , of the same cardinality, such that
sup 1≤i<j≤h dY(yi, yj) − dX(xi, xj) ≤ 3ε. (1.8.5)
Vice-versa it is simple to show [39, Example 11.1.5] that, if we have two metric spaces (X, dX) and (Y, dY) and two finite ε-dense subsets A = {x1. , . . . , xh} ⊆ X and B = {y1. , . . . , yh} ⊆ Y which satisfy
sup 1≤i<j≤h
|dY(yi, yj) − dX(xi, xj)| ≤ ε, (1.8.6) then
dGH(X, Y ) ≤ 3ε. (1.8.7)
Proposition 1.8.7. Let us assume (X, dX) and (Y, dY) are compact metric spaces and, for each k ∈ N, there exists an εk-isometry fk : X → Y , with εk → 0. Then, up to subsequences, fk converge pointwise to an isometry f : (X, dX) → (Y, dY).
Hence, from the previous proposition, it is clear that (1.8.2) holds.
Now we will continue this introduction, giving the definition of the pointed Gromov-Hausdorff distance.
Definition 1.8.8 (Pointed Gromov-Hausdorff distance and convergence). We may define also the pointed Gromov-Hausdorff distance. Given (X, x, dX) and (Y, y, dY) two pointed metric spaces, we say that
dGH((X, x, dX), (Y, y, dY)) .
= inf{dH(X, Y )+d(x, y) : d is an admissibile metric on XtY }. (1.8.8) Consider M? the class of all proper pointed metric spaces. With proper we mean that each closed ball is compact. If (Xi, xi, dXi), (X, x, dX) ∈ M?, we say that (Xi, xi, dXi) → (X, x, dX) in the pointed Gromov-Hausdorff topology if
∀R > 0 :BRdXi(xi), xi, dXi
→BRdX(x), x, dX (1.8.9) where the last convergence is in the pointed Gromov-Hausdorff distance defined above.
For a different definition, one can read [7, Definition 8.1.1]. To better understand what is going on, read Remark 1.8.9 (Realizations of GH-convergence).
1.8. Gromov-Hausdorff distance and convergence
Remark 1.8.9 (Realizations of GH-convergence). A way to think of these convergences is to imagine all these metric spaces in a realization. Under good properties of the spaces involved, i.e. if we are dealing with geodesic and proper spaces, we can have a realization like the following. For a reference, one can read [19, Section 2.1] and reference therein.
• If (Xn, dXn) → (X, dX) in the Gromov-Hausdorff sense, then there is a metric space
(Y, dY) and isometric embeddings in: Xn → Y and i∞: X → Y such that
dYH(in(Xn), i∞(X)) → 0, (1.8.10) where dY
H is the Hausdorff distance associated to dY.
• If (Xn, xn, dXn) → (X, x, dX) in the pointed Gromov-Hausdorff sense, then there is a metric space (Y, dY) and isometric embeddings in : Xn → Y and i∞ : X → Y such that
in(xn) → i∞(x),
∀R > 0 dYH(in(BR(xn)), i∞(BR(x))) → 0, (1.8.11) where both convergences are to be intended according to the metric dY.
Then, in each of the previous two cases, writing that zn ∈ Xn converges to z ∈ X makes sense: having one of this realization in mind, it would mean simply in(zn) → i∞(z) in the metric dY of the realization.
It is also simple to notice that, having this realization, in both the previous cases, for each z ∈ X, there is a sequence zn ∈ Xn such that zn→ z. This follows from (1.8.10) in the first case, and from the second line of (1.8.11) in the second case.
It has to be noticed, finally, that in the cases we are mainly interested in, i.e. pointed Gromov-Hausdorff limits of Riemannian n-manifolds with Ricci uniformly bounded from below, such a realization Y may be taken proper, i.e. such that every closed ball is a compact set (see [19, Equation (2.1)]).
It is worth noticing that if (Xn, xn, dXn) → (X, x, dX), we have that for each z ∈ X there is a sequence zn ∈ Xn such that
(Xn, zn, dXn) → (X, z, dX). (1.8.12)
This is true for the following reason: first of all, as said before, it is simple to find zn → z and moreover the distances dXn(xn, zn) are equi-bounded by a constant D > 0. Then, if we fix R > 0, we have that
BRdXn(zn) ⊆ BdXn
R+dXn(xn,zn)(xn) ⊆ Bd Xn
R+D(xn) (1.8.13)
and we know that BdXn
R+D(xn) → Bd
X
R+D(x) in the Hausdorff distance of the realization, from the second of (1.8.11). Thus we can use the result of the forthcomingLemma 1.8.16
to give, eventually, an estimate from above of dGHBRdXn(zn), BdX
R (z) by a multiple of dGH BdXn R+D(xn), Bd X R+D(x)
which goes to zero and thus obtaining that, for every R > 0, dGH BRdXn(zn), BdX R (z) → 0 (1.8.14)
which is exactly what we wanted to obtain in (1.8.12). It is worth mentioning here also some notes of D. Jansen in which it is proved, in a different way, the previous property and lots of other elementary lemmas about pointed Gromov-Hausdorff convergence [29]. To conclude the remark, the interested reader may read [23] in which connections between different definitions of pointed (measured) Gromov-Hausdorff convergence are explored.
For the sake of completeness, let us introduce the pointed measured Gromov-Hausdorff convergence in this setting. We will not go through the details but a precise definition is in [23, Definition 3.24]. We will give a simplified definition, as in [19, Section 2.1], which is equivalent to the more general definitions assuming, for example, we are dealing with geodesic and proper metric spaces uniformly c-doubling, i.e. for which it holds the doubling condition with an uniform c not depending on the spaces.
Given mn, m Radon measures over Xn and X respectively, we say that
(Xn, xn, dXn, mn) → (X, x, dX, m) (1.8.15)
in the pointed measured Gromov-Hausdorff sense if the two conditions in (1.8.11) are fulfilled and (in)?mnweakly converges to (i∞)?m. This is to say that, for each ϕ continuous with bounded support in Y , we have
ˆ
ϕd(in)?mn → ˆ
ϕd(i∞)?m. (1.8.16)
From now on when we put the term pointed in brackets we are saying that the proper-ties hold also when we consider the class M? with the pointed Gromov-Hausdorff topology defined above. In this setting we can also define a notion of convergence of maps.
Definition 1.8.10 (Convergence of maps). Given (Xi, dXi), (X, dX), (Yi, dYi), (Y, dY)
metric spaces (i = 1, 2, . . . ) with
(Xi, dXi) → (X, dX), (Yi, dYi) → (Y, dY) (1.8.17)
in the (pointed) Gromov-Hausdorff convergence, we say that fi : Xi → Yi converge to f : X → Y if for all xi ∈ Xi → x ∈ X we have
fi(xi) → f (x) (1.8.18)
where the convergences xi → x and the one in (1.8.18) is understood in the sense of some fixed realization, see Remark 1.8.9 (Realizations of GH-convergence).
It is said that a sequence of functions as above is equicontinuous if for every ε > 0 there is a δ > 0 such that for every xi ∈ Xi
fi(Bδ(xi)) ⊆ Bε(fi(xi)), ∀i ≥ 1. (1.8.19) With this definition in mind, we can state an analogous of Ascoli-Arzel`a lemma. Lemma 1.8.11 (Ascoli-Arzel`a lemma for Gromov-Hausdorff convergence). If we have that fi : Xi → Yi are equicontinuous according to (1.8.19) and we have (Xi, dXi) → (X, dX) and (Yi, dYi) → (Y, dY) in the (pointed) Gromov-Hausdorff topology, then there
exists a convergent subsequence of {fi}i≥1 in the sense of Definition 1.8.10(Convergence of maps). In the case we are dealing with pointed metric spaces, we also have to require that the maps fix the base points.
1.8. Gromov-Hausdorff distance and convergence
Now we will state a general proposition from which some compactness theorems in the classes M and M? will follow. First of all, given (X, dX) a metric space, let us define
CapX(ε)= maximum number of disjoint. ε
2-balls in X,
CovX(ε)= minimum number of ε-balls it takes to cover X.. (1.8.20) Later on we will use two inequalities that involve these two quantities, which can be found in [39, page 403]. If we have X and Y compact metric spaces satisfying dGH(X, Y ) < δ, then
CovX(ε + 2δ) ≤ CovY(ε)
CapY(ε + 2δ) ≤ CapX(ε). (1.8.21) Now we come to the general Gromov compactness theorem.
Proposition 1.8.12 (General Gromov compactness theorem). For a class C ⊆ (M, dGH) of compact metric spaces all with diameters bounded by some constant D, the following are equivalent:
• C is pre-compact in the Gromov-Hausdorff topology,
• There exists a function N1(ε) : ( 0, α ) → ( 0, ∞ ) such that CapX(ε) ≤ N1(ε) for all X ∈ C,
• There exists a function N2(ε) : ( 0, α ) → ( 0, ∞ ) such that CovX(ε) ≤ N2(ε) for all X ∈ C.
From this proposition we have these corollaries: the second follows from an application of Theorem 1.5.5 (Volume comparison).
Corollary 1.8.13 (Pointed Gromov compactness theorem). A collection C ⊆ M? is pre-compact in the pointed Gromov-Hausdorff topology if and only if for each R > 0 the collection
n
BRdX(x) : BdX
R (x) ⊆ (X, x, d
X) ∈ Co⊆ (M, dGH) (1.8.22) is pre-compact in the Gromov-Hausdorff topology.
Corollary 1.8.14 (Gromov compactness theorem for manifolds). For any integer n ≥ 2, every k ∈ R and D > 0 the following classes are precompact:
• The collection of closed Riemannian manifolds of dimension n with Ric ≥ (n − 1)k and diam ≤ D,
• The collection of pointed complete Riemannian manifolds of dimension n with Ric ≥ (n − 1)k.
From now on we will use the same notation as in [9, page 47]. We will denote with Ψ(ε1, . . . , εk|c1, . . . , cN) (1.8.23) a positive constant which depends on non-negative parameters ε1, . . . , εk, c1, . . . , cN and which goes to zero if c1, . . . , cN are fixed and ε1+ · · · + εk→ 0.
At this stage we have all the basic stuffs we need to handle with the Gromov Hausdorff convergence. Now let us prove a lemma which will be very useful later on. Heuristically
this lemma tells us that if we have that a ball BL(p) in a Riemannian manifold M is GH-close to a ball BL(0) in Rn, we can recover some geometric structures of Rn on this ball. It is worth noticing that this is a metric result: during the proof we will use only the fact that (M, dM) is a proper metric space and a length space. This means that the following result holds also if we substitute (M, dM) with a metric space satisfying these two conditions: this will be clear also in the statement of Lemma 1.8.16.
Lemma 1.8.15. Consider M an n-dimensional Riemannian manifold. Let p ∈ M and let us fix R >0.
Assume there exist δ >0 and L > R such that
dGH(BL(p), BL(0)) < δ. (1.8.24) Here we intend that the balls are closed, BL(0) is the ball of radius L centered at the origin in Rn and the GH-distance is between those closed balls centered at their centers, as in (1.8.8). Let us consider a metric on the disjoint union BL(p) t BL(0) in which it is realized
dH(BL(p), BL(0)) + d(p, 0) < δ. (1.8.25) From now on we will work using this metric. Consider {ej}nj=1 the canonical basis of Rn and fix, for each j = 1, . . . , n, points qj+, q
−
j ∈ BL(p) such that d(q+
j , Lej) < δ d(qj−, −Lej) < δ. (1.8.26) This can be done because of (1.8.25). Let us define, for each j = 1, . . . , n, the excess functions which will be useful also in the next chapter
EjM(·)= d. M(·, q+ j ) + d M(·, q− j ) − d M(q+ j , q − j ), ERn j (·) . = dRn(·, Lej) + dRn(·, −Lej) − 2L. (1.8.27) Let us define the shifted distance functions in M for each j = 1, . . . , n,
b±j(·)= d. M(·, q± j ) − d
M(p, q±
j ). (1.8.28)
Let us call Φ= (b. −1, . . . , b−n) on BR(p). Then it holds that EjM(p) < 2δ,
dGH(BR(p), BR(p)) < 2δ,
Φ maps BR(p) in BR+Ψ(δ,L−1|n,R)(0)
Φ is a Ψ(δ, L−1|n, R)-GH approximation according to (1.8.3),
(1.8.29)
Proof. For the sake of simplicity let us fix j = 1. We can use the triangle inequality, working in the disjoint union BL(p) t BL(0) with the metric d as in the statement. Then
d(q+1, q −
1) ≥ d(Le1, −Le1) − d(Le1, q +
1 ) − d(−Le1, q −
1) > 2L − 2δ. (1.8.30) Obviously, by the triangle inequality, d(q+
1, q −
1) ≤ 2L so that, with the previous one, we get 0 ≤ 2L − d(q+ 1 , q − 1 ) = d(−Le1, Le1) − d(q + 1, q − 1) < 2δ (1.8.31) If, for the sake of contradiction, d(p, q+
1) ≤ L − 2δ, then d(p, q1−) ≥ d(q + 1, q − 1) − d(p, q + 1 ) > L (1.8.32)
1.8. Gromov-Hausdorff distance and convergence which is impossible as q− 1 ∈ BL(p). Then d(p, q + 1) > L − 2δ and it holds 0 ≤ L − d(p, q+ 1 ) < 2δ (1.8.33)
and, analogously to(1.8.33), we get the same with the minus. From(1.8.30)and (1.8.33), it immediately follows the first of (1.8.29).
To show the first of (1.8.3) it suffices to notice that for x ∈ BR(0), being xj the j-th coordinate in Rn, one has
|dRn(x, ±Lej) − L ± xj| ≤ Ψ(L−1|n, R) (1.8.34) and the fact that it holds the second property of (1.8.29)
dGH(BR(p), BR(0)) < 2δ. (1.8.35) The inequality in(1.8.34)is just a matter of simple geometry in Rn, while to show(1.8.35) one can argue as follows. Fixed some ε > 0 and given any x ∈ BR(p), there exists ¯xε ∈ BL(0) such that d(x, ¯xε) < dH(BL(p), BL(0)) + ε. Then, as dH(BL(p), BL(0)) + d(p, 0) < δ in view of (1.8.25), it is simple to notice that d(¯xε,0) < R + δ + ε by means of triangle inequality. For if it were not true then
d(x, p) ≥ d(¯xε,0) − d(x, ¯xε) − d(p, 0) > R (1.8.36) which is absurd since x ∈ BR(p). From this it follows that BR(p) is contained in the ((dH(BL(p), BL(0))) + ε)-fattening of BR+δ+ε(0). By the symmetry of the previous ar-gument, also BR(0) is contained in the ((dH(BL(p), BL(0))) + ε)-fattening of BR+δ+ε(p). This easily implies that BR(p) is contained in the ((dH(BL(p), BL(0))) + δ + 2ε)-fattening of BR(0) and vice-versa, as we are working with length spaces. Hence
dH(BR(p), BR(0)) ≤ dH(BL(p), BL(0)) + δ + 2ε. (1.8.37) Taking ε → 0,
dH(BR(p), BR(0)) ≤ dH(BL(p), BL(0))) + δ (1.8.38) and taking into account (1.8.25)it can be concluded, from (1.8.38),
dGH(BR(p), BR(0)) < 2δ. (1.8.39) Now we are ready to show the first of (1.8.3). We want to show that
sup (x,y)∈BR(p)×BR(p)
|dRn(Φ(x), Φ(y)) − dM(x, y)| ≤ Ψ. (1.8.40)
Given x and y in BR(p), in view of (1.8.38) and(1.8.25), we can find ¯xand ¯y in BR(0) such that
d(x, ¯x) < 2δ, d(y, ¯y) < 2δ (1.8.41) so that, taking into account (1.8.33), (1.8.41)and (1.8.26)with the sign −, and by using the triangle inequality it holds
|(dM(x, q−i ) − d M
(p, qi−)) − (dR
n
From the previous equation, jointly with (1.8.34), again by using the triangle inequal-ity, one obtains that
|b−i (x) − (¯x)i| ≤ Ψ(δ, L−1|n, R). (1.8.43) and then, being
dRn(Φ(x), Φ(y)) = v u u t n X i=1 (b−i (x) − b−i (y))2 (1.8.44) from (1.8.43) it follows |dRn(Φ(x), Φ(y)) − dRn(¯x,y)| ≤ Ψ(δ, L¯ −1|n, R). (1.8.45)
Then, easily from (1.8.41), it follows
|dRn(¯x,y) − d¯ M(x, y)| < 4δ (1.8.46) and joining (1.8.45) with (1.8.46), by using the triangle inequality, we obtain (1.8.40).
The third property in (1.8.29) follows directly taking y = p in (1.8.40), while to conclude the proof of the fourth property in (1.8.29) it remains to show the second of (1.8.3). This is readily seen and we sketch the idea: given ¯x ∈ BR+Ψ(0), it is sufficient to take x ∈ BR(p) which is Ψ-close to ¯x: this can be done because of the second of (1.8.29). Then (1.8.43) can be obtained as before and it can be exploited in (1.8.44) with ¯xinstead of Φ(y) to show exactly the second of (1.8.3).
With very little modifications in the previous proof, it holds also the following. Lemma 1.8.16. Consider (X, dX) and (Y, dY) two proper length metric spaces. Let x ∈ X and y ∈ Y and fix R >0.
Assume there exist δ >0 and L > R + 2δ such that dGH BLdX(x), BdY L (y) < δ. (1.8.47)
Here we intend that the balls are closed and centered at their centers, as in (1.8.8). Let us consider a metric on the disjoint union BdX
L (x) t Bd
Y
L (y) in which it is realized dH
BLdX(x), BLdY(y)
+ d(x, y) < δ. (1.8.48) Let us assume there is x1 ∈ Bd
X
L (x) such that d(x, x1) = R and take y1 ∈ Bd
Y
L (y) such that d(x1, y1) < δ. Then
d(y, y1) ≤ R + 2δ, dGH BRdX(x1), BdY R (y1) <4δ ∀R ≤ L − R − 2δ. (1.8.49) Proof. Just mimic the first part of Lemma 1.8.15, i.e. from the lines above (1.8.36) to (1.8.39).
For the sake of completeness let us state and give only the idea of the proof of some re-sults concerning how the pointed Gromov-Hausdorff behaves with the product structure. Remember that (dX×Y)2 = (d. X)2+ (dY)2.
1.9. Hausdorff measures
Lemma 1.8.17. Let Xn, dXn and Yn, dYn be proper metric spaces. Let us assume
that, for some xn ∈ Xn and yn ∈ Yn it holds
Xn, xn, dXn → X, x, dX , Yn, yn, dYn → Y, y, dY ,
(1.8.50) for some proper pointed metric spaces X, x, dX and Y, y, dY. Then it holds
Xn× Yn,(xn, yn), dXn×Yn → X × Y, (x, y), dX×Y . (1.8.51) Proof. It follows by using the idea of evaluating the (pointed) GH-distance by means of (pointed) GH-approximations. In particular it follows from the pointed analogous of (1.8.4), and by taking (pointed) GH-approximations defined on finite dense subsets (see
Remark 1.8.6).
Lemma 1.8.18. Let Xn, dXn and Y
n, dYn be proper metric spaces. Let us assume that there exists a pointed proper metric space (Z, z, dZ) such that
Xn× Yn,(xn, yn), dXn×Yn → (Z, z, dZ). (1.8.52)
Then there exists pointed proper metric spaces (X, x, dX) and (Y, y, dY) such that, up to subsequences,
Xn, xn, dXn → X, x, dX ,
Yn, yn, dYn → Y, y, dY , (1.8.53)
and then (Z, dZ) is isometric to X × Y, dX×Y with an isometry which takes z in (x, y). Proof. From (1.8.52) and (1.8.21) it follows that, for every ε and R > 0, both CovBdXn
R (xn)(ε)
and CovBdYn
R (yn)(ε) are uniformly bounded in n. Then, by usingProposition 1.8.12
(Gen-eral Gromov compactness theorem) and Corollary 1.8.13 (Pointed Gromov compactness theorem) it follows (1.8.53).
Then, using Lemma 1.8.17 and the fact that two complete metric spaces arising as the pointed GH-limit of the same sequence are isometric [7, Theorem 8.1.7], also the last part of the statement follows.
1.9. Hausdorff measures
In this subsection we recall some properties of Hausdorff measures which will be useful later on. Some possible references are [21], [4] and [24].
First of all the classical definitions are contained, for example, in [24, Definition 11.1] or in [21, Paragraph 2.10] in a more general setting. For the sake of completeness let us write here the definition of the Hausdorff measure. Given (X, dX) a generic metric space, for A ⊆ X it can be defined
Hkδ(A)=. ωk 2k inf (+∞ X i=0 diam(Si)k : A ⊆ +∞ [ i=1 Si,diam Si < δ ) , (1.9.1)
where the infimum is taken over all possible countable coverings of A with Si generic subsets of X, and ωk= πk2 Γ k 2 + 1 (1.9.2)
which coincides with the volume of the unit ball in Rk when k ≥ 1 is an integer. Here Γ(·) is the usual Gamma function. Then the Hk pre-measure on each subset A ⊆ X is
Hk(A) = lim δ→0H
k
δ(A). (1.9.3)
Remark 1.9.1. When the family of subsets of X over which we take the infimum in (1.9.1) is the family of closed balls, we obtain the spherical Hausdorff measures Hk
s,δ and Hks. It can be shown that [21, Paragraph 2.10.2]
Hk≤ Hsk≤ 2kHks. (1.9.4) We will recall some useful properties which will be useful later on. We will work in a generic metric space (X, dX). The first property comes from the definitions and it is contained in [24, Lemma 11.2], for example.
Lemma 1.9.2. Given A ⊂ X we have the following: for each 0 ≤ k < +∞, Hk(A) = 0 ⇔ Hk
∞(A) = 0. (1.9.5)
The second property is a consequence of the definition of Hk
∞. It is shown in [35, Theorem 2.1] and holds also for Hk
δ.
Lemma 1.9.3. If we consider An, A compact subsets of X and suppose
dXH(An, A) → 0, (1.9.6)
then, for each 0 ≤ k < +∞, Hk
∞(A) ≥ lim sup n→+∞
Hk
∞(An). (1.9.7)
Here it is intended that dX
H is the Hausdorff distance associated to dX. In other words we have that H∞k is upper-semicontinuous with respect to the Hausdorff convergence on compact sets.
The third property is shown in [24, Lemma 11.3] only for subsets in Rn, but the proof holds in a generic metric space which is second-countable. For an analogous statement one can see [21, Theorem 2.10.17].
Lemma 1.9.4. Let us consider A ⊂ X and let us fix 0 ≤ k < +∞. Then Hk x ∈ A: lim sup r→0+ Hk ∞(A ∩ Br(x)) ωkrk <2−k = 0. (1.9.8)
CHAPTER 2
THE ALMOST SPLITTING THEOREM
2.1. Statement of the Theorem
The following result is classic in Riemannian geometry and it was proved by Cheeger and Gromoll in 1971 [14].
Theorem 2.1.1 (Splitting theorem). Let (M, h·, ·i) be a complete Riemannian manifold of dimension n and suppose RicM ≥ 0.
Suppose that M contains a line, i.e. a geodesic which is minimal from every pair of points chosen on it. Then M splits isometrically as
M ∼= R × ˜M (2.1.1)
where ˜M is an (n − 1)-dimensional Riemannian manifold with RicM˜ ≥ 0. The distance considered on R × ˜M is the Hilbertian product distance.
For a proof shorter than the one of Cheeger and Gromoll, [20] or [39, page 300] can also be read. An outline of the proof, which uses the Busemann functions, can be found in [9, pages 35-37].
Remark 2.1.2. By a careful inspection of the proof given in [20] it can also be concluded that, under this isometry, the line in M becomes a fiber R × { ˜m}. It is useful to point it out and, for this purpose, we will briefly remind the proof.
In general, given a ray γ : [ 0, +∞ ) → M , i.e. a geodesic which is minimizing between every two points taken on it, one can construct the Busemann function bγ : M → R just by taking the limit for t → +∞ of the decreasing sequence of functions {bt} defined by
bt(x) = d(x, γ(t)) − t, ∀x ∈ M (2.1.2) Given a line γ : R → M with unit tangent vector, one can consider the Busemann function bγ+ associated to the ray γ+ = γ|[ 0, +∞ ). Hence, it is shown in the proof that.
bγ+ is C∞ and, eventually adding some constants in (2.1.2), ˜M = {bγ. + = 0} is a smooth
Riemannian submanifold of M . One, then, constructs the map Φ : R × ˜M → M as
Φ(t, y)= ϕt(y). (2.1.3)
where d
dtϕt(y) = ∇bγ+(ϕt(y)) and ϕ0(y) = y. It can be shown that this is an isometry and, after having shown that RicM˜ ≥ 0, this concludes the proof.
It can be shown also that |∇bγ+| = 1 on γ, and, by the very definition of bγ, it
holds bγ+(γ(s)) = −s for every s ∈ R. Differentiating this last equality one obtains
h∇bγ+(γ(s)), γ0(s)i = −1 and from the fact that also |γ0(s)| = 1 it follows ∇bγ+(γ(s)) =
−γ0(s). Then flowing from the point γ(0), which is on ˜M, we cover the entire line γ and thus, under the isometry, the line γ is the fiber R × {γ(0)}.
To investigate Ricci-limit spaces, a key point in the works of Cheeger and Colding is to establish a generalization of the splitting theorem that holds for metric spaces Y that are pointed Gromov-Hausdorff limits of n dimensional Riemannian manifolds Mi with Ricci curvature bounded from below RicMi ≥ −(n − 1)δi and δi > 0 with δi → 0. If Y
contains a line, one probably may expect that it is the case that Y splits isometrically as before and it comes out that it is right.
Theorem 2.1.3 (Splitting theorem for limit spaces). Let (Mi, h·, ·ii) be a sequence of Riemannian manifolds of dimension n with RicMi ≥ −(n − 1)δi, with δi >0 and δi → 0.
Assume that there are mi ∈ Mi and y ∈ Y such that (Mi, mi) → (Y, y) in the sense of Definition 1.8.8 (Pointed Gromov-Hausdorff distance and convergence). If Y contains a line, i.e. a curve γ : R → Y such that dY(γ(s), γ(t)) = |t − s| for every t, s ∈ R, then Y splits isometrically as
Y ∼= R × X (2.1.4)
where X is a length space, i.e. a metric space in which the distance between two points is realized as the infimum of the lengths of the curves which connect the two points.
The distance considered on R × X is the Hilbertian product distance.
Remark 2.1.4. We will assume from now on that δi <1 for every i. Indeed, up to ignoring some initial terms, the pointed metric space (Y, y) is the limit of (Mi, mi) satisfying RicMi ≥ −(n − 1)δi with 0 < δi <1 and δi → 0.
Remark 2.1.5. It is important to notice that the line γ is not needed to be a limit line, i.e. the limit of longer and longer segments in Mi. It is also worth mentioning that, as it will follow from the proof, it is not clear whether the line γ in Y , under the isometry, is a fiber R × {x}. This could be true if the segments from pi and qi±, as they will be constructed at the start of the proof of Theorem 2.1.3 in Section 2.6, converge to γ.
The proof is rather long and a sketch could be found in [9, pages 47-52]. We will follow the discussion in this reference and try to fill the details here. Before starting the proof, we will give a detailed outline in the following section.
2.2. Outline of the proof
In this outline of the proof we will be a bit sketchy. The precise statements of all the results we state in this section will be presented in this chapter.
To prove the splitting theorem for limit spaces the first idea is to reduce everything to a quantitative statement on an n-dimensional Riemannian manifold M with Ricci bounded from below.
Let us recall that, given two points q+, q− on M we can define the excess function on M as
Eq+,q−(x)= d(x, q. +) + d(x, q−) − d(q+, q−). (2.2.1)
Now the result to which we can reduce the theorem can be stated: if we take M an n-dimensional Riemannian manifold, we fix R > 0 and consider three points p, q+, q− in M such that for some δ > 0 the following three conditions hold
RicM ≥ −(n − 1)δ, Eq+,q−(p) ≤ δ,
min{d(p, q+), d(p, q−
)} ≥ δ−1 R,
2.2. Outline of the proof
then there exist a metric space X and a point x ∈ X such that
dGH(BR(p), BR((0, x))) ≤ Ψ (2.2.3)
according to (1.8.8). Here BR(p) is the closed ball of center p and radius R in M with the Riemannian distance and BR((0, x)) is the closed ball of center (0, x) and radius R in the product metric space R × X. Here and in the following cases it is understood that each Ψ is a Ψ(δ|n, R) according to (1.8.23) and on the product R × X we consider the metric
dR×X((s, x1), (t, x2)) =p(t − s)2+ dX(x1, x2)2. (2.2.4) In the following sections, the main efforts are made to show this result. Once we have obtained this, which is a sort of almost splitting theorem, the splitting theorem for limit spaces follows by taking, in some sense, the limit of such configurations. This limit argument is contained in the last part of this chapter, precisely in Section 2.6.
Now we will indicate the main steps to show (2.2.3). The core idea is the following: the metric space X can be taken to be a level set (β+)−1(0), with the induced distance, where β+is the harmonic replacement of the shifted distance function b+(·)= d(·, q. +) − d(p, q+) on BλR(p) with λ > 0 satisfying
δ−1 λR R. (2.2.5)
The necessity of taking this harmonic replacement is due to the fact that we only know that the distance function is Lipschitz. The first part of the proof is about to prove some estimates regarding the gradient and the Hessian of this harmonic replacement. After this, one uses this estimates to prove an approximate Pythagorean theorem from which (2.2.3) follows. Precisely the road map is the following:
• By Laplacian comparison it holds, in the sense of barriers,
∆(b+− β+) ≤ Ψ on BλR(p); (2.2.6) • By a little modification to an inequality due to Abresch and Gromoll (see the forthcoming Lemma 2.3.1), we can extend the second of (2.2.2) to the whole ball BλR(p), up to substituting δ with Ψ, and then, by min-max principles, we can give an L∞-estimate of the difference |b+ − β+|: i.e. it can be proved that (see the forthcoming Lemma 2.3.2)
|b+− β+| ≤ Ψ on BλR(p). (2.2.7)
This, jointly with the previous point, gives an estimate on the gradient (see the forthcoming Lemma 2.4.2)
Bλ1R(p)
|∇β+− ∇b+|2 ≤ Ψ, (2.2.8) where λ1 is properly chosen and satisfies λ > λ1 1.
• Using the Bochner formula, the fact that |∇β+| is bounded on BR(p), due to Cheng-Yau gradient estimate, and the previous step, it can be proved that (see the forth-comingLemma 2.4.5)
Bλ2R(p)
where λ2 is properly chosen and satisfies λ1 > λ2 1. In this step the special func-tion with compact support and bounded Laplacian constructed inProposition 1.7.1
is needed.
• These estimates involving the gradient and the Hessian of the harmonic replace-ment β+ are used, jointly with a clever use of the segment inequality, to prove an approximate version of the Pythagorean theorem for points on {β+ = 0}: see the forthcoming Proposition 2.5.1 (Approximate Pythagorean theorem). Precisely it can be shown that if we take w ∈ BR(p) and z one of the closest points to w in {β+= 0}, then for every x ∈ {β+= 0} ∩ Bλ
3R(p), with λ2 > λ3 1 properly
chosen, we have
|d(x, z)2+ d(z, w)2− d(x, w)2| ≤ Ψ. (2.2.10) This part will be delicate and we will need three technical lemmas to perform the proof: Lemma 2.5.2, Lemma 2.5.3 and Lemma 2.5.4.
• Finally we will construct a map F on BR(p) just by sending a point w ∈ BR(p) to the couple (d(w, z), z) where z is one of the closest point to w in {β+ = 0}. We say that the point p is sent to (d(p, x), x) where x is one of the closest point to p in {β+= 0}.
The approximate Pythagorean theorem will be useful to show that F takes values in BR+Ψ((0, x)) and that it is a Ψ-GH approximation, according to (1.8.3): see The-orem 2.6.1 (Almost splitting theorem). This will be sufficient to conclude (2.2.3), due to the pointed analogous of (1.8.4).
2.3. The harmonic replacement of the shifted distance function
Let us introduce the excess function: given two points q+ and q− in M , we will denoteE(·)= d(·, q. +) + d(·, q−) − d(q+, q−) (2.3.1) a function defined on M .
The first lemma we show is a modified version of the celebrated Abresch-Gromoll excess estimate [1, Theorem 2.1] and can, loosely speaking, be summarized as follows: for fixed R, if a Riemannian manifold M ha Ricci curvature bounded from below, and we pick p, q+, q− such that p is far away from q+ and q− and the excess in p with respect to q+ and q−
is sufficiently small, then the excess will be globally sufficiently small in a ball BR(p). The idea of the proof is taken from [1, Theorem 2.1].
Now let us write down the precise hypotheses with which we will work throughout this chapter, and which makes clear the discussion above. We fix R > 0, 0 < δ < 1 (Remark 2.1.4), L > 0 and ε > 0 such that
L >2R + 1, ε < Cn,L,δϕ−δ R, R+1 2 , (2.3.2) where Cn,L,δ = 2(n − 1). 1 L−2R−12 + √ δ and ϕ−δ is defined in (1.5.5).
2.3. The harmonic replacement of the shifted distance function
We will assume, from now on, that
RicM ≥ −(n − 1)δ, min{d(p, q+), d(p, q−
)} ≥ L, E(p) ≤ ε.
(2.3.3)
We will use, throughout this chapter, the Ψ notation defined in (1.8.23). Where it is not indicated, the generic Ψ is intended to be Ψ(ε, L−1, δ|n, R).
Lemma 2.3.1. Let (M, h·, ·i) be a complete Riemannian manifold of dimension n. Let us assume (2.3.2) and (2.3.3) are fulfilled.
Then sup BR(p) E ≤Ψ(ε, L−1, δ|n, R). (2.3.4) R x η R+ 1 2 p q+ q−
