comparison theorem

(de Gruyter 2004

Symmetrization of starlike domains in Riemannian manifolds and a qualitative generalization of Bishop’s volume

comparison theorem

Ryuichi Fukuoka*

(Communicated by G. Gentili)

Abstract.We introduce a new type of symmetrization in starlike domains in Riemannian mani- folds that maintains the Ricci curvature in the radial direction. We prove that this symmetriza- tion is volume increasing. We get, as its direct consequence, a generalization of Bishop’s volume comparison theorem. Moreover, this generalization shows that this kind volume comparison theorem is qualitative in nature, instead of being quantitative. Using this symmetrization, we get some volume upper bounds in terms of some integrals of the Ricci curvature. Finally, we intro- duce a new type of symmetrization in geodesic balls within the injectivity radius, which is vol- ume decreasing.

Key words.Volume comparison, Symmetrization, Ricci curvature.

2000 Mathematics Subject Classification. 53B20, 53C65

1 Introduction

Symmetrization is a very useful tool in mathematics. In particular, symmetrization of Riemannian manifolds is a powerful tool in geometrical analysis. Frequently when we consider a class of objects that shares some features, the maximum or the minimum of a given property is attained at the most symmetrical object in this class. In addi- tion a symmetric object is usually simpler to study, what makes the symmetrization a very interesting tool to consider.

For instance letM_kⁿ be then-dimensional space form with constant sectional cur- vature kAR. LetM_Vⁿ be the set of compactn-dimensional manifolds with smooth boundary inM_kⁿ, with fixed volumeV. Consider the isoperimetric quotient

=n;VðWÞ ¼ AreaðqWÞ VolðWÞ^ðn1Þ=n

* Research partially supported by CAPES, Brazil.

inM_Vⁿ, where AreaðÞdenotes theðn1Þ-dimensional volume and VolðÞdenotes the n-dimensional volume. It is a well known fact that the minimum of=n;V is attained at the geodesic disc with volumeV(see [5] and references therein).

Another well known example is given by the Faber–Krahn inequality. It says that the lowest fundamental tone of a Riemannian manifold inM_Vⁿ is given by the geo- desic disc with volumeV(see also [5] and references therein).

There are several other examples that illustrate this kind of situation: The maxi- mum or the minimum of ‘‘interesting’’ functionals is either attained at the most symmetrical object, or it is not attained (for instance, substitute ‘‘minimum’’ in the examples given above by ‘‘maximum’’).

Let us begin to present this work. LetMⁿ be ann-dimensional Riemannian mani- fold. Denote the tangent space atpAMbyTpMand its unit vectors bySpM. We can define a polar coordinate system with the origin at pin a neighborhood of p. Denote it byðt;yÞ, whereyASpM andtis the radial component. The canonical metric on Sⁿ¹ is denoted by dy² and its volume element bydA. The curve parametrized by arclengthgð;yÞrepresents the geodesic such thatgð0;yÞ ¼ pandg⁰ð0;yÞ ¼y, where the superscript ⁰stands for the derivative in the radial variable. Denote by ~ccðyÞ the cut point of palonggð;yÞ.

We say that a domainDHMⁿis starlike with respect to pADif givenxADthen there exists a unique minimizing geodesicg:½0;c_x !Mconnectingpandxsuch that gð½0;c_xÞHD. It is not di‰cult to see that we can define a global polar coordinate system in starlike domains. Moreover, it can be defined asfexp_pðt;yÞAD;yASpM;

0ct<cðyÞg, wherecðyÞcccðyÞ. In order to be more explicit, we denote a starlike~ domain by Dðp;cÞ. Notice that geodesic balls, not necessarily within the injectivity radius, are starlike domains or the union of starlike domains with some of its closure points. We denote the geodesic ball with center pand radiusrbyBðp;rÞ.

Remark 1.1. Whenever we mention Bishop’s volume comparison theorem, we are referring to the version where the Ricci curvature is bounded from below (see Theo- rem 2.1).

The main purpose of this paper is to introduce two new types of symmetrizations in starlike domains and to study their influence on the volume element in a polar coordinate system. One of these symmetrization is volume increasing and one of its consequences is a generalization of Bishop’s volume comparison theorem. Further- more this generalization shows that this kind of volume comparison theorem is quali- tative in essence instead of being quantitative. We also get some volume upper bounds in terms of some integrals of the Ricci curvature thanks to this symmetrization. The other symmetrization that we introduce in this work is volume decreasing, and fur- ther details about it will be given afterwards.

This paper is divided as follows: In Section 2, we introduce some notation and basic facts. In Section 3, we summarize our work defining the symmetrizations and present- ing the main theorems without proofs. In addition we justify their importance in Rie- mannian geometry. In Section 4, we complete all the details about symmetrizations and we prove the qualitative generalization of Bishop’s volume comparison theorem.

Finally, in Section 5, we prove another volume comparison theorem and some vol- ume upper bounds.

2 Notation and basic facts

Let us introduce some notation and present some basic facts. Calligraphic mathe- matical letters will indicate an object related to the tangent space.

Let ðMⁿ;gÞ be an n-dimensional Riemannian manifold with metric g, ‘ the Levi–Civita connection,RðX;YÞ ¼‘X‘Y‘Y‘X‘½X;Ythe curvature tensor and RicðX;YÞ ¼Pn

i¼1hRðei;XÞY;eii the Ricci tensor, where fe1;. . .;eng is an ortho- normal frame andh;idenotes the metricg. IfjZj ¼1, then RicðZ;ZÞis the Ricci curvature in the directionZ. An important object for the study of the geometry along geodesics in polar coordinates is the Ricci curvature in the radial direction_qt^q. It will be calledradial Ricci curvatureand it will be denoted by Ricrðt;yÞ.

The exponential map exp_p restricted to Dð0;cÞ ¼ fXAT_pM:yASⁿ¹;0ckXk

<cðyÞginduces spherical coordinatesðt;x₁;x₂;. . .;x_n1ÞonDðp;cÞ, wheretis the ra- dial coordinate andx_i, fori¼1;. . .;n1, denotes the angular coordinates. Its rela- tionship with Cartesian coordinatesðx1;. . .;x_nÞis given by

x₁¼rsinx₁; x₂¼rcosx₁sinx₂; x3¼rcosx₁cosx₂sinx₃;

...

xn1¼rcosx₁. . .cosxn2sinxn1; xn¼rcosx₁. . .cosxn2cosxn1: The spherical coordinates will be useful in some calculations.

LetDðp;cÞbe a starlike domain. For eachtA½0;cðyÞÞ, letT_gðt;yÞ^? M be the orthog- onal complement ofg⁰ðt;yÞin the tangent spaceTgðt;yÞM. Define theradial curvature operator Rðt;yÞ:T_gðt;yÞ^? M!T_gðt;^? _yÞM by Rðt;yÞX:¼RðX;g⁰ðt;yÞÞg⁰ðt;yÞ. Observe that Ricrðt;yÞis the trace ofRðt;yÞ.

Fix y. Define the path of linear operators Aðt;yÞ:T_gð0;^? _yÞM!T_gð0;yÞ^? M by Aðt;yÞX¼ ðttÞ¹JðtÞ, whereJðtÞis the Jacobi field alonggð;yÞsatisfyingJð0Þ ¼0 and‘g⁰ð0;yÞJð0Þ ¼X, andtt is the parallel transport fromT_gð0;yÞ^? M toT_gðt;yÞ^? M along gð;yÞ. Define also the path of linear operators Rðt;yÞ:T_gð0;^? _yÞM!T_gð0;yÞ^? M by Rðt;yÞX¼ ðttÞ¹Rðt;yÞðttXÞ. It is well known thatAðt;yÞis the solution of the equa- tionA⁰⁰ðt;yÞ þRðt;yÞAðt;yÞ ¼0 with initial conditionsAð0;yÞ ¼0 andA⁰ð0;yÞ ¼I, where I denotes the identity operator. Moreover, the volume element in polar coor- dinate system is given by ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

detgðt;yÞ

p :dt:dA¼detAðt;yÞ:dt:dA.

Now we recall Bishop’s celebrated volume comparison theorem (see [3] and com- pare with [5]). Denote byS_kthe function

S_kðtÞ ¼

1ffiffi_k

p sinð ffiffiffi pk

tÞ k>0

t k¼0

1ffiffiffiffiffi

pksinhð ffiffiffiffiffiffiffi pk

tÞ k<0:

8>

><

>>

:

Theorem 2.1 (Bishop).Let M be a Riemannian manifold and fix pAM. Consider a geodesic gð;yÞ parametrized by arclength in M such that gð0;yÞ ¼ p. Suppose that the radial Ricci curvature along gð;yÞis greater than or equal to ðn1Þk for every tAð0;cðyÞ.Then

detAðt;yÞ S_kⁿ¹ðtÞ

!0

c0 ð1Þ

onð0;cðyÞÞand

detAðt;yÞcS_kⁿ¹ðtÞ

on½0;cðyÞ.We have equality in(1)at t¼t0Að0;cðyÞÞif and only if Aðt;yÞ ¼S_kðtÞI; Rðt;yÞ ¼kI

for every tAð0;t₀.

Denote the geodesic ball of radiusrin the space form of constant curvaturekby BkðrÞ. A global version of Theorem 2.1 is given below.

Theorem 2.2(Bishop).Let M be a Riemannian manifold and fix pAM.Suppose that the radial Ricci curvature is greater than or equal toðn1Þkin Bðp;rÞ.Then

VolðBðp;rÞÞcVolðBkðrÞÞ ð2Þ

with equality if and only if Bðp;rÞis isometric to BkðrÞ.

We end this section remarking that we will usually simplify the notation if there is no possibility of misunderstandings (for example,AðtÞinstead ofAðt;yÞ).

3 Symmetrizations and the main theorems: a summary

In order to define the symmetrizations, we present some types of starlike domains.

Definition 3.1.We have the following types of metrics on starlike domains, from the more general to the more specific (All metrics are written in a polar coordinate sys- tem).

1. General starlike domains: Metrics of typeds_g²¼dt²þgijðt;yÞdyⁱdy^j.

2. Starlike domains with scalar radial curvature operator: Metrics of typeds_f²¼dt²þ f²ðt;yÞdy².

3. Radially symmetric starlike domains: Metrics of typeds²_h¼dt²þh²ðtÞdy². 4. Starlike domains with constant curvaturek: Metrics of typeds_k²¼dt²þS_k²ðtÞdy².

For the sake of brevity, we call them starlike domains of typeiði¼1;2;3;4Þ. The definition of radially symmetric starlike domain may seem a little bit artificial. In fact, it is more suitable to geodesic balls within the injectivity radius. But we conserve Def- inition 3.1 as it is, in order to emphasize the hierarchy that exists on the metrics in starlike domains.

The reason why the second type of starlike domains is calledstarlike domains with scalar radial curvature operatorwill be explained in Proposition 4.1.

It is a natural idea to create a symmetrization process such that the first step is to transform a general starlike domain into a starlike domain with scalar radial curva- ture operator, the second step is to transform a starlike domain with scalar radial cur- vature operator into a radially symmetric starlike domain and so on. In order to de- fine these symmetrizations, we have to determine what properties we want to keep.

Let us define these symmetrizations. One of them transforms a starlike domain of type 1 into a starlike domain of type 2, and it is calledsymmetrization(of a starlike domain)along the radial geodesics. For the sake of brevity, we call it 1-2 symmetri- zation. It is characterized by transforming a general starlike domain Dgðp;cÞinto a starlike domain with scalar radial curvature operator D_fðp;cÞ with the same radial Ricci curvature. The other symmetrization transforms a geodesic ballB_fðp;rÞof type 2 within the injectivity radius into a geodesic ballB_hðp;r_hÞof type 3, withr_hcr. It is calledsymmetrization(of a geodesic ball of type2 within the injectivity radius)along spheres that are equidistant to the origin, and for the sake of brevity we call it 2-3 symmetrization. The 2-3 symmetrization is characterized by the property that the average of the radial Ricci curvature on qBfðp;tÞ,tAð0;rhÞ, is equal to the corre- spondent average on qBhðp;tÞ. We could create a ‘‘3-4 symmetrization’’, but these two symmetrizations are enough for our purposes. The formalization and all details about these symmetrizations will be done in Section 4.

A remark must be made: The metric of these symmetrizations can loose its smooth- ness atðt¼0Þand become only continuous at this point. But this loss of regularity will not harm the volume calculations.

The following theorem generalizes Bishop’s volume comparison theorem.

Theorem 3.2.LetðMⁿ;gÞbe a Riemannian manifold,Dgðp;cÞHM a starlike domain, and let Dfðp;cÞbe the1-2symmetrization of Dgðp;cÞ.Fixy.If ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

detgðt;yÞ

p :dt:dA and

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi detfðt;yÞ

p :dt:dA are respectively the volume element of Dgðp;cÞ and Dfðp;cÞ in a

polar coordinate system,then

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi detgðt;yÞ p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi detfðt;yÞ p

!0

c0 ð3Þ

onð0;cðyÞÞand

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi detgðt;yÞ

p :dt:dAc ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

detfðt;yÞ

p :dt:dA: ð4Þ

on½0;cðyÞÞ.In particular, VolðDgðp;cÞÞcVolðDfðp;cÞÞ.

Equality is achieved in(3) (as well as in(4))at t₀Að0;cðyÞÞ,if and only if Rgðt;yÞ ¼Rfðt;yÞ

for every tA½0;t0,whereRgandRf are identified in the natural way.

The radial curvature operator has a leading rule to determine the volume element behavior through the equation

A⁰⁰ðt;yÞ þRðt;yÞ:Aðt;yÞ ¼0 Að0;yÞ ¼0

A⁰ð0;yÞ ¼I:

8<

: ð5Þ

The di¤erence between Theorem 2.1 and Theorem 3.2 is that in the former, the solution of (5) is compared to the solution of the equation

A⁰⁰ðt;yÞ þh^{infsA½0;cðyÞÞ}_ðn1Þ^Ricrðs;yÞ:Ii

:Aðt;yÞ ¼0

Að0;yÞ ¼0 A⁰ð0;yÞ ¼I;

8>

<

>: ð6Þ

and in Theorem 3.2, the solution of (5) is compared to the solution of the equation A⁰⁰ðt;yÞ þh^Ricrðt;_ðn1Þ^yÞ:Ii

:Aðt;yÞ ¼0

Að0;yÞ ¼0 A⁰ð0;yÞ ¼I:

8>

<

>: ð7Þ

Therefore Theorem 3.2 is a qualitative generalization of Theorem 2.1.

The symmetrization along radial geodesics allow us to get some upper bounds for the volume element in a polar coordinate system. These estimates are given in terms of some integrals of Ricrðt;yÞ. This is possible for three reasons: First of all, the sym- metrization along radial geodesics is volume increasing. Secondly, we do not loose any information about the radial Ricci curvature alonggð;yÞ, as it happens in Equation (6). Finally, Equation (7) is simple enough to get the desired upper bounds.

Let us be more explicit. ConsiderDðp;cÞa starlike domain. Fixy. Set Ricrðt;yÞ :¼maxð0;Ricrðt;yÞÞ. Bishop’s volume comparison theorem implies that

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi detgðs;yÞ

p c sinhðpffiffiffik ffiffiffi :sÞ pk

n1

ð8Þ wherek¼sup_tA½0;cðyÞÞðRicrðt;yÞÞ(Ifk¼0, then the inequality above has the obvious

meaning). Therefore we can estimate the volume element in polar coordinate system in terms of aL^y norm of ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðRicrÞ

p alonggð;yÞ. In Theorem 3.3, we use Theorem 3.2 in order to get anðL²Þ²version of (8).

Theorem 3.3.LetðMⁿ;gÞbe a Riemannian manifold,D_gðp;cÞHM a starlike domain, and ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

detgðt;yÞ

p :dt:dA its volume element in a polar coordinate system.FixyASⁿ¹

and define kRicrðs;yÞk_L1 :¼Ðs

0Ricrðt;yÞdt. Then there exist constants A1;A2;B1

and B₂such that the following(equivalent)estimates hold:

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi detgðs;yÞ

p c

2 64A₁:s:

sinh ^B1:kRicr

ðs;yÞk_L1:s ðn1Þ

B1:kRicrðs;yÞk_L1:s ðn1Þ

3 75

n1

ð9Þ

and

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi detgðs;yÞ

p cA₂ⁿ¹sⁿ¹e^{kRicrðs;yÞkL1:B2:s}: ð10Þ

IfkRicrðs;yÞk_L1¼0,then(9)has the obvious meaning.

We have the following estimate for geodesic balls as a consequence of Theorem 3.3. The geodesic ball is not necessarily within the injectivity radius.

Corollary 3.4.Let Mⁿbe a complete Riemannian manifold and Bðp;rÞHM a geodesic ball.Then there exist constants A₃;B₃>0such that

Vol½Bðp;rÞcA₃ⁿ¹rⁿ¹ ð

Sⁿ¹

ðr 0

e^{B3:r2:Ricrðt;yÞ}dt:dA: ð11Þ

Finally we have the volume comparison theorem that is related to the 2-3 sym- metrization.

Theorem 3.5.LetðBfðrÞ;dt²þf²ðt;yÞdy²ÞHM be a geodesic ball within the injectivity radius.If B_hðrhÞis the 2-3symmetrization of B_fðrÞ,then we have that Vol½qBhðtÞc Vol½qBfðtÞfor every tA½0;r_hÞ.In particular, Vol½BhðrhÞcVol½BfðrhÞ.If n¼2,then we have that r_h¼r, Vol½qBhðtÞ ¼Vol½qBfðtÞ for every tA½0;rÞ, and Vol½BhðrÞ ¼ Vol½BfðrÞ.This theorem is still valid if B_fðrÞis the1-2symmetrization of another geo- desic ball.

Let us make some remarks about these results. Theorems 3.2 and 3.3 deal with the geometry along geodesics that emanate from some point. We control the curvature in order to get upper bounds for the volume element in polar coordinates. Notice that the radial curvature operator and the radial Ricci curvature are very important to this kind of theory.

Myers’ classical theorem says that if RicðÞdðn1Þalong a geodesic gwith arc- length greater than p, then gis not minimizing (See [10]). In particular, if the Rie-

mannian manifoldM is complete, thenMis compact and its diameter is less than or equal to p. Afterwards Ambrose, Avez, Calabi, Galloway and Markvorsen among others generalized Myers’ theorem imposing weaker conditions on the Ricci curvature (See [1], [2], [4], [8], [9]). These works show that if we have some ‘‘positiveness’’ on the Ricci curvature along a geodesic, then the solution of Equation (5) becomes singular after some time, and the geodesic is no longer minimizing after that. Theorem 3.3 is similar to these works because we can get a upper bound for the volume element in terms of some lower bounds of the Ricci curvature. The di¤erence is that the upper bound for the volume element does not go to zero as in these former works.

In order to generalize Myers’ theorem, Ambrose, Calabi and Markvorsen (see [1], [4], [9]) use conditions on the integral of the Ricci curvature along geodesics. Notice that Theorem 3.3 is an integral version of Equation (8), although it is not a full gen- eralization. We will make some comments about the lack of sharpness of Theorem 3.3 in Remark 5.4.

Controlling several geometric properties of Riemannian manifolds through some integrals of the curvature has been an important issue nowadays. For instance, many works have used some local-global integral of the lowest eigenvalue of the Ricci tensor to study geometrical and topological properties of Riemannian manifolds (See [6], [7], [11], [12], [13] among others). In particular, we can get bounds for the diameter and a generalization of Bishop’s volume comparison theorem using these integral invariants (See [11], [12], [13]).

4 Symmetrization of starlike domains and a generalization of Bishop’s volume comparison theorem

In this section, we describe the symmetrization process on starlike domains. In Sub- section 4.1, we define the 1-2 symmetrization and we get a qualitative generalization of Bishop’s volume comparison theorem. In Subsection 4.2, we define the 2-3 sym- metrization.

4.1 Symmetrization along radial geodesics and a qualitative generalization of Bishop’s volume comparison theorem.We begin justifying the namestarlike domain with scalar radial curvature operator.

Proposition 4.1. Let ðDfðp;cÞ;dt²þf²ðt;yÞ:dy²Þ be a starlike domain with radial scalar curvature operator. Then,for every ðt;yÞAD_fðp;cÞ fpg, its radial curvature operator is given by

Rðt;yÞ ¼ f⁰⁰ðt;yÞ

fðt;yÞ I; ð12Þ

where I denotes the identity operator.

Proof.Fixðt;yÞAðDfðp;cÞ fpgÞ. In spherical coordinatesðr;x₁;. . .;xn1Þ, we have that

ds²_f ¼dt²þf²ðt;yÞ:dx²₁þf²ðt;yÞ:cos²x₁:dx₂²

þf²ðt;yÞ:cos²x₁:cos²x₂:dx₃²þ þ f²ðt;yÞ:cos²x₁. . .cos²x_n2:dx_n1² : Observe that we can always choose spherical coordinates without singularities at ðt;yÞ. Now we can calculate the Christo¤el symbols and the components of the cur- vature operator atðt;yÞexplicitly, and the result follows. r Now we describe the 1-2 symmetrization. Denote the space of symmetricðn1Þ ðn1Þmatrices overRbyM_n1 and the starshaped Euclidean domain with respect to the originfðt;yÞAEⁿ;yASⁿ¹;0ct<cðyÞgbyDð0;cÞ.

Theorem 4.2.Let D_gðp;cÞbe a general starlike domain.For eachyfixed,let fðt;yÞbe the solution of the equation f⁰⁰ðt;yÞ þ^Ricrðt;_ðn1Þ^yÞfðt;yÞ ¼0satisfying the initial conditions fð0;yÞ ¼0 and f⁰ð0;yÞ ¼1. Consider the punctured starshaped Euclidean domain Dð0;cÞ f0gendowed with the symmetric2-form

ds²_f ¼dt²þf²ðt;yÞdy²; ð13Þ and denote it by D_fðp;cÞ ft¼0g.Then ds_f²is a smooth metric in½Dfðp;cÞ ft¼0g extendable to a continuous metric at ðt¼0ÞAD_fðp;cÞ. Moreover, D_fðp;cÞ has the same radial Ricci curvature as D_gðp;cÞ.

Proof.We will divide the proof in three parts:

First part: The 2-formds_f²is a positive definite symmetric 2-form onD(0,c)C{0}.It is a consequence of Lemma 4.3 below. Notice that its proof is similar to the proof of Bishop’s volume comparison theorem (compare [5]).

Lemma 4.3.LetR:½0;rÞ !Mn1be a continuous map andA:½0;rÞ !Mn1the so- lution of the matricial ordinary di¤erential equationA⁰⁰ðtÞ þRðtÞ:AðtÞ ¼0with initial conditionsAð0Þ ¼0andA⁰ð0Þ ¼I.Consider the symmetrization of this problem,that is, A_f⁰⁰ðtÞ þRfðtÞ:AfðtÞ ¼0 with initial conditions Afð0Þ ¼0 and A_f⁰ð0Þ ¼I, where RfðtÞ:¼trace½RðtÞ=ðn1Þ:I.IfdetAðtÞis non-singular for every tAð0;rÞ,then

detAðtÞ detAfðtÞ

0

c0 ð14Þ

onð0;rÞand

detAðtÞcdetAfðtÞ: ð15Þ

on½0;rÞ.In particular, detAfðtÞis also non-singular onð0;rÞ.

Equality is achieved in(14) (as well as in(15))at t0Að0;rÞif and only ifR¼Rf on

½0;t₀.

Proof of Lemma 4.3. Consider TAM_n1. The Cauchy–Schwarz inequality implies that

traceðT²ÞdðtraceðTÞÞ²

n1 ð16Þ

with equality if and only ifT is a scalar multiple of the identity.

Define U:¼A⁰A¹. Then U is self-adjoint and satisfies the matricial Riccati equation

U⁰þU²þR¼0: ð17Þ

By (16) we have that

0¼ ðtraceðUÞÞ⁰þtraceðU²Þ þtraceRdðtraceðUÞÞ⁰þðtraceðUÞÞ²

n1 þtraceðRfÞ andf:¼traceðUÞ ¼ ðdetAÞ⁰=detAsatisfies the di¤erential inequality

f⁰þ f²

n1þtraceðRfÞc0: ð18Þ

Let us studyAf. Ifjis the solution of j⁰⁰ðtÞ þtraceðRfðtÞÞ

n1 jðtÞ ¼0 ð19Þ

with initial conditionsjð0Þ ¼0 andj⁰ð0Þ ¼1, thenjⁿ¹ðtÞ ¼detAfðtÞ. Set c:¼ðdetAfÞ⁰

ðdetAfÞ ¼ ðn1Þj⁰

j: ð20Þ

Using (19) and (20) we have that

c⁰þ c²

n1þtraceðRfÞ ¼0: ð21Þ

Let ð0;bÞHð0;rÞ be the maximal interval such that det A_fðtÞ>0 for every tAð0;bÞ. We will compare fandcin ð0;bÞ. They satisfy (18) and (21) respectively and limt!0^þfðtÞ ¼limt!0^þcðtÞ ¼ þy. In order to compare them neart¼0 (let us say, in the intervalð0;e), consider its inversesff~¼1=fandcc~¼1=c. They satisfy re- spectively

f~

f⁰ðtÞd 1

n1þtraceðRfðtÞÞ:ff~²ðtÞ ð22Þ

and

c~

c⁰ðtÞ ¼ 1

n1þtraceðRfðtÞÞ:cc~²ðtÞ ð23Þ withfð0Þ ¼cð0Þ ¼0. Subtracting (22) from (23) we have that:

ðccðtÞ ~ ffðtÞÞ~ ⁰ctraceðRfðtÞÞðccðtÞ þ~ ffðtÞÞð~ ccðtÞ ~ ffðtÞÞ:~ Seta1ðtÞ:¼ ðccðtÞ ~ ffðtÞÞ~ andb1ðtÞ:¼traceðRfðtÞÞðccðtÞ þ~ ffðtÞÞ. Then~

d

dtða1ðtÞe Ð^t

0b1ðzÞdz

Þ ¼e Ð^t

0b1ðzÞdz

ða₁⁰ðtÞ b₁ðtÞa1ðtÞÞc0 and it follows that

a₁ðtÞe Ð^t

0b1ðzÞdz

c0)a₁ðtÞc0

with equality if and only ifRðzÞ ¼RfðzÞfor everyzA½0;t. ThereforefðtÞccðtÞin ð0;e, withfðt0Þ ¼cðt0Þif and only ifRðtÞ ¼RfðtÞfor everytA½0;t₀.

The comparison betweenfandcin½e;bÞfollows in a similar fashion. Subtracting (21) from (18), we have that

ðfcÞ⁰cðcþfÞ

n1 ðfcÞ:

Seta2ðtÞ:¼ ðfðtÞ cðtÞÞandb2ðtÞ:¼ ðcðtÞ þfðtÞÞ=ðn1Þ. Then d

dtða2ðtÞ:eÐ^t

eb2ðzÞdzÞ ¼e Ð^t

eb2ðzÞdzða₂⁰ðtÞ a2ðtÞb2ðtÞÞc0 hence

a₂ðtÞca₂ðeÞe Ð^t

eb2ðzÞdz

with equality if and only ifRðzÞ ¼RfðzÞfor everyzA½e;t. ThereforefðtÞccðtÞ þ a₂ðeÞe

Ð^t

eb2ðzÞdz

in½e;rÞ, withfðt0Þ ¼cðt0Þif and only iffðeÞ ¼cðeÞ, andRðtÞ ¼RfðtÞ for everytA½0;t₀.

Joining the estimates in ð0;e and ½e;bÞ, we have thatfðtÞccðtÞin ð0;bÞ, with fðt0Þ ¼cðt0Þif and only ifRðtÞ ¼RfðtÞfor everytA½0;t₀.

Thus

detðAfðtÞÞ detðAðtÞÞ

detAðtÞ detAfðtÞ

0

¼ðdetAðtÞÞ⁰

detAðtÞ ðdetAfðtÞÞ⁰ detAfðtÞ c0

what implies that

detAðtÞ detAfðtÞ

0

c0:

Therefore detAðtÞcdetAfðtÞinð0;bÞdue to

t!0þlim

detAðtÞ detAfðtÞ¼1;

and consequently we have thatb¼r, which settles Lemma 4.3. r Equation (15) implies that (13) is positive definite in Dð0;cÞ f0g, which settles the first part of the proof of Theorem 4.2.

Second part: The 2-form ds²_f is a smooth metric defined in Df(p,c)C{tF0}, con- tinuously extendable at (tF0)ADf(p,c).The smoothness ofds_f²¼dt²þ f²ðt;yÞ:dy² inDfðp;cÞ ft¼0gfollows from a classical result on ordinary di¤erential equations (smooth dependence of f in terms of the parametery).

We prove now thatds_f² can be continuously extended toðt¼0Þ. LetX andY be two continuous vector fields in the ballBð0;rÞHDð0;cÞ, wherer>0 is a su‰ciently small positive number. Consider the Euclidean metric ds_E²¼dt²þt²:dy² in Bð0;rÞ.

DecomposingX andYin its radial and angular components inBð0;rÞ ft¼0g, we have that

X ¼X_tþX_y and Y¼Y_tþY_y: ð24Þ The Euclidean scalar producthX;Yi_Ecan be written as

hX;Yi

E ¼hX_t;Y_ti

EþhX_y;Y_yi

E: ð25Þ

Now consider the 2-formds²_f ¼dt²þf²ðt;yÞdy²inBð0;rÞ ft¼0g. Denoteds_f² by h;i_f. We will prove that hX;Yi_f is continuously extendable at ðt¼0Þ, with limt!0hXðt;yÞ;Yðt;yÞi_f ¼hXð0Þ;Yð0Þi_E.

The decomposition ofX andY in its radial and angular part with respect tods_f² is also given byX ¼X_tþX_y andY ¼Y_tþY_y. Indeed the radial-angular decompo- sition of the tangent spaces coincide for ds²_f andds_E². Thus the scalar product of X andY inds_f²is given by

hX;Yi_f ¼hXt;Yti_f þhXy;Yyi_f: ð26Þ Now observe that

hXt;Xti_E¼hXt;Xti_f

and

hXy;Xyi_f ¼ f²ðt;yÞ

t² hXy;Xyi_E: Hence

hX;Yi_f ¼hX_t;Y_ti_Eþ f²ðt;yÞ

t² hX_y;Y_yi_E: ð27Þ For fixedy, the Taylor series of fðt;yÞ is given by fðt;yÞ ¼tþ f⁰⁰ðty;yÞt²=2 for some tyAð0;tÞ. We know that f⁰⁰ðt;yÞ ¼ Ricrðt;yÞfðt;yÞ=ðn1Þ is bounded in Bð0;rÞ ft¼0g. Hence limt!0f²ðt;yÞ=t²¼1 uniformly with respect toy, what im- plies that limt!0hXðt;yÞ;Yðt;yÞi_f ¼hXð0Þ;Yð0Þi_E. Thereforeds_f² can be extended continuously toðt¼0Þ.

Third part: The starlike domain Df(p,c) has the same radial Ricci curvature as Dg(p,c).It follows from the definition of f and Proposition 4.1. This settles Theorem

4.2. r

Therefore the symmetrization along radial geodesics is a well defined operation.

The only problem, as observed in the introduction, is that the metric is not necessarily smooth at the origin, but this is a minor problem because it does not harm the vol- ume calculations.

Definition 4.4.The starlike domainðDfðp;cÞ;dt²þf²ðt;yÞdy²Þconstructed in The- orem 4.2 is called the 1-2 symmetrization ofDgðp;cÞ.

We have the following qualitative generalization of Bishop’s volume comparison theorem as a direct consequence of Proposition 4.1, Theorem 4.2 and Lemma 4.3.

Theorem 4.5.LetðMⁿ;gÞbe a Riemannian manifold,D_gðp;cÞHM a starlike domain, and let D_fðp;cÞbe the1-2symmetrization of D_gðp;cÞ.Fixy.If ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

detgðt;yÞ

p :dt:dA and

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi detfðt;yÞ

p :dt:dA are respectively the volume element of D_gðp;cÞand D_fðp;cÞin a polar

coordinate system,then

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi detgðt;yÞ p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi detfðt;yÞ p

!0

c0 ð28Þ

onð0;cðyÞÞand

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi detgðt;yÞ

p :dt:dAc ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

detfðt;yÞ

p :dt:dA: ð29Þ

on½0;cðyÞÞ.

In particular, VolðDgðp;cÞÞcVolðDfðp;cÞÞ.

Equality is achieved in(28) (as well as in(29))at t₀Að0;rÞ,if and only if Rgðt;yÞ ¼Rfðt;yÞ

for every tA½0;t₀,whereRgandRf are identified in the natural way.

Remark 4.6.A geodesic ballBðp;rÞHM is a starlike domainD_gðp;cÞwith some of its closure points included. Therefore we can generalize Theorem 2.2 replacing (2) by

VolðBðp;rÞÞcVolðDfðp;cÞÞ;

whereD_fðp;cÞis the 1-2 symmetrization ofD_gðp;cÞ.

4.2 Symmetrization along spheres that are equidistant to the origin. Let ðBfðrÞ;

dt²þ f²ðr;yÞdy²Þbe a geodesic ball within the injectivity radius. Define Ricr_fðtÞas the average of the radial Ricci curvature inqB_fðtÞ. Observe that RicrfðtÞcan be ex- tended continuously toðt¼0Þas the scalar curvature ofB_fðrÞat the origin, even if B_fðrÞis a 1-2 symmetrization of another geodesic ball. We will create a radially sym- metric geodesic ballðBhðrhÞ;dt²þh²ðtÞdy²Þ,r_hcr, such that the average of the ra- dial Ricci curvature onqBhðtÞis equal to RicrfðtÞfor everytA½0;rhÞ.

Using Proposition 4.1, the radial Ricci curvature of a radially symmetric geodesic ball with metricds_h²¼dt²þh²ðtÞdy²is given by

RicrhðtÞ ¼ ðn1Þh⁰⁰ðtÞ hðtÞ ;

which does not depend on y. Thus the expression above is also the average of the radial Ricci curvature onqBhðtÞ. Thereforehmust satisfy

h⁰⁰ðtÞ þRicr_fðtÞ

n1 hðtÞ ¼0; hð0Þ ¼0; h⁰ð0Þ ¼1: ð30Þ The existence and uniqueness ofhis assured by the theory of ordinary di¤erential equations. The 2-formds²_h¼dt²þh²ðtÞdy² is obviously smooth in BhðrÞ ft¼0g and it can be extended continuously toðt¼0Þ, because we can prove that limt!0ds_h²

¼ds_E²in the same fashion as in the second part of the proof of Theorem 4.2. Finally, restrict the domain ofh to the maximal interval½0;r_hÞJ½0;rÞsuch that hðtÞ>0 for everytAð0;r_hÞ, and we have the following definition:

Definition 4.7. The geodesic ballðBhðrhÞ;dt²þh²ðtÞdy²Þis called the 2-3symmetri- zationofBfðrÞ.

Remark 4.8. The 2-3 symmetrization can be extended to starlike domains such that

VolðqDfðp;tÞÞis a smooth function oft. But we will restrict this symmetrization to geodesic balls within the injectivity radius due to the artificiality of the general situa- tion.

5 Volume estimates

In Subsection 5.1, we get some upper bounds for the volume using the 1-2 symmet- rization. Afterwards, in Subsection 5.2, we prove a volume comparison theorem re- lated to the 2-3 symmetrization.

5.1 Upper bounds for the volume related to the 1-2 symmetrization.

Theorem 5.1.LetðMⁿ;gÞbe a Riemannian manifold,Dgðp;cÞHM a starlike domain, and ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

detgðt;yÞ

p :dt:dA its volume element in a polar coordinate system.FixyASⁿ¹

and define kRicrðs;yÞk_L1 :¼Ðs

0Ricrðt;yÞdt. Then there exist constants A₁;A₂;B₁ and B₂such that the following(equivalent)estimates hold:

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi detgðs;yÞ

p c

2 64A₁:s:

sinh ^B1:kRicr

ðs;yÞk_L1:s ðn1Þ

B1:kRicrðs;yÞk_L1:s ðn1Þ

3 75

n1

ð31Þ

and

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi detgðs;yÞ

p cA₂ⁿ¹sⁿ¹e^{kRicrðs;yÞkL1:B2:s}: ð32Þ

IfkRicrðs;yÞk_L1¼0,then(31)has the obvious meaning.

Proof.We begin with some estimates on the solution of the equation f⁰⁰ðtÞ þKðtÞ:fðtÞ ¼0

fð0Þ ¼0 f⁰ð0Þ ¼1 8>

<

>: ð33Þ

defined on the interval½0;s, whereK is a continuous function on½0;sand fðtÞ>0 inð0;s. SetKðtÞ ¼maxðKðtÞ;0Þ. We are looking for an upper bound of fðsÞin terms ofkKk_L1. Thus we can consider

ff~⁰⁰ðtÞ KðtÞ:ff~ðtÞ ¼0 ff~ð0Þ ¼0

ff~⁰ð0Þ ¼1 8>

<

>:

instead of (33) because fcff~. Observe that ff~is increasing and convex on½0;s.

Assume that kKk_L100 (otherwise there is nothing to prove). Take s0¼0<

s1< <sN1<sNARsuch thatDs:¼siþ1si¼1=ð4kKk_L1ÞandsN1<scsN. Suppose thatDs<s, which is equivalent to supposing thatNd2 (The other case will

be considered afterwards). Fix½si;s_iþ1for somei¼0;. . .;N2. We intend to esti- mate ff~ðsiþ1Þand ff~⁰ðsiþ1Þin terms of ff~ðsiÞand ff~⁰ðsiÞ.

We have that 1

2ðff~⁰ðsiþ1ÞÞ²1

2ðff~⁰ðsiÞÞ² ¼1

2ðff~⁰ðtÞÞ²j_s^s_i^iþ1 ¼1 2

ðsiþ1

si

ððff~⁰ðtÞÞ²Þ⁰:dt

¼ ðsiþ1

si

ff~⁰⁰ðtÞ:ff~⁰ðtÞ:dt¼ ðsiþ1

si

KðtÞ:ff~ðtÞ:ff~⁰ðtÞ:dtcff~ðsiþ1Þff~⁰ðsiþ1ÞkKk_L1; which gives the following quadratic inequality in terms of ff~⁰ðsiþ1Þ:

ðff~⁰ðsiþ1ÞÞ²2ff~ðsiþ1ÞkKk_L1ff~⁰ðsiþ1Þ ðff~⁰ðsiÞÞ²c0: ð34Þ Solving (34), we have the estimate

ff~⁰ðsiþ1Þc2ff~ðsiþ1ÞkKk_L1þff~⁰ðsiÞ: ð35Þ We know that ff~is convex, what gives

ff~ðsiþ1Þ ff~ðsiÞ

Ds cff~⁰ðsiþ1Þ: ð36Þ Joining (35), (36) andDs¼1=ð4kKk_L1Þ, we have after some calculations that

ff~ðsiþ1Þc 1 2kKk_L1

ff~⁰ðsiÞ þ2ff~ðsiÞ: ð37Þ

Combining (35) and (37), we can finally estimate ff~ðsiþ1Þand ff~⁰ðsiþ1Þin terms of ff~ðsiÞand ff~⁰ðsiÞ:

ff~ðsiþ1Þc2ff~ðsiÞ þ 1 kKk_L1

ff~⁰ðsiÞ ð38Þ

ff~⁰ðsiþ1Þc4kKk_L1ff~ðsiÞ þ2ff~⁰ðsiÞ: ð39Þ A priori, the estimates (38) and (39) are valid only fori¼0;. . .;N2. We claim that we can estimate ff~ðsÞand ff~⁰ðsÞfrom ff~ðsN1Þand ff~⁰ðsN1Þusing (38) and (39).

Indeed, we only have to repeat all the calculations replacing Ds¼1=ð4kKk_L1Þby Dsc1=ð4kKk_L¹Þ.

Now we use (38) and (39) N times in order to estimate ff~ðsÞ from ff~ð0Þ ¼0 and ff~⁰ð0Þ ¼1. This is a Linear Algebra problem: The matrix

M¼ 2 _kK¹k_L1

4kKk_L1 2

" #

has eigenvalues 0 and 4, and their respective eigenvectors are

v₀¼ _2kK¹k_L1

1

" #

and v₄ ¼

1 2kKk_L1

1

" #

:

The vector

"

ff~ð0Þ ff~⁰ð0Þ

#

¼ 0 1 is written as¹₂v₀þ¹₂v₄. AfterNiterations, we get

fðsÞcff~ðsÞc M^N: 1 2v0þ1

2v4

1

¼ 4^N1 kKk_L1

c4^ð4kKkL1sÞ4s; ð40Þ where ½₁ represents the first line of the vector. Thus (40) is the desired estimate of (33) ifDs<s.

IfDsds, then we make the same estimates directly on the interval½0;sinstead of making on each interval½si;siþ1. As a result, Inequality (37) gives ff~ðsÞc2s, which is included in (40). Hence (40) is the desired estimate of (33).

Let us return to the theorem. In order to get an upper bound for the volume ele- ment ofD_gðp;cÞ, we can consider its 1-2 symmetrizationD_fðp;cÞinstead of the origi- nal domain (due to Theorem 4.5). If we write the metric ofD_fðp;cÞasds_f²¼dt²þ f²ðt;yÞdy, then f is exactly the solution of (33) whenKðtÞ is replaced by RicrðtÞ=

ðn1Þ. But the volume element can be written as fⁿ¹ðt;yÞ:dt:dA, and (32) follows.

In order to get (31), apply (32) in the formulax:e^xcsinh 2x. Finally, estimates (31)

and (32) are equivalent because sinhxcx:e^x. r

Let us make some remarks about Theorem 5.1.

Remark 5.2.We can get the explicit estimatesA1¼8,A2¼4,B1¼8 ln 4 andB2 ¼ 4 ln 4 using the proof above, but they certainly are not close to the sharpest ones.

Remark 5.3.Let us point out the importance of the termðs:kRicrðs;yÞk_L1Þin this kind of estimates.

Consider a starlike domain ðDfðp;cÞ;dt²þf²ðt;yÞdy²Þ. Its volume element in a polar coordinate system is given by fⁿ¹ðt;yÞ:dt:dA, where fð;yÞ:½0;t !Ris the solution of the equation

f⁰⁰ðt;yÞ þRicrðt;yÞ

n1 :fðt;yÞ ¼0 satisfying the initial conditions fð0;yÞ ¼0 and f⁰ð0;yÞ ¼1.

We want to compareD_fðp;cÞandðDflðp;l:cÞ;dt²þ f_l²ðt;yÞdy²Þ, wherel>0 and f_lðt;yÞ:¼l:fðt=l;yÞ. We have that

f_l⁰⁰ðt;yÞ þ 1 l²

Ricrðt=l;yÞ

ðn1Þ flðt;yÞ ¼0:

If we identifyD_fðp;cÞandD_flðp;l:cÞvia dilation by l, then we can see that every pair of identified points has the same value ofðs:kRicrðs;yÞk_L1Þ. Moreover the vol- ume element ofDflðp;l:cÞin polar coordinate system islⁿ¹times the correspondent volume element ofDfðp;cÞ. Therefore the estimate (31) is good because it considers this kind of symmetries.

Remark 5.4.Unfortunately, estimates (31) and (32) are not sharp because their growth rate whensgoes to infinity are larger than the growth rate of the volume element of geodesic balls in space forms with constant negative curvature. We could try to im- prove (31) considering estimates like

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi detgðs;yÞ

p c

2 64A1:

sinh ^B1:kRicr

ðs;yÞk_L1:s ðn1Þ

B1:kRicrðs;yÞk_L1:s ðn1Þ

3 75

n1

; ð*Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi detgðs;yÞ

p c

2 64A₁:

sinh ^B1:kRicr

ðs;yÞk_L1

ðn1Þ

B1:kRicrðs;yÞk_L1

ðn1Þ

3 75

n1

ð**Þ

or

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi detgðs;yÞ

p c

2 64A₁:s:

sinh ^B1:kRicr

ðs;yÞk_L1 ðn1Þ

B1:kRicrðs;yÞk_L1

ðn1Þ

3 75

n1

; ð***Þ

but they do not work: Estimates (*) and (**) do not work because we can take a suitable sequence of starlike domains with negative curvatureDflðp;l:cÞ,l!y(see Remark 5.3), and see that (*) and (**) fail. Estimate (***) fails because for everyA1

andB1, we can choose a space form with small negative curvature (in absolute value) such that (***) does not hold for a su‰ciently large geodesic ball in it. These exam- ples show that it is not easy to improve (31) using similar estimates that depend on kRicrðs;yÞk_L1.

A natural candidate for a sharp estimate is an inequality of the form ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

detgðt;yÞ

p cA₁:s:

sinh B1:

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Ricrðs;yÞ

p

n1

L¹

B₁:

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Ricrðs;yÞ

p

n1

_L1

:

As a consequence of Theorem 5.1, now we prove the following upper bound for the volume of a geodesic ball, which is not necessarily within the injectivity radius.

Corollary 5.5.Let Mⁿbe a complete Riemannian manifold and Bðp;rÞHM a geodesic ball.Then there exist constants A₃;B₃>0such that

Vol½Bðp;rÞcA₃ⁿ¹rⁿ¹ ð

Sⁿ¹

ðr 0

e^{B3:r2:Ricrðt;yÞ}dt:dA: ð41Þ

Proof.Using Formula (32), we have the following estimate:

Vol½Bðp;rÞc ð

Sⁿ¹

ðccðyÞ~ 0

A₂ⁿ¹sⁿ¹e^B2:s:

Ð^s

0Ricrðt;yÞdt

:ds:dA:

SubstitutingsinÐs

0Ricrðt;yÞdtandccðyÞ~ byr, and using the Ho¨lder inequality, we have that

Vol½Bðp;rÞcA₂ⁿ¹ ð

Sⁿ¹

ðr 0

s²ⁿ²ds 1=2ðr

0

e^2:B2:s Ð^r

0Ricrðt;yÞdt

ds 1=2

dA:

Using the inequalityÐr

0e^asdscre^ar, we have that Vol½Bðp;rÞcA₃ⁿ¹rⁿ

ð

Sⁿ¹

ðe Ð^r

0Ricrðt;yÞ:B3:r:dt

ÞdA;

which can be written as

Vol½Bðp;rÞcA₃ⁿ¹rⁿ ð

Sⁿ¹

e^ð Ð^r

0B3:r²:Ricrðt;yÞdt=rÞ

dA:

Finally use the Jensen inequality to get Vol½Bðp;rÞcA₃ⁿ¹rⁿ¹

ð

Sⁿ¹

ðr 0

e^{B3:r2:Ricrðt;yÞ}dt:dA

and the result follows. r

5.2 Volume comparison theorem related to the 2-3 symmetrization.

Theorem 5.6.Let ðBfðrÞ;dt²þf²ðt;yÞdy²ÞHM be a geodesic ball within the injec- tivity radius.If BhðrhÞis the2-3symmetrization of BfðrÞ,then we have thatVol½qBhðtÞ cVol½qBfðtÞ for every tA½0;rhÞ. In particular, Vol½BhðrhÞcVol½BfðrhÞ. If n¼2, then we have that rh¼r, Vol½qBhðtÞ ¼Vol½qBfðtÞfor every tA½0;rÞ,andVol½BhðrÞ

¼Vol½BfðrÞ.This theorem is still valid if BfðrÞis the 1-2symmetrization of another geodesic ball.

Proof. Assumend3 (The case n¼2 is simpler and it can be proved in the same fashion as in the casend3).

The proof will follow an indirect approach. We construct a radially symmetric geo- desic ballðBuðrÞ;dt²þu²ðtÞdy²Þsuch that Vol½qBuðtÞ ¼Vol½qBfðtÞfor everytA½0;rÞ.

Then we compareB_u withB_h.

Let us see thatðBuðrÞ;dt²þu²ðtÞdy²Þis a well defined object. The explicit expres- sion ofuis

uⁿ¹ðtÞ ¼ Ð

Sⁿ¹ fⁿ¹ðt;yÞdA Ð

Sⁿ¹dA : ð42Þ

We will show thatdt²þu²ðtÞdy² is a smooth metric inðBuðrÞ ft¼0gÞthat ad- mits a continuous extension atðt¼0Þin the same fashion as in the second part of the proof of Theorem 4.2.

The smoothness in ðBuðrÞ ft¼0gÞis straightforward. Let us see the continuity atðt¼0Þ. By (42) we have that

uðtÞ ¼ Ð

Sⁿ¹ fⁿ¹ðt;yÞdA Ð

Sⁿ¹dA

^1=ðn1Þ

:

Take the Taylor series of f. We get

uðtÞ ¼ Ð

Sⁿ¹ðtþf⁰⁰ðty;yÞ:t²=2Þⁿ¹ðt;yÞdA Ð

Sⁿ¹dA

!1=ðn1Þ

¼t Ð

Sⁿ¹ð1þf⁰⁰ðty;yÞ:t=2Þⁿ¹ðt;yÞdA Ð

Sⁿ¹dA

!1=ðn1Þ

where t_yAð0;tÞ.

Now take two continuous vector fieldsX andY and make exactly the same cal- culations as in the second part of Theorem 4. Instead of (27), we have that

hX;Yi_u¼hX_t;Y_ti_Eþu²ðtÞ

t² hX_y;Y_yi_E: ð43Þ and now is clear that dt²þu²ðtÞdy² converges to ds_E² when t goes to 0. Thus dt²þu²ðtÞdy² is continuous atðt¼0Þ.

The next step is to compare the average of the radial Ricci curvature on qBfðtÞ with its correspondent onqBuðtÞ. Taking derivatives with respect totin (42), we have that

uⁿ²ðtÞ:u⁰ðtÞ ¼ Ð

Sⁿ¹ fⁿ²ðt;yÞ:f⁰ðt;yÞdA Ð

Sⁿ¹dA : ð44Þ

Taking another derivative int, we get

ðn2Þ:uⁿ³ðtÞðu⁰ðtÞÞ²þuⁿ²ðtÞu⁰⁰ðtÞ

¼ Ð

Sⁿ¹½ðn2Þfⁿ³ðt;yÞðf⁰ðt;yÞÞ²þfⁿ²ðt;yÞf⁰⁰ðt;yÞdA Ð

Sⁿ¹dA : ð45Þ

Using (42), (44) and (45) we can isolateu⁰⁰from the rest:

u⁰⁰ðtÞ ¼ ð2nÞðÐ

Sⁿ¹ fⁿ²ðt;yÞf⁰ðt;yÞdAÞ² ðÐ

Sⁿ¹ fⁿ¹ðt;yÞdAÞ^{ð2n3Þ=ðn1Þ}ðÐ

Sⁿ¹dAÞ^1=ðn1Þ þðÐ

Sⁿ¹ fⁿ³ðt;yÞ½f⁰ðt;yÞ²:dAÞðÐ

Sⁿ¹ fⁿ¹ðt;yÞdAÞ ðÐ

Sⁿ¹ fⁿ¹ðt;yÞdAÞ^{ð2n3Þ=ðn1Þ}ðÐ

Sⁿ¹dAÞ^1=ðn1Þ þðÐ

Sⁿ¹ fⁿ²ðt;yÞf⁰⁰ðt;yÞdAÞðÐ

Sⁿ¹ fⁿ¹ðt;yÞdAÞ ðÐ

Sⁿ¹ fⁿ¹ðt;yÞdAÞ^{ð2n3Þ=ðn1Þ}ðÐ

Sⁿ¹dAÞ^1=ðn1Þ : ð46Þ

Now we can calculate the radial Ricci curvature RicruðtÞofBuðrÞ:

Ricr_uðtÞ ¼ ðn1Þu⁰⁰ðtÞ uðtÞ

¼ðn1Þðn2ÞðÐ

Sⁿ¹ fⁿ²ðt;yÞf⁰ðt;yÞdAÞ² ðÐ

Sⁿ¹ fⁿ¹ðt;yÞdAÞ² ð47Þ ðn1Þðn2ÞðÐ

Sⁿ¹ fⁿ³ðt;yÞðf⁰ðt;yÞÞ²dAÞ Ð

Sⁿ¹ fⁿ¹ðt;yÞdA ð48Þ ðn1ÞðÐ

Sⁿ¹ fⁿ²ðt;yÞf⁰⁰ðt;yÞdAÞ Ð

Sⁿ¹ fⁿ¹ðt;yÞdA : ð49Þ Observe that (49) is the average of the radial Ricci curvature ofB_fðrÞon the geo- desic sphere of radius t. We claim that the sum of the terms (47) and (48) is non- positive. In fact, the Ho¨lder inequality gives

ð

Sⁿ¹

fⁿ²ðt;yÞf⁰ðt;yÞdA 2

c ð

Sⁿ¹

fⁿ³ðt;yÞðf⁰ðt;yÞÞ²dA ð

Sⁿ¹

fⁿ¹ðt;yÞdA:

Therefore the average of the radial Ricci curvature onqBuðtÞis less or equal than the correspondent average onqBfðtÞfor everytA½0;rÞ.

Finally we compare Vol½qBhðtÞand Vol½qBfðtÞ.

LetðBhðrhÞ;dt²þh²ðtÞdy²Þbe the 2-3 symmetrization ofBfðr;yÞ. The calculations made just before and the definition of the 2-3 symmetrization implies that the geo-