On the locally conformally flat hyper surfaces with nonnegative scalar curvature in r5

Suppose M4 is a complete noncompact locally conformally flat hyper-surfacewith nonnegative scalar curvature immersed in R5.. For an open surface M , the total Gaussian curvature is bound

ON THE LOCALLY CONFORMALLY FLAT

HYPER-SURFACES WITH NON-NEGATIVE SCALAR

ZHOU JIURU

NATIONAL UNIVERSITY OF SINGAPORE

2013

ON THE LOCALLY CONFORMALLY FLAT

HYPER-SURFACES WITH NON-NEGATIVE SCALAR

ZHOU JIURU (M Sc., Nanjing University)

A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

DEPARTMENT OF MATHEMATICS NATIONAL UNIVERSITY OF SINGAPORE

2013

Copyright c

All rights reserved

I hereby declare that this thesis was composed in its entirety by myself and that thework contained therein is my own, except where explicitly stated otherwise in the text.Besides, I understand that I have duly acknowledged all the sources of informationwhich have been used in the thesis

Finally, this thesis has also not been submitted for any degree in any universitypreviously

Zhou Jiuru

It is my honor to give my sincerest gratitude to my supervisor, Professor Xu wang, who is always kind and full of humor In the past few years, he has taught me theprofessional knowledge and shown me the virtue and principle of a researcher His var-ious knowledge and charming personality will benefit me for my whole life He alwaysshares his own ideas and experience in research without reservation, and is patient toexplain any question, easy or tough I also appreciate Prof Xu for his unselfish help ofthis thesis, and without his guidance, it would have not been finished

Xing-During the time when I am studying in Department of Mathematics, National versity of Singapore, I have learnt a lot, and here I would like to thank Dr Han Fei forsharing his own research experience and many interesting mathematical stories I feeldeeply grateful to my friends and classmates, especially Ngo Quoc Anh, Zhang Hong,Cai Ruilun, Ye Shengkui I would also like to thank the University and the departmentfor support

Uni-Special thanks should also be given to my master degree’s supervisor, Dr Mei aqiang for building up my mathematical background I thank Prof Wang Hongyu forhis help at the beginning of my postgraduate life and his support during the year when

Ji-I stayed in Yangzhou University Ji-I also thank my friends Xu Haifeng, Zhu Peng, GuPeng for their help

Finally, for all the people who have ever cared me, helped me, I would like to offer

my undying gratitude

To my parents and my wife

Declaration i

Acknowledgements iii

Dedication v

Contents vii

Summary ix

List of notations and conventions xi

1 Introduction 1

1.1 Studies on open surfaces 1

1.2 Studies on locally conformally flat open four dimensional manifolds 2

1.3 Results of this paper 3

2 Backgrounds and Preparation 7

2.1 Riemann connection, Curvatures and second fundamental form 7

2.2 Qn curvature and Qn curvature equation 9

2.3 Chern-Gauss-Bonnet formula 11

3 High Dimensional Chern-Gauss-Bonnet formula 15

3.1 Chern-Gauss-Bonnet formula for Rn 16

3.2 Geometric conditions for metric to be normal 25

3.3 Chern-Gauss-Bonnet formula in Local version 28

3.4 Chern-Gauss-Bonnet formula for conformally flat manifolds 33

4 Controlling the number of ends by the mean curvature 37

5 Mean curvature and embedding 45

5.1 Decomposition of the conformal factor 45

Contents Contents

5.2 Immersion and Embedding 51

6 Conclusions and further work 53

Appendices 55

References 59

viii

This thesis studies the geometry and topology of manifolds from an extrinsic point

of view Suppose M4 is a complete noncompact locally conformally flat hyper-surfacewith nonnegative scalar curvature immersed in R5 Given some conditions on the secondfundamental form and the mean curvature, we should show that if the L4 norm of themean curvature of M , i.e (RM|H|4 dvM)1 is bounded by some constant which doesnot depend on the manifold M , then M is embedded in R5 This result should be ageneralization of S M¨uller and V ˇSver´ak’s result on two dimensional manifolds whichimmersed in Rn

List of Notations and Conventions

∇ Riemann connection

Rm Riemann curvature tensor

Ric Ricci curvature tensor

R Scalar curvature

Br(x) A ball centered at point x with radius r

B Unit ball centered at the origin of Euclidean space

ωn Volume of the unit sphere Sn in the Euclidean space (Rn+1, |dx|2)

Introduction

A central problem in global differential geometry is connections between the etry and the topology of a manifold One of the most important results of such style isthe Chern-Gauss-Bonnet formula for closed Riemannian manifolds (Chern, 1944, [8]).Actually, before Chern’s work, some mathematicians thought that differential geometrywas a dead end Although there were plenty of satisfying results in classical differentialgeometry, all were based on local analysis, which blocked the depth development ofdifferential geometry Afterwards, it is S.S Chern who brought a new life to differentialgeometry and had used analysis, topology, algebra to study differential geometry from

geom-a globgeom-al point of view, geom-and geom-also estgeom-ablished globgeom-al differentigeom-al geometry

For modern differential geometry, a powerful branch is geometric analysis leading

by S.T Yau It mainly uses differential equations to study differential geometry, andhas created excellent work for many mathematical problems such as Yang-Mills fields,Calabi-Yau manifolds, Ricci flow The Poincar´e conjecture, which is one of the sevenMillennium Prize Problems, was solved in 2002 by using Ricci flow Therefore, moreattention should be paid to consider using differential equations in studying differentialgeometry

In the rest of the chapter, we will introduce a problem in differential geometry, andalso a brief review of several results done by various mathematicians, which providesthe background and motivation of this thesis

1.1 Studies on open surfaces

As mentioned previously, the Chern-Gauss-Bonnet formula is a very important result

in differential geometry Actually, after Chern’s work, many mathematicians tried togeneralize this formula to open manifolds, and studied the total curvature to openmanifolds

For an open surface M , the total Gaussian curvature is bounded by the Euler number

of the surface up to a multiplication of a constant, as long as the Gaussian curvature

is absolutely integrable, i.e

Z

M

K dvM ≤ 2πχ(M ), (1.1)and such an open surface can be conformally compactified by attaching finite points(Huber, 1957, [17]) Furthermore, the deficit between the Euler number of the surface

Z

M

|A|2dvM < +∞,then the Chern-Gauss-Bonnet formula (1.2) also holds, and in this case, the totalisoperimetric number is equal to the total number of ends of the open surface (countedwith multiplicity) (M¨uller and ˇSver´ak, 1995, [25]), i.e.,

χ(M ) − 1

2πZ

where l is still the number of ends and mi is the multiplicity of each end Furthermore,

if RM|A|2dvM < 8π for n = 3, or RM|A|2dvM ≤ 4π for n ≥ 4, then M is ded (M¨uller and ˇSver´ak, 1995, [25]) These conclusions indicate that some geometricassumptions can deduce topologic results, but they are just on surfaces, i.e two dimen-sional manifolds Since we have no idea if they also hold on general cases, our researchshould be concentrated on general manifolds

embed-1.2 Studies on locally conformally flat open four dimensional

manifolds

For four dimensional manifolds, Huber’s result on the upper bound of the totalGaussian curvature can be generalized to complete four dimensional manifolds of pos-itive sectional curvature outside a compact set (R Greene and H Wu, 1976, [16]), butthe deficit between the Euler number and the total Gauss curvature of the manifolds

is hard to deduce This gap is filled only after Branson creating Qn curvature

As we all know, isothermal coordinates always exist on a surface, which means asurface is always locally conformally flat Therefore, it is very natural to study the lo-cally conformally flat manifolds Actually, for a locally conformally flat four dimensionalclose manifold M , the total Gaussian curvature is equal to the total Q4 curvature up to

a multiplication of a constant For locally conformally flat open four dimensional folds with finitely many simple ends whose scalar curvature is nonnegative at each endand the Q4 curvature is integrable, we can use the Q4 curvature equation to determinethe deficit between the Euler number of M and the total Q4 curvature (Chang, Qingand Yang, 2000, [7]) With appropriate conditions on scalar curvature, Ricci curvaturetensor and Q4 curvature, they obtain that such manifold can be compactified by ad-joining a finite number of points (Chang, Qing and Yang, 2000, [7]) This work provides

mani-a method to study the locmani-ally conformmani-ally flmani-at mmani-anifolds However, for genermani-al openmanifolds, we still do not have enough knowledge about the total Gaussian curvature

1.3 Results of this paper 3

1.3 Results of this paper

Based on the analysis of previous work, applications of the Chern-Gauss-Bonnetformula need to be studied for locally conformally flat four dimensional manifolds.Therefore, in this thesis, we try to generalize M¨uller and ˇSver´ak’s work ([25]) on surfaces

to locally conformally flat four dimensional manifolds In [25], they showed the followingresults,

Theorem 1.1 Let M → Rn be a complete, connected, non-compact surface immersedinto Rn Assume that either

we will show the following theorem,

Theorem 1 Let M → R5 be a complete, simply connected, noncompact, locally formally flat hyper-surface immersed into R5 with 16H2− |A|2 to be non-negative and

con-∆(16H2− |A|2) ∈ H1(M ) Assume that

(Z

M

|H|4dv)1 < C2, (1.3)where

C2 = 113C1

.Then M is embedded

For the constant C1 in Theorem 1, please refer to Chapter 4

This theorem shows that some weak conditions on the mean curvature and ond fundamental form, which are extrinsic quantities, can control the topology of themanifold

sec-First let us give some standard notations Let M be a complete, connected, compact, oriented, locally conformally flat four dimensional manifold immersed into

non-R5, i.e M is a locally conformally flat hyper-surface of R5 We denote the secondfundamental form of M by A, and Q4 curvature of M by Q4 Choose a conformalparametrization

f : Ω∗→ Σ ⊂ M → R5,where

Ω∗ = {x ∈ R4, |x| > 1},and Σ is a neighborhood of an end of M Under this conformal parametrization, wedenote the conformal metric to be g = e2ug0, where g0 is the standard flat metric.With the conditions in Theorem 1, we obtain,

4 1 Introduction

Theorem 2 Suppose H4 and ∆(16H2− |A|2) ∈ H1(M ), R = 16H2− |A|2 ≥ 0 Then

in local coordinates (Ω∗, e2ug0), the metric e2ug0 is normal Furthermore, the conformalfactor has the following decomposition,

u(x) = u0(x) + α log |x| + h(x),where

lim

x→∞u0(x) = 0,and h is a biharmonic function on Ω∗∪ {∞}

For the definition of normal metric, please refer to Chapter 5 By the decomposition

of the conformal factor, we can prove that the parametrization function f behaves like

xm in some sense

Theorem 3 Choose a conformal parametrization

f : Ω∗ → Σ ⊂ M → R5.Then we have the following limit,

λ = lim

x→∞h(x)

Finally, we can prove Theorem 1

As shown in Theorem 2, given weak assumptions on the mean curvature and thesecond fundamental form, we have a nice decomposition of the conformal factor u Theresults show that the conformal factor u behaves like a log function at infinity Hence itbecomes possible to study the conformal factor Furthermore, with this decomposition,the deficit between the Euler number of M and the total Q4 curvature is equal to thetotal number of ends of M This is similar to the result in [25], where they claimed inCor 4.2.5 that if a surface is immersed into a Euclidean space RnwithRM|A|2dv < ∞,the deficit between the Euler number and the total Gaussian curvature is the totalnumber of ends of the given surface This decomposition is also used to study theconformal parametrization function f , which can be seen in Theorem 3

Theorem 1 is the key result of this thesis, which says that under suitable conditions

on the mean curvature and the second fundamental form, the immersion of the manifoldbecomes an embedding The main step to obtain this result is to use conditions onthe total integral of the mean curvature to find the decomposition of the conformalfactor After that, we find the behavior of the conformal parametrization function, i.e.the conformal parametrization function behaves like xm Then we use the integrabilitycondition on the mean curvature to control the number of ends of the manifold Finally,the immersion is actually an embedding

We work on this result, because it shows some connections between the topologyand the geometry of a manifold, which is one of the central problems in differentialgeometry

The following section shows the structure of this thesis

1.3 Results of this paper 5

Chapter 2 provides some necessary material and conventions in differential geometry.Chapter 3 discusses the high dimensional Chern-Gauss-Bonnet theorem which is due

to X W Xu [33] Chapter 4 shows the way to use the mean curvature to control thenumber of ends of a manifold with nonnegative scalar curvature Chapter 5 presentsthe main result of this thesis, i.e using the mean curvature to control the topology oflocally conformally flat four dimensional hyper-surfaces

Backgrounds and Preparation

In this chapter, we are going to introduce necessary materials and clarify conventionswhich we need for the rest of the thesis Most of these can be found in standardtextbooks on differential geometry or Riemannian geometry

2.1 Riemann connection, Curvatures and second fundamental form

Following [9], given a complete Riemannian manifold (M, g, ∇) with Riemann metric

g and the induced Riemann connection ∇, the Riemann curvature tensor is defined bythe following formulas,

R(X, Y )Z = ∇X∇YZ − ∇Y∇XZ − ∇[X,Y ]Z, (2.1)Rm(X, Y, Z, W ) = hR(X, Y )Z, W i (2.2)

In local coordinates, (2.1) and (2.2) is equivalent to the following equations,

R(∂i, ∂j)∂k= Rlijk∂l,

Rijkl= hR(∂i, ∂j)∂k, ∂li = gmlRmijk (2.3)According to (2.3), by taking the trace of Riemann tensor, we get Ricci curvaturetensor Ric Furthermore, if we take the trace of Ricci curvature tensor, we obtain scalarcurvature R Hence, in local coordinates,

Ricij = gkmRkijm,

R = gijRicij.Now we use moving frames to deal with the Riemann connection and curvaturetensors Suppose (M, g, ∇) is a Riemannian manifold, {ei} is a local frame on someopen subset U ⊂ M , and let {ωi} be the dual coframe Then there exist one forms {ωji}such that

∇ei = ωijej (2.4)Then

Ωij = dωij− ωik∧ ωkj = 1

2R

j iklωk∧ ωl (2.5)

is the curvature form

8 2 Backgrounds and Preparation

By lowering down the index, i.e.,

Suppose Mn is a differentiable manifold Mn is immersed into the Euclidean space(Rn+1, |dx|2), i.e Mnis a hyper-surface of Rn+1 We will denote by ˜∇ the Riemannianconnection on Rn+1induced by the canonical metric |dx|2, and denote by ∇ the inducedRiemann connection on Mn

Suppose X, Y are tangent vector fields on M , the second fundamental form A isdefined to be the normal projection of the Riemann connection,

A(X, Y ) = ( ˜∇XY )⊥.For any point p ∈ M and the unit normal vector field N on M , the second fun-damental form A determines a self-adjoint linear map AN : TpM → TpM , i.e theWeingarten operator defined as follows:

hAN(X), Y i = hA(X, Y ), N i

For any point p ∈ M , we denote by λ1, λ2, , λn the eigenvalues of the symmetrictransformation AN of the tangent space TpM These eigenvalues λ1, λ2, , λn arecalled the principal curvatures at p The Gaussian curvature K is defined to be thedeterminant of the second fundamental form,

Kn= λ1· λ2· · · λn−1· λn (2.7)

For the hyper-surface M → Rn+1, suppose Sn ⊂ Rn+1 is the unit sphere centered

at the original point of Rn+1, the Gauss map

G : M → Sn⊂ Rn+1

maps every point of the hyper-surface to the normal vector at that point,

∀ p ∈ M, G(p) = N (p),where N (p) is the unit normal vector at p With this definition, we have the followingWeingarten equation,

hdGp(X), Y i = −hG(p), A(X, Y )i,for tangent vector fields X, Y of M

Hence, by taking an orthonormal frame {e1, e2, · · · , en} of the tangent bundle of M ,

we have the norm

2.2 Qn curvature and Qn curvature equation 9The scalar second fundamental form h is the symmetric 2-tensor on M defined by

h(X, Y ) = hA(X, Y ), N i

In local coordinates, we have the local form of the Gauss equation,

Rijkl = hilhjk− hikhjl (2.8)Hence, after contracting (2.8) once and twice respectively, the Ricci curvature tensorand scalar curvature can be represented in terms of the second fundamental form,

Rjk = nH · hjk− hlkhjl,

R = n2H2− |A|2,where

H = 1n

n

X

i=1

hii= 1n

n

X

i=1

λi

is defined to be the mean curvature

2.2 Qn curvature and Qn curvature equation

Next, let us consider the high order curvature which is a generalization of the scalarcurvature in two dimensional case, i.e the Qn curvature

Suppose (Mn, g0) is a complete Riemannian manifold For conformal geometry, sider the conformal metric g = e2ug0 on M ; then an operator A on the manifold M issaid to be conformally covariant of bidegree (a, b) if

con-Ag(φ) = e−buAg 0(eauφ), (2.9)for φ ∈ C∞(M )

The simplest example is the Laplace-Beltrami operator ∆ for two dimensional ifold, where

It is a conformally covariant operator of bidegree (0, 2) More precisely, it satisfies

∆g = e−2u∆g 0.Furthermore, for this operator, we have the following Gaussian curvature equation

∆g0u + Kge2u= Kg0, (2.10)where K is the Gauss curvature

It is quite natural to ask whether there exists a high order conformally covariantoperator Pn, and whether there exists a local curvature invariant Qn satisfying thefollowing conformal transformation formula,

enuQn,g = Qn,g 0 + Pn,g 0u, (2.11)for n > 2

Thanks to C R Graham, R Jenne, L Mason and G Sparling, they show thefollowing result in [15],

10 2 Backgrounds and Preparation

Theorem 2.1 For Riemannian manifold (Mn, g), let m be a positive even integer isfying

sat-m =

(any positive even integer, n odd

m ≤ n, n even

Then there exists a conformally covariant operator Pnon C∞(M ) of degree (m−n2 ,m+n2 ),such that the leading symbol of Pn is the same as the symbol of ∆n2 In particular, on(Rn, |dx|2), Pn= ∆n2

For a special case when n is even, and m = n, the conformally covariant operator Pn

is of bidegree (0, n) In [2], T Branson proved that there exists Qncurvature satisfying(2.11), where Qncan be represented by the metric g, the connection ∇ and the curvaturetensor Rm Then (2.11) is said to be the Qn curvature equation

Suppose (M, g) is a complete Riemannian manifold, if for every point p ∈ M , thereexists a local coordinate system (xi, Up) such that the metric g is conformal to the flatmetric, we write

gij = e2uδij, (2.12)where u is some function defined on the neighborhood Up of p Then the manifold M issaid to be locally conformally flat Since for a flat metric, the curvature tensor vanishes,the Qn curvature for g0 also vanishes Hence, for a locally conformally flat manifold(M, g), we have the Qn curvature equation

For n = 2, it is obvious that

P2= ∆,and

Q2 = K = R

2.For n = 4, we have the following forth order differential operator, i.e the Paneitzoperator

P4 = ∆2+ δ(2

3Rg − 2 Ric)d,where δ is the divergence operator, R is the scalar curvature, Ric is the Ricci curvaturetensor, and d is the differential Q4 curvature of a four dimensional manifold is definedby

E = Ric −1

4Rg.

It is well-known that for an orientable, closed even dimensional Riemannian manifold

Mn with n = 2m, Chern-Gauss-Bonnet theorem states

0, i1, · · · , in are not all distinct

In the following, we are going to recall some results of Chern-Gauss-Bonnet theorem

on open manifolds

For an n dimensional differentiable manifold M , consider the Kulkarni-Nomizu uct

prod-◦ : S2M × S2M → CM,where S2M = T∗M ⊗ST∗M is the bundle of symmetric 2-tensor, and CM is the bundle

of curvature tensors, defined by:

(α ◦ β)ijkl= αilβjk+ αjkβil− αikβjl− αjlβik.The Riemann curvature tensor can be decomposed into three orthogonal part,

Property 2.2 A Riemannian manifold (Mn, g) with n ≥ 4 is locally conformally flat

⇐⇒ the Weyl tensor W = 0

12 2 Backgrounds and Preparation

Notice that we have defined the Gaussian curvature Kn in (2.7) Actually, there is

= 16σ2(S) ω1∧ ω2∧ ω3∧ ω4 (2.21)

By (2.19) and (2.21), we get

K4 = 2

3σ2(S). (2.22)Now we calculate that

by divergence theorem, the Chern-Gauss-Bonnet formula says

18π2

(a) The scalar curvature is non-negative at infinity at each end

(b) The Q4 curvature is integrable

µi = lim

r→∞

R

∂B r (0)e3uidσ(x)

4

4(2π2)1 R

B r (0)\Be4u idx,and l is the number of the ends

High Dimensional Chern-Gauss-Bonnet formula

In this chapter, we want to introduce Chern-Gauss-Bonnet theorem for high evendimensional locally conformally flat open manifolds For more general case, please see[33] Since this result and the method of the proof will be applied to prove the maintheorem of this thesis, we will provide the argument for completeness of this thesis Inorder to state the results, let us present some definitions and notations

Definition 3.1 Suppose (M, g) is a complete open locally conformally flat n sional manifold such that

(Ei, gi) = (Rn\B, e2uig0),for some function ui Such manifolds will be called complete locally conformally flat ndimensional manifolds with simple ends

Actually, there are many examples of such manifolds For example, complete metricswith constant positive scalar curvature on Sn\{pi}k

i=1, i.e Sn with k points deletedconstructed by R Schoen [28] and R Mazzeo and F Pacard [24]

The high dimension Chern-Gauss-Bonnet theorem can be stated in the following,Theorem 3.2 Suppose (M, g) is an open complete locally conformally flat n = 2mdimensional manifold with simple ends Let l be the number of ends Assume that(a) the scalar curvature is non-negative at infinity on each end;

(b) the Qn curvature is absolutely integrable

Then, we have the following formula,

16 3 High Dimensional Chern-Gauss-Bonnet formulawhere

3.1 Chern-Gauss-Bonnet formula for Rn

In the following, we consider the even dimensional Euclidean space Rn Similar to[7], we define normal metric for high dimensional manifolds,

Definition 3.3 A conformal metric e2ug0 on the Euclidean space Rn with the Qn vature to be absolutely integrable is said to be normal if the function u has the followingdecomposition,

|x − y|



Qn(y)enu(y)dy + C0,

where

Qn(y) = e−nu(y)[(−∆)n2u](y),

C0 is a constant and ωn is the volume of the unit sphere Sn in the Euclidean space

∂B r (0)

rk−n+1(1 + r∂u

∂r)

n−k−1ekudσ,for k = 1, 2, , n − 1

By the above definition, for all 1 ≤ j ≤ n − 1 and 1 ≤ k ≤ n − j, we define theisoperimetric ratio to be,

Ck,k+j(r) = V

k+j j(n−1)

k

(nωn)n−11 V

k j(n−1)

k+j

We want to prove the following theorem,

Theorem 3.4 Suppose e2ug0is a smooth complete normal metric on Rnwith absolutelyintegrable Qn curvature Then

3.1 Chern-Gauss-Bonnet formula for Rn 17Notice that the above theorem was obtained both by X.W Xu [33] and C.B Ndiaye,

J Xiao [26] using different method In order to prove Theorem 3.4, we need severallemmas

Lemma 3.5 Suppose e2ug0 is a complete normal metric Then the following identityholds,

lim

r→∞(1 + r∂ ¯u

∂r) = 1 −

2(n − 1)!ωn

Z

Rn

Qn(y)enu(y)dy

+ 2(n − 1)!ωn

Z

Rn

Qn(y)enu(y)dy

+ 2(n − 1)!ωn

Qn(y)enu(y)dy (3.2)

By the formula that

d

dr−Z

∂B r (x 0 )

udσ = −Z

∂B r (x 0 )

du

drdσ,which is proved in the Appendices, we can see in order to finish the proof of Lem 3.5,

we only need to show that the spherical average integral of the last term in (3.2) tends

|Qn(y)|enu(y)dy

Since Qn is absolutely integrable, ∀ ε > 0, ∃ a sufficiently large R0> 0 such that

18 3 High Dimensional Chern-Gauss-Bonnet formula

−Z

∂B r (0)

1

|x − y|n−2dσx

# k n−2

since |x−y|1n−2 is a harmonic function

By (3.3), (3.4) and (3.5), for r = |x| ≥ R1, we have,

|−

Z

∂B r (0)

(Z

Rn

"

−Z

|y|≤R0

"

−Z

Z

|y|≥R0

"

−Z

With k = 1 in (3.6), take spherical average of (3.2) and let r → ∞, we get (3.1) 2

Lemma 3.6 Suppose the metric e2ug0 on Rn is a normal metric Then for any k > 0,

we have

−Z

∂B r (0)

ekudσ = ek ¯ueo(1), (3.7)where o(1) → 0 as r → ∞

Proof Since e2ug0 is a normal metric, we decompose u as follows,

u(x)

= 2(n − 1)!ωn

|x|



Qn(y)enu(y)dy + C0

+ 2(n − 1)!ωn

3.1 Chern-Gauss-Bonnet formula for Rn 19

We can see that f (|x|) is a radial symmetric function which is the major part of uwhen |x| → ∞ For sufficiently large |x|, if |x|12 ≤ |y| ≤ 1

2|x|, then1

2|x| ≤ |x| − |y| ≤ |x − y| ≤ |x| + |y| ≤

3

2|x|,Hence,



|x|

|x − y|



|Qn(y)|enu(y)dy

+ 2(n − 1)!ωn



|x|

|x − y|



|Qn(y)|enu(y)dy

≤ C · log |x|

1 2

|Qn(y)|enu(y)dy, (3.9)

where C, C1 do not depend on x

Hence,

|u1(x)| = o(1) as |x| = r → ∞ (3.10)Next, we want to show that the average integral of u2 has the estimate that ¯u2(r) =o(1) as r → ∞ In order to prove it, we rewrite ¯u2 by Fubini theorem,

∂B r (0)

log

... 35

20 High Dimensional Chern-Gauss-Bonnet formula

On the other hand, it also follows by Jenson’s inequality and the fact that |x − y| ≤... class="page_container" data-page="37">

22 High Dimensional Chern-Gauss-Bonnet formula

Notice that the last step in (3.24) is the potential estimate (7.32) in [14] on page 159

by choosing...

In order to see this, by Jenson’s inequality,

exp



−Z

= −Z

Now we estimate the first integral of the last term of (3.22) If |y| 6= r, then by theestimate