ON THE LOCALLY CONFORMALLY FLAT
HYPER-SURFACES WITH NON-NEGATIVE SCALAR
CURVATURE IN R
5
ZHOU JIURU
NATIONAL UNIVERSITY OF SINGAPORE
2013

ON THE LOCALLY CONFORMALLY FLAT
HYPER-SURFACES WITH NON-NEGATIVE SCALAR
CURVATURE IN R
5
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
 2013 by Zhou Jiuru.
All rights reserved.
Declaration
I hereby declare that this thesis was composed in its entirety by myself and that the
work 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 information
which have been used in the thesis.
Finally, this thesis has also not been submitted for any degree in any university
previously.
Zhou Jiuru

Acknowledgements
It is my honor to give my sincerest gratitude to my supervisor, Professor Xu Xing-
wang, who is always kind and full of humor. In the past few years, he has taught me the
professional 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 always
shares his own ideas and experience in research without reservation, and is patient to
explain any question, easy or tough. I also appreciate Prof. Xu for his unselfish help of
this thesis, and without his guidance, it would have not been finished.
During the time when I am studying in Department of Mathematics, National Uni-
versity of Singapore, I have learnt a lot, and here I would like to thank Dr. Han Fei for
sharing his own research experience and many interesting mathematical stories. I feel
deeply 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 department
for support.
Special thanks should also be given to my master degree’s supervisor, Dr. Mei Ji-
aqiang for building up my mathematical background. I thank Prof. Wang Hongyu for
his help at the beginning of my postgraduate life and his support during the year when
I stayed in Yangzhou University. I also thank my friends Xu Haifeng, Zhu Peng, Gu
Peng for their help.
Finally, for all the people who have ever cared me, helped me, I would like to offer
my undying gratitude.

Dedication
To my parents and my wife

Contents
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 Q
n
curvature and Q
n
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 R
n
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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
Summary
This thesis studies the geometry and topology of manifolds from an extrinsic point
of view. Suppose M
4
is a complete noncompact locally conformally flat hyper-surface
with nonnegative scalar curvature immersed in R
5
. Given some conditions on the second
fundamental form and the mean curvature, we should show that if the L
4
norm of the
mean curvature of M, i.e. (

M
|H|
4
dv
M
)
1
4
is bounded by some constant which does
not depend on the manifold M, then M is embedded in R
5
. This result should be a
generalization of S. M¨uller and V.
ˇ
Sver´ak’s result on two dimensional manifolds which
immersed in R

n
.

List of Notations and Conventions
∇ Riemann connection
Rm Riemann curvature tensor
Ric Ricci curvature tensor
R Scalar curvature
B
r
(x) A ball centered at point x with radius r
B Unit ball centered at the origin of Euclidean space
ω
n
Volume of the unit sphere S
n
in the Euclidean space (R
n+1
, |dx|
2
)
−

B
f dx Average integral of f given by
−

B
f dx =
1

|B|

B
f dx
1
Introduction
A central problem in global differential geometry is connections between the geom-
etry and the topology of a manifold. One of the most important results of such style is
the Chern-Gauss-Bonnet formula for closed Riemannian manifolds (Chern, 1944, [8]).
Actually, before Chern’s work, some mathematicians thought that differential geometry
was a dead end. Although there were plenty of satisfying results in classical differential
geometry, all were based on local analysis, which blocked the depth development of
differential geometry. Afterwards, it is S.S. Chern who brought a new life to differential
geometry and had used analysis, topology, algebra to study differential geometry from
a global point of view, and also established global differential 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, and
has 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 seven
Millennium Prize Problems, was solved in 2002 by using Ricci flow. Therefore, more
attention should be paid to consider using differential equations in studying differential
geometry.
In the rest of the chapter, we will introduce a problem in differential geometry, and
also a brief review of several results done by various mathematicians, which provides
the 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 to
generalize this formula to open manifolds, and studied the total curvature to open
manifolds.

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.

M
K dv
M
≤ 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
2 1 Introduction
and the total Gaussian curvature is just the total isoperimetric number (Finn, 1965,
[13]), i.e.,
χ(M) −
1
2π

M
K dv
M
=
l

i
µ
i
, (1.2)
where l is the number of ends of M , and µ
i
is the isoperimetric ratio of each end.

Therefore, the Chern-Gauss-Bonnet formula is excellently generalized on open surfaces.
However, these are all intrinsic properties. For extrinsic properties, suppose an open
surface M with finite number of ends is immersed into an n dimensional Euclidean
space R
n
, we consider the total integral of the second fundamental form.
Suppose

M
|A|
2
dv
M
< +∞,
then the Chern-Gauss-Bonnet formula (1.2) also holds, and in this case, the total
isoperimetric number is equal to the total number of ends of the open surface (counted
with multiplicity) (M¨uller and
ˇ
Sver´ak, 1995, [25]), i.e.,
χ(M) −
1
2π

M
K dv
M
=
l

i

m
i
,
where l is still the number of ends and m
i
is the multiplicity of each end. Furthermore,
if

M
|A|
2
dv
M
< 8π for n = 3, or

M
|A|
2
dv
M
≤ 4π for n ≥ 4, then M is embed-
ded (M¨uller and
ˇ
Sver´ak, 1995, [25]). These conclusions indicate that some geometric
assumptions 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 research
should be concentrated on general manifolds.
1.2 Studies on locally conformally flat open four dimensional
manifolds
For four dimensional manifolds, Huber’s result on the upper bound of the total

Gaussian 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]), but
the 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 Q
n
curvature.
As we all know, isothermal coordinates always exist on a surface, which means a
surface 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 dimensional
close manifold M, the total Gaussian curvature is equal to the total Q
4
curvature up to
a multiplication of a constant. For locally conformally flat open four dimensional mani-
folds with finitely many simple ends whose scalar curvature is nonnegative at each end
and the Q
4
curvature is integrable, we can use the Q
4
curvature equation to determine
the deficit between the Euler number of M and the total Q
4
curvature (Chang, Qing
and Yang, 2000, [7]). With appropriate conditions on scalar curvature, Ricci curvature
tensor and Q
4
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
a method to study the locally conformally flat manifolds. However, for general open
manifolds, 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-Bonnet
formula need to be studied for locally conformally flat four dimensional manifolds.
Therefore, in this thesis, we try to generalize M¨uller and
ˇ
Sver´ak’s work ([25]) on surfaces
to locally conformally flat four dimensional manifolds. In [25], they showed the following
results,
Theorem 1.1 Let M → R
n
be a complete, connected, non-compact surface immersed
into R
n
. Assume that either

M
|A|
2
dv
M
< 8π, n = 3,
or

M
|A|
2
dv
M
≤ 4π, n ≥ 4.
Then M is embedded.

In this thesis, we study the topology and geometry of locally conformally flat open
four dimensional manifolds immersed into the Euclidean space R
5
. More specifically,
we will show the following theorem,
Theorem 1 Let M → R
5
be a complete, simply connected, noncompact, locally con-
formally flat hyper-surface immersed into R
5
with 16H
2
− |A|
2
to be non-negative and
∆(16H
2
− |A|
2
) ∈ H
1
(M). Assume that
(

M
|H|
4
dv)
1
4

< C
2
, (1.3)
where
C
2
=
1
13C
1
.
Then M is embedded.
For the constant C
1
in Theorem 1, please refer to Chapter 4.
This theorem shows that some weak conditions on the mean curvature and sec-
ond fundamental form, which are extrinsic quantities, can control the topology of the
manifold.
First let us give some standard notations. Let M be a complete, connected, non-
compact, oriented, locally conformally flat four dimensional manifold immersed into
R
5
, i.e. M is a locally conformally flat hyper-surface of R
5
. We denote the second
fundamental form of M by A, and Q
4
curvature of M by Q
4
. Choose a conformal

parametrization
f : Ω
∗
→ Σ ⊂ M → R
5
,
where
Ω
∗
= {x ∈ R
4
, |x| > 1},
and Σ is a neighborhood of an end of M. Under this conformal parametrization, we
denote the conformal metric to be g = e
2u
g
0
, where g
0
is the standard flat metric.
With the conditions in Theorem 1, we obtain,
4 1 Introduction
Theorem 2 Suppose H
4
and ∆(16H
2
− |A|
2
) ∈ H
1

(M), R = 16H
2
− |A|
2
≥ 0. Then
in local coordinates (Ω
∗
, e
2u
g
0
), the metric e
2u
g
0
is normal. Furthermore, the conformal
factor has the following decomposition,
u(x) = u
0
(x) + α log |x| + h(x),
where
lim
x→∞
u
0
(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
x
m
in some sense.
Theorem 3 Choose a conformal parametrization
f : Ω
∗
→ Σ ⊂ M → R
5
.
Then we have the following limit,
lim
x→∞
|f(x)|
|x|
α+1
=
e
λ
α + 1
,
where
λ = lim
x→∞
h(x).
Finally, we can prove Theorem 1.
As shown in Theorem 2, given weak assumptions on the mean curvature and the
second fundamental form, we have a nice decomposition of the conformal factor u. The
results show that the conformal factor u behaves like a log function at infinity. Hence it
becomes possible to study the conformal factor. Furthermore, with this decomposition,

the deficit between the Euler number of M and the total Q
4
curvature is equal to the
total number of ends of M . This is similar to the result in [25], where they claimed in
Cor 4.2.5 that if a surface is immersed into a Euclidean space R
n
with

M
|A|
2
dv < ∞,
the deficit between the Euler number and the total Gaussian curvature is the total
number of ends of the given surface. This decomposition is also used to study the
conformal 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 manifold
becomes an embedding. The main step to obtain this result is to use conditions on
the total integral of the mean curvature to find the decomposition of the conformal
factor. After that, we find the behavior of the conformal parametrization function, i.e.
the conformal parametrization function behaves like x
m
. Then we use the integrability
condition 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 topology
and the geometry of a manifold, which is one of the central problems in differential
geometry.
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 the
number of ends of a manifold with nonnegative scalar curvature. Chapter 5 presents
the main result of this thesis, i.e. using the mean curvature to control the topology of
locally conformally flat four dimensional hyper-surfaces.