# Multi-channel Hybrid Access Femtocells:
A Stochastic Geometric Analysis

Yi Zhong and Wenyi Zhang, _Senior Member, IEEE_ The authors are with
Department of Electronic Engineering and Information Science, University of
Science and Technology of China, Hefei 230027, China (email:
<EMAIL_ADDRESS>wenyizha@ustc.edu.cn). The research has been
supported by the National Basic Research Program of China (973 Program)
through grant 2012CB316004, National Natural Science Foundation of China
through grant 61071095, Research Fund for the Doctoral Program of Higher
Education of China through grant 20103402120023, and by MIIT of China through
grants 2010ZX03003-002 and 2011ZX03001-006-01.

###### Abstract

For two-tier networks consisting of macrocells and femtocells, the channel
access mechanism can be configured to be open access, closed access, or hybrid
access. Hybrid access arises as a compromise between open and closed access
mechanisms, in which a fraction of available spectrum resource is shared to
nonsubscribers while the remaining reserved for subscribers. This paper
focuses on a hybrid access mechanism for multi-channel femtocells which employ
orthogonal spectrum access schemes. Considering a randomized channel
assignment strategy, we analyze the performance in the downlink. Using
stochastic geometry as technical tools, we model the distribution of
femtocells as Poisson point process or Neyman-Scott cluster process and derive
the distributions of signal-to-interference-plus-noise ratios, and mean
achievable rates, of both nonsubscribers and subscribers. The established
expressions are amenable to numerical evaluation, and shed key insights into
the performance tradeoff between subscribers and nonsubscribers. The
analytical results are corroborated by numerical simulations.

###### Index Terms:

Channel management, femtocell, hybrid access, Neyman-Scott cluster process,
spatial Poisson process, two-scale approximation, two-tier network

##  I Introduction

In current cellular network services, about 50% of phone calls and 70% of data
services take place indoors [1]. For such indoor use cases, network coverage
is a critical issue. One way to improve the indoor performance is to deploy
the so-called femtocell access points (FAPs) besides macrocell base stations
(MBSs). Femtocells are small cellular base stations, typically designed for
use in home or small business [2][3]. The use of femtocells not only benefits
the users, but also the operators. As the distance between transmitter and
receiver is reduced, users will enjoy high quality links and power savings.
Furthermore, the reduced transmission range also creates more spatial reuse
and reduces electromagnetic interference.

Among the many challenges faced by femtocells, and more generally, two-tier
networks, is the issue of interference; see Figure 1. The two-tier
interference problem differs from that in traditional single-tier networks in
several important aspects: First, due to the limitations of access mechanism,
a user equipment (UE) may not be able to connect to the access point which
offers the best service. Second, since femtocells connect to operator’s core
network via subscribers’ private ISP, coordination between macrocells and
femtocells and among femtocells is limited. Finally, compared to planned
macrocell deployments, femtocells are usually deployed in an ad hoc manner,
and the randomly placed femtocells make it difficult to manage the
interference. In two-tier networks, interference can be categorized into two
types: (a) cross-tier, referring to the interference from one tier to the
other; (b) co-tier, referring to the interference within a tier.

Figure 1: Downlink two-tier network model for hybrid access femtocell.

In this paper, we consider two-tier networks based on multicarrier techniques,
for example those deploying LTE or WiMAX standards, which use orthogonal
frequency-division multiple access (OFDMA) techniques. In multicarrier
systems, the available spectrum is divided into orthogonal subcarriers, which
are then grouped into multiple subchannels, assigned to different users. Due
to the flexibility in channel assignment, the interference may be alleviated.

The access mechanism of femtocells (see, e.g., [4]) is a key factor that
affects the performance of two-tier networks, and generally can be classified
as follows, where we call the UEs registered to a femtocell as subscribers,
and those not registered to any femtocell as nonsubscribers.

  * •

Closed access: An FAP only allows its subscribers to connect.

  * •

Open access: An FAP allows all its covered UEs, no matter registered or not,
to connect.

  * •

Hybrid access: An FAP allows its covered nonsubscribers to connect via a
subset of its available subchannels, and reserves the remaining subchannels
for its subscribers.

Hybrid access [5] is an intermediate approach, in which a fraction of resource
is allocated to nonsubscribers. By doing so, nonsubscribers near an FAP may
handover into the femtocell to avoid high interference; meanwhile, with
certain amount of resource reserved for subscribers, the performance of
subscribers may be well assured even in the presence of nonsubscribers.

In hybrid access, a central issue is how to allocate the resource between
subscribers and nonsubscribers. Previous studies [6] [7] indicate that hybrid
access improves the network performance at the cost of reduced performance for
subscribers, therefore suggesting a tradeoff between the performance of
nonsubscribers and subscribers. In this paper, we consider a hybrid access
mechanism that uses a randomized channel assignment strategy, and analyze the
performance in the downlink of both macrocells and femtocells. We employ
stochastic geometry to characterize the spatial distributions of users as well
as access points; see, e.g., [8] and references therein for its recent
applications in wireless networks. In order to make the work integral, we will
carry out the analysis in two different cases. As a general assumption, we
first assume that the FAPs are distributed as a Poisson point process (PPP).
Then, we switch to the case when the FAPs are distributed as a Neyman-Scott
cluster process. The cluster process is likely to be more realistic because
the FAPs are typically deployed in populous locations, like commercial or
residential area. Accordingly, we derive the key performance indicators
including mean achievable rates and distributions of the signal-to-
interference-plus-noise ratios (SINRs) of both nonsubscribers and subscribers.
In our study, we establish general integral expressions for the performance
indicators, and closed form expressions under specific model parameters. With
the obtained results, we reveal how the performance of subscribers and
nonsubscribers trades off each other.

The introduction of stochastic geometry in the analysis of wireless network is
not our original, and an overview of related works is as follows. In [9], the
authors proposed to study key performance indicators for cellular networks,
such as coverage probabilities and mean achievable rates. In [10], the
considered scheme divides the spectrum resource into two orthogonal parts
which are assigned to macrocells and femtocells, respectively, with femtocells
being closed access. In [11], the authors considered two-tier femtocell
networks using time-hopped CDMA, examining the uplink outage probability and
the interference avoidance capability. In [12], the success probabilities
under Rayleigh fading for both macrocells and femtocells are derived in uplink
and downlink respectively. In [13] and [14], stochastic geometry tools are
applied in the coexistence analysis of cognitive radio networks. In [15] and
[16], the authors studied the performance of various femtocell access
mechanisms, under substantially different system models from ours. More
explicitly, the work in [15], which does not make use of stochastic geometry,
focused on the uplink with only one MBS and one FAP in the model. Though the
work in [16] also applies the Laplace transform of interference in the
derivation, the substantially difference of our work lies in that we take the
load (measured by the number of UEs in a cell) into consideration, utilizing
the size distribution of Voronoi cells to derive the distribution of the load.
Moreover, we model the mechanism for sharing sub-channels in multi-channel
systems, and propose to use a two-scale approximation which substantially
simplifies the analysis. All the works mentioned above are based on the PPP
assumption. The works applying the clustered model can be found in [17] which
derived the success probability for transmission in clustered ad-hoc networks
and in [18] which discussed the property of interference with clustered
interferers.

The main contribution of our work is detailed as follows. The existing works
mostly ignore the network load which is a key factor that affects the
distribution of interfering access points (APs). For example, when the load is
uniformly distributed in the plane, the APs with larger coverage may
experience more load, thus leading to more interference to the network.
Moreover, the optimal proportion of shared resources of a femtocell also
depends on the distribution of network load. In our work, we focus on the
performance analysis in the context of multi-channel systems, in which case
not all sub-channels are occupied and not all APs cause interference to a
given subchannel. We evaluate the two-tier interference when the FAPs are
distributed as the PPP and the Neyman-Scott cluster respectively. In addition,
we propose two-scale approximation to simplify the analysis and verify the
effectiveness of the approximation by simulation.

The remaining part of the paper is organized as follows. Section II describes
the two-tier network model, the channel assignment strategy, and the hybrid
access mechanism. Based on a two-scale approximation for the spatial
distributions of FAPs, Section III analyzes the statistical behavior of UEs,
deriving the distributions of the number of UEs connecting to either an FAP or
an MBS, as well as the probabilities of a subchannel being used by either an
FAP or an MBS. Built upon those statistics, Section IV and V establish
expressions for the distributions of SINRs, and mean achievable rates in the
cases when the FAPs are distributed as PPP and Neyman-Scott cluster
respectively. Section VII illustrates the aforementioned analysis by numerical
results, which are also corroborated by simulations. Finally, Section VIII
concludes the paper.

##  II Network Model

###  II-A Hybrid Access Femtocells

In the two-tier network, we consider two types of access points, MBSs and
FAPs. The MBSs constitute the macrocell tier, and they induce a Voronoi
tessellation of the plane (see Figure 2). When a UE attempts to access the
macrocell network, it chooses to connect to the MBS in the Voronoi cell in
which the UE is situated. An FAP provides network access to UEs in its
vicinity, and we assume that all FAPs have a covering radius of $R_{f}$.
Within the covered circular area of each FAP are two types of UEs, called
subscribers and inside nonsubscribers. Inside nonsubscribers are those UEs who
gather around an FAP without subscribing to its service; for example,
transient customers in a shop or a restaurant. Besides those two types of UEs,
we also consider a third type of UEs, outside nonsubscribers, who are
uniformly scattered over the whole plane, corresponding to those regular
macrocell network users.

(a) FAPs are distributed as PPP.

(b) FAPs are distributed as Neyman-Scott cluster.

Figure 2: The Voronoi macrocell topology, in which each Voronoi cell is the
coverage area of a macrocell and each small circle represents a femtocell.

The available spectrum is evenly divided into $M$ subchannels, which are to be
shared by both macrocell tier and femtocell tier. Each FAP is configured to
allocate a fixed number, $M_{s}$, of subchannels for its covered inside
nonsubscribers. These $M_{s}$ subchannels are called shared subchannels, and
the remaining $M_{r}=M-M_{s}$ subchannels are called reserved subchannels as
they are reserved for the subscribers. In the considered hybrid access
mechanism, each FAP selects its shared subchannels randomly, and independently
of other FAPs. We assume that each UE, whether subscriber or nonsubscriber,
needs one subchannel for transmitting. When a UE accesses an MBS or an FAP,
the serving subchannel is selected randomly (see Figure 3).

Figure 3: Spectrum allocation in each hybrid access femtocell. All the $M_{s}$
shared subchannels are randomly selected by each FAP.

The hybrid access mechanism operates as follows.

  * •

A subscriber accesses to one of the $M_{r}$ reserved subchannels of its
corresponding FAP. When there are more than $M_{r}$ subscribers in an FAP,
they are served by time-sharing with equal time proportion.

  * •

An inside nonsubscriber accesses to one of the $M_{s}$ shared subchannels of
its covering FAP. When there are more than $M_{s}$ inside nonsubscribers in an
FAP, they are served by time-sharing with equal time proportion.

  * •

An outside nonsubscriber accesses the MBS located in the Voronoi cell in which
the outside nonsubscriber is situated. When there are more than $M$ outside
nonsubscribers in the Voronoi cell, they are served by time-sharing with equal
proportion.

###  II-B Mathematical Model and Two-scale Approximation

To formulate the aforementioned hybrid access scenario mathematically, we
model the spatial distributions of the nodes using spatial point processes as
follows. The MBSs constitute a homogeneous Poisson point process (PPP)
$\Phi_{m}$ of intensity $\lambda_{m}$ on the plane. The distribution of FAPs
will be divided into two cases for discussion:

  * •

Case 1: the FAPs constitute another homogeneous PPP $\Phi_{f}$ of intensity
$\lambda_{f}$.

  * •

Case 2: the FAPs are distributed as a Neyman-Scott cluster process $\Phi_{f}$
[19]. The center of the clusters are assumed to be distributed according to a
stationary PPP $\Phi_{p}$ of intensity $\lambda_{p}$, which is called the
parent process. For each cluster center $x\in\Phi_{p}$, the FAPs are
distributed according to an independent PPP $\Phi^{x}$ of intensity
$\lambda_{c}$ in the circular covered area of radius $R_{c}$ around the center
$x$. The complete distribution of all FAPs is given as

$\Phi_{f}=\bigcup_{x\in\Phi_{p}}\Phi^{x}.$ (1)

In this case, the number of FAPs in a typical cluster is a Poisson random with
parameter $\pi R_{c}^{2}\lambda_{c}$ and the intensity of all FAPs is
$\lambda_{f}=\pi R_{c}^{2}\lambda_{c}\lambda_{p}$.

In the circular covered area of radius $R_{f}$ of each FAP, the subscribers
are distributed according to a homogeneous PPP of intensity $\lambda_{s}$, and
the inside nonsubscribers are distributed according to another homogeneous PPP
of intensity $\lambda_{\mathrm{in}}$. Outside nonsubscribers constitute on the
whole plane a homogeneous PPP of intensity $\lambda_{\mathrm{out}}$. All the
PPPs are mutually independent.

In this paper, we focus on the downlink performance. The transmit power is set
to a constant value $P_{m}$ for an MBS, and $P_{f}$ for an FAP. For the sake
of convenience, we adopt a standard path loss propagation model with path loss
exponent $\alpha>2$. Regarding fading, we assume that the link between the
serving access point (either an MBS or an FAP) and the served UE experiences
Rayleigh fading with parameter $\mu$. The received signal power of a UE at a
distance $r$ from its serving access point therefore is $P_{m}hr^{-\alpha}$
(MBS) or $P_{f}hr^{-\alpha}$ (FAP) where $h\sim\mathrm{Exp}(\mu)$. The fading
of interference links may follow an arbitrary probability distribution, and is
denoted by $g$. Furthermore, considering the typical scenario of indoor
femtocell deployment, we introduce a wall isolation at the boundary of each
FAP coverage area, which incurs a wall penetration loss factor $W<1$. For all
receivers, the noise power is $\sigma^{2}$.

The different point processes corresponding to different entities in the
network interact in a complicated way, thus making a rigorous statistical
analysis extremely difficult. For example, an inside nonsubscriber may be
covered by more than one FAPs, thus leading to the delicate issue of FAP
selection, and furthermore rendering the subchannel usage distributions among
FAPs and MBSs intrinsically correlated. To overcome the technical difficulties
due to spatial interactions, in the subsequent analysis we propose a two-scale
approximation for the network model, motivated by the fact that the covered
area of an FAP is significantly smaller than that of an MBS. The two-scale
approximation consists of two views, the macro-scale view and the micro-scale
view. The macro-scale view concerns an observer outside the coverage area of
an FAP, and in that view the whole coverage area of the FAP shrinks to a
single point, marked by the numbers of subscribers and inside nonsubscribers
therein. The micro-scale view concerns an observer inside the coverage area of
an FAP, and in that view the coverage area is still circular with radius
$R_{f}$ in which the subscribers and inside nonsubscribers are spatially
distributed. By such a two-scale approximation, an inside nonsubscriber can
only be covered by a unique FAP, and the coverage area of an FAP can only be
within a unique Voronoi cell of an MBS. Meanwhile, at the cell edge the
outside nonsubscribers are clearly divided by the boundary and the subscribers
and inside nonsubscribers are all attached to the corresponding FAPs which are
also clearly divided. These consequences substantially simplify the
performance analysis. In Section VII, we validate the two-scale approximation
method through comparing analytical results and simulation results, for
network parameters of practical interest.

##  III Statistics of UEs and Subchannels

In this section, we characterize the distributions of UEs connecting to
different types of access points, and the distributions of used subchannels in
MBSs and FAPs. The analysis is based on a snapshot of the network model, and
the obtained results will then be applied for characterizing the distributions
of SINRs and achievable rates in Section IV.

###  III-A Distributions of UEs

Let $U_{s}$ be the number of subscribers accessing a given FAP and from our
model we have $U_{s}\sim\mathrm{Poisson}(\lambda_{s}\pi R_{f}^{2})$.
Similarly, let $U_{\mathrm{in}}$ be the number of inside nonsubscribers
accessing a given FAP, and we have
$U_{\mathrm{in}}\sim\mathrm{Poisson}(\lambda_{\mathrm{in}}\pi R_{f}^{2})$.

The number of outside nonsubscribers who access a given MBS, denoted by
$U_{\mathrm{out}}$, is characterized as follows. We note that the macrocell
coverage area is a Voronoi cell, and denote by $S$ the area of the Voronoi
cell. There is no known closed form expression of the probability density
function (pdf) of $S$, whereas a simple approximation [20] has proven
sufficiently accurate for practical purposes. Considering scaling, the
approximate pdf of the size of a macrocell coverage area is given by

$f(S)=\frac{343}{15}\sqrt{\frac{7}{2\pi}}(S\lambda_{m})^{\frac{5}{2}}\exp(-\frac{7}{2}S\lambda_{m})\lambda_{m}.$
(2)

Conditioning upon $S$, the number of outside nonsubscribers is a Poisson
random variable with mean $\lambda_{\mathrm{out}}S$. The probability
generating function of the unconditioned $U_{\mathrm{out}}$ is thus given by

$G(z)=\int_{0}^{\infty}\exp\Big{(}\lambda_{\mathrm{out}}(z-1)S\Big{)}f(S)dS.$
(3)

Plugging in the approximate pdf of $S$ and simplifying the integral, we get

$G(z)=\frac{343}{8}\sqrt{\frac{7}{2}}\Big{(}\frac{7}{2}-\frac{\lambda_{\mathrm{out}}}{\lambda_{m}}(z-1)\Big{)}^{-\frac{7}{2}}.$
(4)

The distribution of $U_{\mathrm{out}}$ is therefore given by the derivatives
of $G(z)$,

$\mathbb{P}\\{U_{\mathrm{out}}=i\\}=\frac{G^{(i)}(0)}{i!},\quad i=0,1,\ldots.$
(5)

###  III-B Distributions of Subchannel Usage

Since the subchannels are uniformly and independently selected by each FAP, it
suffices to analyze an arbitrary one of them. Let us examine the probability
that a given subchannel is used by an MBS or an FAP. First, we evaluate the
average number of subchannels used by an MBS or an FAP, and then we normalize
the average number by the total number of subchannels, $M$.

The probability that a subchannel is used by an FAP is

$\displaystyle
P_{\mathrm{busy},f}=\frac{1}{M}\Big{(}\sum_{i=0}^{\infty}\min\\{i,M_{r}\\}\mathbb{P}\\{U_{s}=i\\}$
$\displaystyle+\sum_{j=0}^{\infty}\min\\{j,M_{s}\\}\mathbb{P}\\{U_{\mathrm{in}}=j\\}\Big{)}.$
(6)

For a Poisson random variable $N\sim\mathrm{Poisson}(\lambda)$, its cumulative
distribution function (cdf) is

$\sum_{i=0}^{n}\mathbb{P}\\{N=i\\}=\sum_{i=0}^{n}\frac{\lambda^{i}}{i!}e^{-\lambda}=\frac{\Gamma(n+1,\lambda)}{n!},$
(7)

where $\Gamma(s,x)=\int_{x}^{\infty}t^{s-1}e^{-t}\mathrm{d}t$ is the
incomplete gamma function. Using (7) to simplify $P_{\mathrm{busy},f}$, we get

$\displaystyle
P_{\mathrm{busy},f}=1+\frac{M_{r}}{M}\frac{1}{M_{r}!}\Big{(}\lambda_{s}\pi
R_{f}^{2}\Gamma(M_{r},\lambda_{s}\pi R_{f}^{2})$
$\displaystyle-\Gamma(M_{r}+1,\lambda_{s}\pi R_{f}^{2})\Big{)}$
$\displaystyle+\frac{M_{s}}{M}\frac{1}{M_{s}!}\Big{(}\lambda_{\mathrm{in}}\pi
R_{f}^{2}\Gamma(M_{s},\lambda_{\mathrm{in}}\pi R_{f}^{2})$
$\displaystyle-\Gamma(M_{s}+1,\lambda_{\mathrm{in}}\pi R_{f}^{2})\Big{)}.$ (8)

The probability that a subchannel is used by an MBS is

$\displaystyle
P_{\mathrm{busy},m}=\frac{1}{M}\sum_{i=0}^{\infty}\min\\{i,M\\}\mathbb{P}\\{U_{\mathrm{out}}=i\\},$
(9)

where $\mathbb{P}\\{U_{\mathrm{out}}=i\\}$ is given by (5).

The spatial point process of FAPs that use a given subchannel is the
independent thinning of the original process of FAPs $\Phi_{f}$ by the
probability $P_{\mathrm{busy},f}$, denoted by $\Phi_{f}^{\prime}$. The term
“independent thinning” means that $\Phi_{f}^{\prime}$ can be viewed as
obtained from $\Phi_{f}$ by independently removing points with probability
$1-P_{\mathrm{busy},f}$. When the FAPs are distributed as a homogeneous PPP,
the resulting point process is still a homogeneous PPP with intensity
$\lambda_{f}^{\prime}=\lambda_{f}P_{\mathrm{busy},f}$. For the case when the
FAPs are distributed as Neyman-Scott cluster process, the resulting point
process is a Neyman-Scott cluster process with intensity
$\lambda_{f}^{\prime}=\lambda_{f}P_{\mathrm{busy},f}$. Moreover, the intensity
of the parent process is still $\lambda_{p}$ and the intensity of the FAPs in
each cluster is reduced to
$\lambda_{c}^{\prime}=\lambda_{c}P_{\mathrm{busy},f}$. As for the MBSs, the
correlations between the sizes of neighboring cells may lead to the dependence
in the thinning of the original PPP of MBSs. However, in order to facilitate
the analysis, we assume that the independent thinning assumption still holds.
Therefore, the spatial process of MBSs that use a given subchannel is the
independent thinning of the original PPP of MBSs $\Phi_{m}$ by the probability
$P_{\mathrm{busy},m}$, denoted by $\Phi_{m}^{\prime}$ with intensity
$\lambda_{m}^{\prime}=\lambda_{m}P_{\mathrm{busy},m}$. Meanwhile, the
computation of $P_{\mathrm{busy},m}$ and the SINR distribution can also be
decoupled. The simulation results validate that our independent thinning
assumption is rather reliable. These two independently thinned point processes
will prove useful in the subsequent analysis.

##  IV Performance With Poisson Distribution Of FAPs

In this section, we derive the distributions of the SINRs and the mean
achievable rates of UEs served by MBS and FAP respectively when the FAPs are
distributed as the PPP. For each type of UEs, we begin with general settings,
and then simplify the general results under specific parameters to gain
insights. The mean achievable rates are the averaged instantaneous achievable
rates over both channel fading and spatial distributions of UEs and access
points.

###  IV-A Macrocell UEs

####  IV-A1 General Case

For an active UE served by an MBS, it must be occupying one subchannel of the
MBS. The following theorem gives the cdf of SINR and the mean achievable rate
of each active macrocell UE in general case.

###### Theorem 1

The cdf of the SINR of a macrocell UE, denoted by
$Z_{m}(T)=\mathbb{P}\\{\mathrm{SINR}\leq T\\}$, is given by

$\displaystyle
Z_{m}(T)=1-\pi\lambda_{m}\\!\\!\\!\int_{0}^{\infty}\\!\\!\\!\exp\Big{(}\pi
v\lambda_{m}^{\prime}\Big{(}1-\beta(T,\alpha)$
$\displaystyle-\frac{1}{P_{\mathrm{busy},m}}\Big{)}-\frac{\mu
Tv^{\frac{\alpha}{2}}\sigma^{2}}{P_{m}}\Big{)}\mathrm{d}v.$ (10)

The mean achievable rate of a macrocell UE is given by

$\displaystyle\tau_{m}=\pi\lambda_{m}\\!\\!\\!\int_{0}^{\infty}\\!\\!\\!\int_{0}^{\infty}\\!\\!\\!\exp\Big{(}\pi
v\lambda_{m}^{\prime}\Big{(}1-\beta(e^{t}-1,\alpha)$
$\displaystyle-\frac{1}{P_{\mathrm{busy},m}}\Big{)}-\frac{\mu
v^{\frac{\alpha}{2}}\sigma^{2}(e^{t}-1)}{P_{m}}\Big{)}\mathrm{d}t\mathrm{d}v.$
(11)

In (10) and (11), $\beta(T,\alpha)$ is given by

$\displaystyle\beta(T,\alpha)=\frac{2(\mu
T)^{\frac{2}{\alpha}}}{\alpha}\mathbb{E}_{g}\Big{(}g^{\frac{2}{\alpha}}\Big{(}\Gamma(-\frac{2}{\alpha},\mu
Tg)$
$\displaystyle-(1+\frac{\lambda_{f}^{\prime}(WP_{f})^{\frac{2}{\alpha}}}{\lambda_{m}^{\prime}P_{m}^{\frac{2}{\alpha}}})\Gamma(-\frac{2}{\alpha})\Big{)}\Big{)}.$
(12)

The proof of Theorem 1 is in Appendix A.

In Theorem 1, $Z_{m}(T)$ in (10) gives the probability that the SINR is below
a given target level $T$, and $\tau_{m}$ in (11) gives the mean achievable
rate of a macrocell UE. The integrals in (10) and (11) can be evaluated by
numerical methods, and furthermore, they can be simplified to concise forms in
the special case when the interference experiencing Rayleigh fading and the
path loss exponent being $\alpha=4$ with no noise, $\sigma^{2}=0$.

####  IV-A2 Special Case When Interference Experiences Rayleigh Fading

Here we consider the case where the interference experiences Rayleigh
distribution with mean $\mu$, i.e. $g\sim\mathrm{Exp}(\mu)$. In this case, the
results are as follows.

###### Corollary 1

When the interference follows Rayleigh fading, the cdf of the SINR is

$\displaystyle\\!\\!\\!Z_{m}(T)$
$\displaystyle\\!\\!\\!\\!\\!\\!=\\!\\!\\!\\!\\!\\!$ $\displaystyle
1-\pi\lambda_{m}\\!\\!\\!\int_{0}^{\infty}\\!\\!\\!\exp\Big{(}-\pi
v\lambda_{m}-\pi v\lambda_{m}^{\prime}\varphi(T,\alpha)$ (13)
$\displaystyle-\pi
v\lambda_{f}^{\prime}\Big{(}\frac{P_{f}WT}{P_{m}}\Big{)}^{2/\alpha}\Gamma(1+\frac{2}{\alpha})\Gamma(1-\frac{2}{\alpha})$
$\displaystyle-\frac{\mu Tv^{\alpha/2}\sigma^{2}}{P_{m}}\Big{)}\mathrm{d}v.$

The mean achievable rate is

$\displaystyle\\!\\!\\!\tau_{m}$
$\displaystyle\\!\\!\\!\\!\\!\\!=\\!\\!\\!\\!\\!\\!$
$\displaystyle\pi\lambda_{m}\int_{0}^{\infty}\\!\\!\\!\int_{0}^{\infty}\\!\\!\\!\exp\Big{(}-\pi
v\lambda_{m}-\pi v\lambda_{m}^{\prime}\varphi(e^{t}-1,\alpha)$ (14)
$\displaystyle-\pi
v\lambda_{f}^{\prime}\Big{(}\frac{P_{f}W(e^{t}-1)}{P_{m}}\Big{)}^{2/\alpha}\Gamma(1+\frac{2}{\alpha})\Gamma(1-\frac{2}{\alpha})$
$\displaystyle-\frac{\mu
v^{\alpha/2}\sigma^{2}(e^{t}-1)}{P_{m}}\Big{)}\mathrm{d}v\mathrm{d}t,$

where

$\varphi(T,\alpha)=T^{2/\alpha}\int_{T^{-2/\alpha}}^{\infty}\frac{1}{1+u^{\alpha/2}}\mathrm{d}u.$
(15)

The results in Corollary 1 are further simplified in the following special
cases.

Specifically, when $\alpha=4$ we obtain

$\displaystyle Z_{m}(T)$ $\displaystyle\\!\\!\\!\\!\\!\\!=\\!\\!\\!\\!\\!\\!$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!1-\frac{1}{1+\sqrt{T}\Big{(}\arctan\sqrt{T}+\frac{\pi}{2}\frac{\lambda_{f}^{\prime}}{\lambda_{m}^{\prime}}\sqrt{\frac{WP_{f}}{P_{m}}}\Big{)}P_{\mathrm{busy},m}}.$

The mean achievable rate of a macrocell UE is simplified into

$\tau_{m}=\int_{0}^{\frac{\pi}{2}}\frac{2}{\tan
y+(\frac{\pi}{2}-y)P_{\mathrm{busy},m}+\frac{\pi}{2}\frac{\lambda_{f}^{\prime}}{\lambda_{m}}\sqrt{\frac{WP_{f}}{P_{m}}}}\mathrm{d}y.$
(17)

From a practical perspective, it is desirable to shape $Z_{m}(T)$ to make it
small for small values of $T$. From (LABEL:equ:SINR_Simple), we see that there
are two approaches to shape $Z_{m}(T)$. First, $Z_{m}(T)$ decreases as
$P_{\mathrm{busy},m}$, the probability that a subchannel is used by an MBS,
decreases. This may be interpreted as an effect of frequency reuse. Second,
$Z_{m}(T)$ decreases as the whole term,
$\frac{\lambda_{f}^{\prime}}{\lambda_{m}}\sqrt{\frac{WP_{f}}{P_{m}}}$,
decreases, which corresponds to a number of network parameters, representing
the effect due to the deployment of the femtocell tier.

###  IV-B Femtocell UEs

####  IV-B1 General Case

A UE served by an FAP occupies one subchannel of the FAP. The following
theorem gives the cdf of the SINR and the mean achievable rate of each active
femtocell UE in general case.

###### Theorem 2

The cdf of the SINR of a femtocell UE in general case is given by

$\displaystyle
Z_{f}(T)=1-\frac{1}{R_{f}^{2}}\int_{0}^{R_{f}^{2}}\exp\Big{(}-\rho(\alpha)T^{2/\alpha}v$
$\displaystyle-\frac{\mu Tv^{\alpha/2}\sigma^{2}}{P_{f}}\Big{)}\mathrm{d}v.$
(18)

The mean achievable rate of a femtocell UE is given by

$\displaystyle\tau_{f}=\frac{1}{R_{f}^{2}}\int_{0}^{R_{f}^{2}}\\!\\!\\!\int_{0}^{\infty}\\!\\!\exp\Big{(}-\rho(\alpha)(e^{t}-1)^{2/\alpha}v$
$\displaystyle-\frac{\mu
v^{\alpha/2}\sigma^{2}(e^{t}-1)}{P_{f}}\Big{)}\mathrm{d}t\mathrm{d}v.$ (19)

in (18) and (2), $\rho(\alpha)$ is given by

$\displaystyle\rho(\alpha)=-\frac{2\pi\mu^{2/\alpha}}{\alpha}\Gamma\Big{(}-\frac{2}{\alpha}\Big{)}\Big{(}\lambda_{m}^{\prime}\Big{(}\frac{WP_{m}}{P_{f}}\Big{)}^{2/\alpha}$
$\displaystyle+\lambda_{f}^{\prime}W^{4/\alpha}\Big{)}\mathbb{E}_{g}(g^{2/\alpha}).$
(20)

The proof of Theorem 2 is in Appendix B.

####  IV-B2 Special Case When Interference Experiences Rayleigh Fading

For Rayleigh fading, the results are essentially the same as that in the
general fading case. The only additional simplification is the evaluation of
$\rho(\alpha)$. We just give the result in the very special case when
$\sigma^{2}=0$ and $\alpha=4$.

When the interference experiences Rayleigh fading, we have
$\mathbb{E}_{g}(g^{\frac{1}{2}})=\mu\int_{0}^{\infty}g^{\frac{1}{2}}e^{-\mu
g}\mathrm{d}g=\frac{1}{2}\sqrt{\frac{\pi}{\mu}}$, and consequently,

$\rho(4)=\frac{\pi^{2}}{2}\Big{(}\lambda_{m}^{\prime}\sqrt{\frac{WP_{m}}{P_{f}}}+\lambda_{f}^{\prime}W\Big{)}.$
(21)

The SINR distribution is then given by

$Z_{f}(T)=1-\frac{1-e^{-\rho(4)\sqrt{T}R_{f}^{2}}}{\rho(4)\sqrt{T}R_{f}^{2}}.$
(22)

Let $y=\rho(4)R_{f}^{2}\sqrt{e^{t}-1}$, the mean achievable rate is simplified
into

$\tau_{f}=2\int_{0}^{\infty}\frac{1-e^{-y}}{y^{2}+\rho^{2}(4)R_{f}^{4}}\mathrm{d}y.$
(23)

These expressions are convenient for numerical evaluation.

##  V Performance With Clustered FAPs

In this section, we derive the distributions of SINRs and the mean achievable
rates in the case when the FAPs are distributed as the Neyman-Scott cluster
process. To facilitate the analysis, we only focus on the case when the
interference links experience exponential fading.

###  V-A Macrocell UEs

The following theorem gives the cdf of SINR and the mean achievable rate of
each active macrocell UE.

###### Theorem 3

In the case when the FAPs are distributed as a Neyman-Scott cluster process
and the interference experiences Rayleigh fading, the cdf of the SINR of a
macrocell UE, denoted by $Z_{m}(T)=\mathbb{P}\\{\mathrm{SINR}\leq T\\}$, is
given by

$\displaystyle
Z_{m}(T)=1-\pi\lambda_{m}\\!\\!\\!\int_{0}^{\infty}\\!\\!\\!\exp\bigg{(}-\pi
v\lambda_{m}$
$\displaystyle-\lambda_{p}\int_{R^{2}}\Big{(}1-\eta\Big{(}\frac{Tv^{\alpha/2}WP_{f}}{P_{m}},x\Big{)}\Big{)}\mathrm{d}x$
$\displaystyle-\frac{\mu Tv^{\alpha/2}\sigma^{2}}{P_{m}}-\pi
v\lambda_{m}^{\prime}\varphi(T,\alpha)\bigg{)}\mathrm{d}v.$ (24)

The mean achievable rate of a macrocell UE is

$\displaystyle\tau_{m}=\pi\lambda_{m}\int_{0}^{\infty}\\!\\!\\!\int_{0}^{\infty}\\!\\!\\!\exp\bigg{(}-\pi
v\lambda_{m}$
$\displaystyle-\lambda_{p}\int_{R^{2}}\Big{(}1-\eta\Big{(}\frac{(e^{t}-1)v^{\alpha/2}WP_{f}}{P_{m}},x\Big{)}\Big{)}\mathrm{d}x$
$\displaystyle-\frac{\mu(e^{t}-1)v^{\alpha/2}\sigma^{2}}{P_{m}}-\pi
v\lambda_{m}^{\prime}\varphi(e^{t}-1,\alpha)\bigg{)}\mathrm{d}v\mathrm{d}t,$
(25)

where $\varphi(T,\alpha)$ is given by (15). Let $C(o,R_{c})$ be the circle
centered at the origin with radius $R_{c}$ and $\eta(s,x)$ is given as

$\eta(s,x)=\exp\Big{(}\int_{C(o,R_{c})}\frac{-\lambda_{c}^{\prime}}{1+\frac{1}{s}|x+y|^{\alpha}}\mathrm{d}y\Big{)}.$
(26)

The proof of Theorem 3 is in Appendix C.

###  V-B Femtocell UEs

The following theorem gives the cdf of SINR and the mean achievable rate of
each active femtocell UE.

###### Theorem 4

In the case when the FAPs are distributed as a Neyman-Scott cluster process
and the interference experiences Rayleigh fading, the cdf of the SINR of a
femtocell UE, denoted by $Z_{f}(T)=\mathbb{P}\\{\mathrm{SINR}\leq T\\}$, is
given by

$\displaystyle Z_{f}(T)=1-\frac{2}{\pi
R_{c}^{2}R_{f}^{2}}\\!\\!\int_{0}^{R_{f}}\\!\\!\exp\bigg{(}-\frac{\mu
Tr^{\alpha}\sigma^{2}}{P_{f}}$ $\displaystyle-\pi
r^{2}\lambda_{m}^{\prime}\Big{(}\frac{WTP_{m}}{P_{f}}\Big{)}^{2/\alpha}\Gamma(1+\frac{2}{\alpha})\Gamma(1-\frac{2}{\alpha})$
$\displaystyle-\lambda_{p}\int_{R^{2}}\Big{(}1-\eta(Tr^{\alpha}W^{2},x)\Big{)}\mathrm{d}x\bigg{)}$
$\displaystyle\bigg{(}\int_{C(o,R_{c})}\eta(Tr^{\alpha}W^{2},y-z)\mathrm{d}y\bigg{)}r\mathrm{d}r.$
(27)

The mean achievable rate of a femtocell UE is given by

$\displaystyle\tau_{f}=\frac{2}{\pi
R_{c}^{2}R_{f}^{2}}\\!\\!\int_{0}^{\infty}\\!\\!\int_{0}^{R_{f}}\\!\\!\exp\bigg{(}-\frac{\mu(e^{t}-1)r^{\alpha}\sigma^{2}}{P_{f}}-$
$\displaystyle\pi
r^{2}\lambda_{m}^{\prime}\Big{(}\frac{W(e^{t}-1)P_{m}}{P_{f}}\Big{)}^{2/\alpha}\Gamma(1+\frac{2}{\alpha})\Gamma(1-\frac{2}{\alpha})$
$\displaystyle-\lambda_{p}\int_{R^{2}}\Big{(}1-\eta\big{(}(e^{t}-1)r^{\alpha}W^{2},x\big{)}\Big{)}\mathrm{d}x\bigg{)}$
$\displaystyle\bigg{(}\int_{C(o,R_{c})}\eta\Big{(}(e^{t}-1)r^{\alpha}W^{2},y-z\Big{)}\mathrm{d}y\bigg{)}r\mathrm{d}r\mathrm{d}t,$
(28)

where $z=(r,0)$ and $\eta(s,x)$ is given by (26).

The proof of Theorem 4 is in Appendix D.

##  VI Mean Achievable Rates of Nonsubscribers and Subscribers

There are two types of nonsubscribers, outside nonsubscribers who access MBSs
and inside nonsubscribers who access FAPs. When the number of outside
nonsubscribers in a macrocell is no greater than the total number of
subchannels (i.e., $U_{\mathrm{out}}\leq M$), each nonsubscriber UE
exclusively occupies a subchannel, and its mean achievable rate is $\tau_{m}$.
However, when $U_{\mathrm{out}}>M$, those $U_{\mathrm{out}}$ UEs share the $M$
subchannels with mean achievable rate $\frac{M}{U_{\mathrm{out}}}\tau_{m}$.
Since the evaluation is conditioned upon the existence of at least one UE, the
mean achievable rate of an outside nonsubscriber UE is given by

$\\!\\!\\!\\!\\!\\!\tau_{\mathrm{out}}=\frac{\sum_{i=1}^{M}\mathbb{P}\\{U_{\mathrm{out}}=i\\}+\sum_{i=M+1}^{\infty}\mathbb{P}\\{U_{\mathrm{out}}=i\\}\frac{M}{i}}{1-\mathbb{P}\\{U_{\mathrm{out}}=0\\}}\tau_{m}.$
(29)

Similarly, the mean achievable rate of an inside nonsubscriber is given by

$\\!\\!\\!\\!\\!\\!\tau_{\mathrm{in}}=\frac{\sum_{j=1}^{M_{s}}\mathbb{P}\\{U_{\mathrm{in}}=j\\}+\sum_{j=M_{s}+1}^{\infty}\mathbb{P}\\{U_{\mathrm{in}}=j\\}\frac{M_{s}}{j}}{1-\mathbb{P}\\{U_{\mathrm{in}}=0\\}}\tau_{f}.$
(30)

When averaged over both inside and outside nonsubscribers, the overall mean
achievable rate of nonsubscriber is obtained as

$\\!\\!\\!\\!\tau_{n}=\frac{\lambda_{out}\tau_{\mathrm{out}}+\lambda_{f}\lambda_{in}\pi
R_{f}^{2}\tau_{\mathrm{in}}}{\lambda_{out}+\lambda_{f}\lambda_{in}\pi
R_{f}^{2}}.$ (31)

Regarding subscribers, note that they are exclusively served by FAPs. Similar
to the analysis of nonsubscribers, the mean achievable rate of a subscriber is
given by

$\tau_{s}=\frac{\sum_{i=1}^{M_{r}}\mathbb{P}\\{U_{s}=i\\}+\sum_{i=M_{r}+1}^{\infty}\mathbb{P}\\{U_{s}=i\\}\frac{M_{r}}{i}}{1-\mathbb{P}\\{U_{s}=0\\}}\tau_{f}.$
(32)

##  VII Numerical Results

The numerical results are obtained according to both the analytical results we
have derived and Monte Carlo simulation. The default configurations of system
model are as follows (also see Table I). The total number of subchannels is
$M=20$ and the coverage radius of each femtocell is $R_{f}=10$m. The transmit
power of FAP is $P_{f}=13$dBm, and that of MBS is $P_{m}=39$dBm. We set the
path loss exponent as $\alpha=4$, with all links experiencing Rayleigh fading
of normalized $\mu=1$. The wall penetration loss is set as $W=-6$dB. We focus
on the interference-limited regime, and for simplicity we ignore the noise
power (i.e., $\sigma^{2}=0$). The intensity of MBSs is set as
$\lambda_{m}=0.00001$ and of FAPs $\lambda_{f}=0.0001$. In the clustered case,
the intensity of parent process is set as $\lambda_{p}=0.00001$ and of FAPs in
each cluster $\lambda_{c}=0.00127$. By setting the radius of each cluster as
$R_{c}=50$m, we get the intensity of FAPs as $\lambda_{f}=0.0001$. So the
average coverage area of an MBS is roughly equal to a circle with a radius of
$180$m, and on average there are ten FAPs within the coverage area of an MBS.
Unless otherwise specified, the subscribers and inside nonsubscribers are
distributed within an FAP coverage are with intensities
$\lambda_{s}=\lambda_{\mathrm{in}}=0.015$. The intensity of outside
nonsubscribers is set as $\lambda_{\mathrm{out}}=0.0001$.

TABLE I: SYSTEM PARAMETERS Symbol | Description | Value
---|---|---
$\Phi_{m},\Phi_{f}$ | Point processes defining the MBSs and FAPs | N/A
$\lambda_{m}$ | Density of MBSs | 0.00001 MBS/m2
$\lambda_{f}$ | Density of FAPs | 0.0001 FAP/m2
$\lambda_{p}$ | Density of parent process for the clustered FAPs | 0.00001 center/m2
$\lambda_{c}$ | Density of FAPs in each cluster | 0.00127 FAPs/m2
$R_{c}$ | Radius of each cluster |  $50$m
$R_{f}$ | Radius of femtocell |  $10$m
$\lambda_{\mathrm{out}}$ | Densities of outside nonsubscribers | 0.0001 user/m2
$\lambda_{s},\lambda_{\mathrm{in}}$ | Densities of subscribers and inside nonsubscribers | 0.015 user/m2
$P_{m},P_{f}$ | Transmit power at MBS and FAP |  $39$dBm,$13$dBm
$M$ | Number of subchannels at each access point | $20$
$M_{s},M_{r}$ | Number of subchannels shared and reserved by each femtocell | Not fixed
$\alpha$ | Path loss exponent | $4$
$\mu$ | Rayleigh fading parameter |  $1$ (normalized)
$W$ | Wall penetration loss |  $-6$dB
$\sigma^{2}$ | Noise power |  $0$ (interference-limited regime)
$P_{\mathrm{busy,m}},P_{\mathrm{busy,f}}$ | Probabilities that a given subchannel is used by an MBS and an FAP | Not fixed
$\Phi_{m}^{\prime},\Phi_{f}^{\prime}$ | Point processes defining MBSs and FAPs that interfere a given subchannel | N/A
$\lambda_{m}^{\prime},\lambda_{f}^{\prime}$ | Densities of MBSs and FAPs that interfere a given subchannel | Not fixed
$U_{\mathrm{out}},U_{\mathrm{in}}$ | Numbers of nonsubscribers that access a given MBS and a given FAP | Not fixed
$U_{s}$ | Numbers of subscribers that access a given FAP | Not fixed
$\tau_{f},\tau_{m}$ | Mean achievable rates of femtocell UEs and macrocell UEs | Not fixed
$\tau_{s},\tau_{n}$ | Mean achievable rates of subscribers and nonsubscribers | Not fixed

Figure 4 displays the SINR distributions of macrocell UEs and femtocell UEs,
when the number of shared subchannels is set as $M_{s}=10$. We plot in dotted
curves the analytical results, and we also plot the empirical cdfs obtained
from Monte Carlo simulation. The curves reveal that the simulation results
match the analytical results well, thus corroborating the accuracy of our
theoretical analysis. From the SINR distributions, we observe that while the
macrocell UEs experience a fair amount of interference, due to the shrinking
cell size, the interference for femtocell UEs is substantially alleviated. We
also observe that the performance of femtocell UEs is worse in the clustered
case than in the Poisson case when the intensity of the FAPs is set as the
same value; however, the performance of the macrocell UEs is just the reverse.
This result reveals that the gathering of FAPs leads to more interference to
femtocell UEs and at the same time reduces the chance that a macrocell UE
being interfered by the nearby FAPs.

Figure 4: Cdfs of SINRs for macrocell UEs and femtocell UEs, when $M_{s}=10$.

Figure 5 displays the mean achievable rates of macrocell UEs and femtocell UEs
as the outside nonsubscribers intensity $\lambda_{\mathrm{out}}$ increases. We
observe that the mean achievable rates of macrocell and femtocell UEs drop
initially and then tend to be stable. To interpret this behavior, we note that
as the intensity of outside nonsubscribers begins to increase, more
subchannels become occupied by MBSs, incurring more macrocell tier
interference; however, when the intensity of outside nonsubscribers is
sufficiently large, almost all the subchannels are persistently occupied by
MBSs with the UEs served by time-sharing, and then the interference saturates
thus leading to stable performance for UEs. In the case when the FAPs are
clustered, we observe that the mean achievable rate of femtocell UEs only
mildly decreases with the increasing of $\lambda_{\mathrm{out}}$, suggesting
that the performance of femtocell UEs is mainly limited by the interference
from the nearby FAPs and has little correlation with the intensity of
interfering MBSs.

Figure 5: Mean achievable rates of macrocell UEs and femtocell UEs as
functions of the intensity of outside nonsubscribers $\lambda_{\mathrm{out}}$.

Figure 6 displays the mean achievable rates of nonsubscribers and subscribers
as functions of the number of shared subchannels, in which the rate of
nonsubscribers is averaged over both outside and inside nonsubscribers. When
few subchannels are to be shared by each FAP (i.e., small $M_{s}$), the mean
achievable rate of nonsubscribers is small while subscribers enjoy a good
spectral efficiency. On the contrary, when most of the subchannels are shared
to nonsubscribers, the mean achievable rate of subscribers deteriorates
seriously. Nevertheless, we observe from the figure that there exists a stable
compromise at which the rates of both subscribers and nonsubscribers do not
drop much from their maxima. For the default configuration in our numerical
study, the stable compromise in the PPP case as well as in the clustered case
occurs when the value of $M_{s}$ lies in the range of $[7,12]$, corresponding
to a reasonably wide tuning range for system designer when provisioning the
resource.

Figure 6: Performance of nonsubscribers and subscribers as a function of the
number of shared subchannals $M_{s}$.

Figure 7 displays the mean achievable rates of nonsubscribers and subscribers
with different proportions of inside nonsubscribers and subscribers in
femtocells. For a fair comparison, we fix the sum intensity of the two types
of femtocell UEs, as $\lambda_{s}+\lambda_{\mathrm{in}}=0.03$. Figure 7(a)
gives the performance in the case when most of the femtocell UEs are
nonsubscribers, while Figure 7(b) corresponds to the opponent case. From the
curves, we observe that in order to achieve a good performance for both types
of UEs, the number of shared subchannels $M_{s}$ should be adjusted based on
the intensity of inside nonsubscribers. Moreover, we also find that the tuning
range of $M_{s}$ in the PPP case is almost the same as that in the clustered
case; thus illustrating that it is the load of the network rather than the
spatial distribution of FAPs that contributes the major impact on the choice
of $M_{s}$.

(a) $\lambda_{s}=0.003$ and $\lambda_{\mathrm{in}}=0.027$.

(b) $\lambda_{s}=0.027$ and $\lambda_{\mathrm{in}}=0.003$.

Figure 7: Performance of nonsubscribers and subscribers with changing
proportions of inside nonsubscribers and subscribers.

##  VIII Conclusions

In this paper, we explored the application of stochastic geometry in the
analysis of hybrid access mechanisms for multi-channel two-tier networks,
focusing on the evaluation of the tradeoff between nonsubscribers and
subscribers. We characterized several key statistics of UEs and subchannels,
and established SINR distribution and mean achievable rate for each type of
UEs.

Our analysis revealed the interaction among the various parameters in the
network model, and thus shed useful insights into the choice of network
parameters and the provisioning of resource in system design. From our
numerical study, we observe that although there is an apparent conflict
between the interests of nonsubscribers and subscribers, there usually exists
a reasonably wide tuning range over which nonsubscribers and subscribers
attain a stable compromise at which the rates of both subscribers and
nonsubscribers do not drop much from their maxima. We also found that although
the spatial distributions of FAPs are different, the tuning range is almost
the same when the intensities of different types of UEs are fixed.

## Appendix A

Assume that the typical marcocell UE is located at the origin $o$, and let $r$
be the distance between it and its serving MBS. Since an outside nonsubscriber
always chooses its nearest MBS to access, the cdf of $r$ is obtained by

$\displaystyle\mathbb{P}\\{r\leq R\\}$ $\displaystyle=$ $\displaystyle
1-\mathbb{P}\\{\mathrm{no\ MBS\ closer\ than\ }R\\}$ (33) $\displaystyle=$
$\displaystyle 1-e^{-\lambda_{m}\pi R^{2}}.$

Then, the pdf of $r$ is $f(r)=e^{-\lambda_{m}\pi r^{2}}2\pi\lambda_{m}r$.
Assuming that the considered UE is at distance $r$ from its serving MBS,
$g_{\cdot}$ is the fading of an interference link, and $R_{\cdot}$ is the
distance between the UE and an interfering access point, the SINR experienced
by the UE is $\mathrm{SINR}=\frac{P_{m}hr^{-\alpha}}{I_{m}+I_{f}+\sigma^{2}}$,
where
$I_{m}=\sum_{i\in\Phi_{m}^{\prime}\setminus\\{b_{0}\\}}P_{m}g_{i}R_{i}^{-\alpha}$
is the interference from the macrocell tier (excluding the serving MBS itself
which is denoted by $b_{0}$) and
$I_{f}=\sum_{j\in\Phi_{f}^{\prime}}WP_{f}g_{j}R_{j}^{-\alpha}$ is the
interference from the femtocell tier with the wall penetration loss taken into
account.

Thus, the cdf of the SINR is given by

$\displaystyle Z_{m}(T)=1-\mathbb{P}\\{\mathrm{SINR}>T\\}$ (34)
$\displaystyle\\!\\!\\!\\!=\\!\\!\\!\\!$ $\displaystyle
1-\\!\int_{0}^{\infty}\\!\\!\\!\\!2\pi\lambda_{m}re^{-\pi\lambda_{m}r^{2}}\mathbb{P}\Big{\\{}\frac{P_{m}hr^{-\alpha}}{I_{m}+I_{f}+\sigma^{2}}>T\Big{\\}}\mathrm{d}r$
$\displaystyle\\!\\!\\!\\!=\\!\\!\\!\\!$ $\displaystyle
1-\\!\int_{0}^{\infty}\\!\\!\\!\\!2\pi\lambda_{m}re^{-\pi\lambda_{m}r^{2}}\mathbb{P}\Big{\\{}h>\frac{Tr^{\alpha}}{P_{m}}(I_{m}+I_{f}+\sigma^{2})\Big{\\}}\mathrm{d}r$
$\displaystyle\\!\\!\\!\\!=\\!\\!\\!\\!$ $\displaystyle
1-\\!\int_{0}^{\infty}\\!\\!\\!\\!2\pi\lambda_{m}re^{-\pi\lambda_{m}r^{2}}\mathbb{E}\Big{\\{}e^{-\frac{\mu
Tr^{\alpha}}{P_{m}}(I_{m}+I_{f}+\sigma^{2})}\Big{\\}}\mathrm{d}r$
$\displaystyle\\!\\!\\!\\!=\\!\\!\\!\\!$ $\displaystyle
1-\\!\\!\int_{0}^{\infty}\\!\\!\\!\\!\\!2\pi\lambda_{m}re^{-\pi\lambda_{m}r^{2}-\frac{\mu
Tr^{\alpha}\sigma^{2}}{P_{m}}}\mathcal{L}_{I_{m}+I_{f}}\Big{(}\frac{\mu
Tr^{\alpha}}{P_{m}}\Big{)}\mathrm{d}r.$

Now, we evaluate the Laplace transform for the interference conditioning on
the fact that the typical UE is served by the nearest MBS $b_{0}$ which is at
the distance $r$. First, we derive the Laplace transform for the interference
of MBSs, denoted as $I_{m}$.

$\displaystyle\mathcal{L}_{I_{m}}(s)$ $\displaystyle=$ $\displaystyle
E_{\Phi_{m}^{\prime}}\Big{\\{}\exp(-s\\!\\!\\!\\!\sum_{i\in\Phi_{m}^{\prime}\setminus\\{b_{0}\\}}\\!\\!\\!\\!P_{m}g_{i}R_{i}^{-\alpha})\Big{\\}}$
(35) $\displaystyle=$ $\displaystyle
E_{\Phi_{m}^{\prime}}\Big{(}\\!\\!\\!\\!\prod_{i\in\Phi_{m}^{\prime}\setminus\\{b_{0}\\}}\\!\\!\\!\\!E_{g_{i}}\\{\exp(-sg_{i}R_{i}^{-\alpha}P_{m})\\}\Big{)}$
$\displaystyle=$ $\displaystyle
E_{\Phi_{m}^{\prime}}\Big{(}\\!\\!\\!\\!\prod_{i\in\Phi_{m}^{\prime}\setminus\\{b_{0}\\}}\\!\\!\\!\\!\mathcal{L}_{g}(sR_{i}^{-\alpha}P_{m})\Big{)},$

The probability generating functional of PPP $\Phi$ in the region $D$ with
intensity $\lambda$, denoted by
$G_{p}(v)=\mathbb{E}\\{\prod_{x\in\Phi}v(x)\\}$, is given by [19] as follows

$G_{p}(v)=\exp\Big{(}-\lambda\int_{D}(1-v(x))\mathrm{d}x\Big{)}.$ (36)

Let $C(o,r)$ be the circle centered at the origin $o$ with radius $r$. As
there is no MBS in the circle $C(o,r)$, we have
$\Phi_{m}^{\prime}(C(o,r))=\emptyset$. This implies that the interfering MBSs
are distributed as PPP on the space $R^{2}$ exclusive of the region $C(o,r)$.
Let $D=R^{2}\setminus C(o,r)$ and $v(x)=\mathcal{L}_{g}(s|x|^{-\alpha}P_{m})$,
by applying the probability generating functional of PPP we get

$\displaystyle\mathcal{L}_{I_{m}}(s)=\exp\Big{(}-2\pi\lambda_{m}^{\prime}\int_{r}^{\infty}(1-\mathcal{L}_{g}(sx^{-\alpha}P_{m}))x\mathrm{d}x\Big{)}$
$\displaystyle=\exp\Big{(}-2\pi\lambda_{m}^{\prime}\underbrace{\int_{0}^{\infty}\\!\\!\int_{r}^{\infty}\\!\\!(1-e^{-sx^{-\alpha}P_{m}g})f(g)x\mathrm{d}x\mathrm{d}g}_{(A)}\Big{)}.$
(37)

where the last equation follows from the exchange of the integral order.

Let $y=sP_{m}gx^{-\alpha}$ and integrate by parts, from the properties of
Gamma function we obtain

$\displaystyle(A)=-\frac{r^{2}}{2}+\frac{1}{\alpha}(sP_{m})^{\frac{2}{\alpha}}\mathbb{E}_{g}\Big{\\{}$
$\displaystyle
g^{\frac{2}{\alpha}}\Big{(}\Gamma(-\frac{2}{\alpha},sP_{m}gr^{-\alpha})-\Gamma(-\frac{2}{\alpha})\Big{)}\Big{\\}}.$
(38)

The evaluation of the Laplace transform for $I_{f}$ is almost the same except
that the interfering FAPs are distributed on the whole space $R^{2}$. Similar
to (37), we get

$\displaystyle\mathcal{L}_{I_{f}}(s)$ $\displaystyle\\!\\!\\!\\!=\\!\\!\\!\\!$
$\displaystyle\exp\Big{(}-2\pi\lambda_{f}^{\prime}\int_{0}^{\infty}\\!\\!(1-\mathcal{L}_{g}(sx^{-\alpha}WP_{f}))x\mathrm{d}x\Big{)}$
(39) $\displaystyle\\!\\!\\!\\!=\\!\\!\\!\\!$
$\displaystyle\frac{1}{2}(sWP_{f})^{\frac{2}{\alpha}}\Gamma\Big{(}1-\frac{2}{\alpha}\Big{)}\mathbb{E}_{g}\Big{\\{}g^{\frac{2}{\alpha}}\Big{\\}}.$

Substituting $\mathcal{L}_{I_{m}}(s)$ and $\mathcal{L}_{I_{f}}(s)$ into
$Z_{m}(T)$ hence leads to (10).

Now we evaluate the mean achievable rate. Since for a positive random variable
$X$, $\mathbb{E}\\{X\\}=\int_{t>0}\mathbb{P}\\{X>t\\}\mathrm{d}t$, we have

$\displaystyle\tau_{m}$ $\displaystyle=$
$\displaystyle\mathbb{E}\\{\ln(1+\mathrm{SINR})\\}$ (40) $\displaystyle=$
$\displaystyle\int_{0}^{\infty}\mathbb{P}\Big{\\{}\ln(1+\mathrm{SINR})>t\Big{\\}}\mathrm{d}t$
$\displaystyle=$
$\displaystyle\int_{0}^{\infty}\mathbb{P}\Big{\\{}\mathrm{SINR}>e^{t}-1\Big{\\}}\mathrm{d}t$
$\displaystyle=$
$\displaystyle\int_{0}^{\infty}(1-Z_{m}(e^{t}-1))\mathrm{d}t.$

Plugging $Z_{m}(T)$ into (40), we arrive at (11) and thus establish Theorem 1.

## Appendix B

Assume that the typical UE is located at the origin $o$. Let $r$ be the
distance between a femtocell UE and its serving FAP. Because the femtocell UEs
are uniformly distributed in the circular coverage area of radius $R_{f}$ of
each FAP, the pdf of $r$ is given by $f(r)={2r}/{R_{f}^{2}}$.

Denote by $I_{m}=\sum_{i\in\Phi_{m}^{\prime}}WP_{m}g_{i}R_{i}^{-\alpha}$ and
$I_{f}=\sum_{j\in\Phi_{f}^{\prime}\setminus\\{b_{0}\\}}W^{2}P_{f}g_{j}R_{j}^{-\alpha}$
the interference strengths from MBSs and FAPs respectively. Similar to the
derivation of (34), we have

$Z_{f}(T)=1-\int_{0}^{R_{f}}\frac{2r}{R_{f}^{2}}e^{-\frac{\mu
Tr^{\alpha}\sigma^{2}}{P_{f}}}\mathcal{L}_{I_{m}+I_{f}}\Big{(}\frac{\mu
Tr^{\alpha}}{P_{f}}\Big{)}\mathrm{d}r.$ (41)

Since $\Phi_{m}^{\prime}$ is a homogeneous PPP with intensity
$\lambda_{m}^{\prime}$, we obtain the Laplace transform for $I_{m}$ similar as
the derivation of (39)

$\\!\\!\\!\\!\mathbb{E}\big{\\{}e^{-sI_{m}}\big{\\}}=\exp\Big{(}-\pi\lambda_{m}^{\prime}(sWP_{m})^{\frac{2}{\alpha}}\Gamma\Big{(}1-\frac{2}{\alpha}\Big{)}\mathbb{E}_{g}(g^{\frac{2}{\alpha}})\Big{)}.$
(42)

The FAPs are distributed as a homogeneous PPP; however, the serving FAP is not
included when calculating the interference. By the Slivnyak-Mecke Theorem, the
reduced Palm distribution of the Poisson p.p. is equal to its original
distribution. Thus, the Laplace transform for $I_{f}$ can still be obtained
similar as the derivation of (39)

$\\!\\!\\!\\!\mathbb{E}\big{\\{}e^{-sI_{f}}\big{\\}}=\exp\Big{(}-\pi\lambda_{f}^{\prime}(sW^{2}P_{f})^{\frac{2}{\alpha}}\Gamma\Big{(}1-\frac{2}{\alpha}\Big{)}\mathbb{E}_{g}(g^{\frac{2}{\alpha}})\Big{)}.$
(43)

In the above, it is noteworthy that the interference from an FAP penetrates
two walls thus the loss becoming $W^{2}$ instead of $W$. Substituting the
Laplace transform for $I_{m}$ and $I_{f}$ into (41) with $v=r^{2}$, we get the
SINR distribution, and similar to (40), we get the mean achievable rate.

## Appendix C

The derivation is exactly the same as Appendix A till the equation (34). The
essential distinction of the derivation lies in that the Laplace transform for
the interference is different from that in the PPP case. Let
$I_{m}=\sum_{i\in\Phi_{m}^{\prime}\setminus\\{b_{0}\\}}P_{m}g_{i}R_{i}^{-\alpha}$
and $I_{f}=\sum_{j\in\Phi_{f}^{\prime}}WP_{f}g_{j}R_{j}^{-\alpha}$. Since
there is no MBS in the disk $C(o,r)$, we have
$\Phi_{m}^{\prime}(C(o,r))=\emptyset$. Referring to the previous derivation in
the PPP case, we obtain the Laplace transform for $I_{m}$ as follows

$\mathbb{E}\big{\\{}e^{-sI_{m}}\big{\\}}=\exp\Big{(}-\pi\lambda_{m}^{\prime}r^{2}\varphi\Big{(}\frac{sP_{m}}{\mu
r^{\alpha}},\alpha\Big{)}\Big{)}$ (44)

The FAPs are distributed as a Neyman-Scott cluster process and the generating
functional $G(v)=\mathbb{E}(\prod_{x\in\Phi}v(x))$ is given by [19, Page 157]

$\displaystyle G(v)=\exp\Big{(}-\lambda_{p}\int_{R^{2}}\Big{(}1-$
$\displaystyle\exp\Big{(}-\lambda_{c}\int_{C(o,R_{c})}(1-v(x+y))\mathrm{d}y\Big{)}\Big{)}\mathrm{d}x\Big{)}$
(45)

Similar to the derivation of (35), we get the Laplace transform for $I_{f}$

$\displaystyle\mathcal{L}_{I_{f}}(s)$ $\displaystyle\\!\\!\\!\\!=\\!\\!\\!\\!$
$\displaystyle\mathbb{E}\Big{\\{}\prod_{j\in\Phi_{f}^{\prime}}\mathcal{L}_{g}(sR_{j}^{-\alpha}WP_{f})\Big{\\}}$
(46) $\displaystyle\\!\\!\\!\\!=\\!\\!\\!\\!$
$\displaystyle\mathbb{E}\Big{\\{}\prod_{j\in\Phi_{f}^{\prime}}\frac{\mu}{\mu+sR_{j}^{-\alpha}WP_{f}}\Big{\\}}$

Let $v(x)=\frac{\mu}{\mu+sP_{f}W|x|^{-\alpha}}$ and plugging into the
generating functional of Neyman-Scott cluster process (45), we get the Laplace
transform for $I_{f}$. Having derived the Laplace transform for $I_{m}$ and
$I_{f}$, similar to the derivations in Appendix A, we obtain the results in
Theorem 3.

## Appendix D

Different from the above proofs, we assume that the serving FAP rather than
the typical UE is located at the origin. The typical UE is distributed in the
circle centered at the origin with radius $R_{f}$. Let
$I_{m}=\sum_{i\in\Phi_{m}^{\prime}}WP_{m}g_{i}R_{i}^{-\alpha}$ and
$I_{f}=\sum_{j\in\Phi_{f}^{\prime}\setminus\\{b_{0}\\}}W^{2}P_{f}g_{j}R_{j}^{-\alpha}$.

First, we evaluate the Laplace transform for the interference conditioned on
the fact that the typical UE is located at distance $r$ from the serving FAP
located at the origin. Without loss of generality, we assume that the typical
UE is located at $z=(r,0)$. Since $\Phi_{m}^{\prime}$ is a homogeneous PPP
with intensity $\lambda_{m}^{\prime}$, we obtain the Laplace transform for
$I_{m}$ similar as the derivation of (39)

$\\!\\!\\!\\!\\!\\!\mathbb{E}\big{\\{}e^{-sI_{m}}\big{\\}}=\exp\Big{(}-\pi\lambda_{m}^{\prime}\Big{(}\frac{sWP_{m}}{\mu}\Big{)}^{\frac{2}{\alpha}}\Gamma(1+\frac{2}{\alpha})\Gamma(1-\frac{2}{\alpha})\Big{)}$
(47)

The FAPs are distributed as a Neyman-Scott cluster process. Since the serving
FAP, located at the origin, is not included when calculating the interference,
we should consider the reduced Palm distribution of the cluster process when
evaluating the Laplace transform for $I_{f}$. Let
$G_{o}^{!}(v)=\mathbb{E}_{o}^{!}(\prod_{x\in\Phi}v(x))$ denotes the generating
functional of the reduced Palm distribution of the cluster process. The
notation $\mathbb{E}_{o}^{!}(\cdot)$ denotes the conditional expectation for
the point process given that there is a point of the process at the origin but
without including the point. The conditional generating functional
$G_{o}^{!}(v)$ is given by the Lemma 1 in [17] as follows

$\displaystyle G_{o}^{!}(v)=\frac{1}{\pi R_{c}^{2}}G(v)\int_{C(o,R_{c})}$
$\displaystyle\exp\Big{(}-\lambda_{c}\int_{C(o,R_{c})}(1-v(x-y))\mathrm{d}x\Big{)}\mathrm{d}y$
(48)

where $G(v)$ is the generating functional of Neyman-Scott cluster process and
is given by (45).

Referring to (46), let $v(x)=\frac{\mu}{\mu+sP_{f}W|x-z|^{-\alpha}}$. Plugging
$v(x)$ into the conditional generating functional of Neyman-Scott cluster
process (48), we get the Laplace transform for $I_{f}$. Having derived the
Laplace transform for $I_{m}$ and $I_{f}$, similar to the derivation in
Appendix B, we obtain the results in Theorem 4.

## Acknowledgement

The authors wish to thank the anonymous reviewers for their valuable comments
on this work.

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# Some results on space-like self-shrinkers

Huaqiao Liu and Y. L. Xin  School of Mathematics and Information Sciences,
Henan University, Kaifeng 475004, China<EMAIL_ADDRESS>Institute of
Mathematics, Fudan University, Shanghai 200433, China<EMAIL_ADDRESS>
###### Abstract.

We study space-like self-shrinkers of dimension $n$ in pseudo-Euclidean space
$\mathbb{R}^{m+n}_{m}$with index $m$. We derive drift Laplacian of the basic
geometric quantities and obtain their volume estimates in pseudo-distance
function. Finally, we prove rigidity results under minor growth conditions in
terms of the mean curvature or the image of Gauss maps .

###### 1991 Mathematics Subject Classification:

58E20,53A10.

The authors are supported partially by NSFC

##  1\. Introduction

Let $\mathbb{R}^{m+n}_{m}$ be an $(m+n)$-dimensional pseudo-Euclidean space
with the index $m$. The indefinite flat metric on $\mathbb{R}^{m+n}_{m}$ is
defined by
$ds^{2}=\sum_{i=1}^{n}(dx^{i})^{2}-\sum_{\alpha=n+1}^{m+n}(dx^{\alpha})^{2}.$
In what follows we agree with the following range of indices

$A,\,B,\,C,\cdots=1,\cdots,m+n;\,i,\,j,\,k\cdots=1,\cdots,n;$
$s,t=1,\,\cdots,m;\,\alpha,\,\beta,\cdots=n+1,\cdots,m+n.$

Let $F:M\to\mathbb{R}^{m+n}_{m}$ be a space-like $n$-dimensional submanifold
in $\mathbb{R}^{m+n}_{m}$ with the second fundamental form $B$ defined by
$B_{XY}\mathop{=}\limits^{def.}\left(\bar{\nabla}_{X}Y\right)^{N}$ for
$X,Y\in\Gamma(TM)$. We denote $(\cdots)^{T}$ and $(\cdots)^{N}$ for the
orthogonal projections into the tangent bundle $TM$ and the normal bundle
$NM$, respectively. For $\nu\in\Gamma(NM)$ we define the shape operator
$A^{\nu}:TM\to TM$ by $A^{\nu}(V)=-(\bar{\nabla}_{V}\nu)^{T}.$ Taking the
trace of $B$ gives the mean curvature vector $H$ of $M$ in
$\mathbb{R}^{m+n}_{m}$ and
$H\mathop{=}\limits^{def.}\text{trace}(B)=B_{e_{i}e_{i}},$ where $\\{e_{i}\\}$
is a local orthonormal frame field of $M.$ Here and in the sequel we use the
summation convention. The mean curvature vector is time-like, and a cross-
section of the normal bundle.

We now consider a one-parameter family $F_{t}=F(\cdot,t)$ of immersions
$F_{t}:M\to\mathbb{R}^{m+n}_{m}$ with the corresponding images
$M_{t}=F_{t}(M)$ such that

(1.1) $\begin{split}\frac{d\,}{d\,t}F(x,t)&=H(x,t),\qquad x\in M\\\
F(x,0)&=F(x)\end{split}$

are satisfied, where $H(x,t)$ is the mean curvature vector of $M_{t}$ at
$F(x,t).$ There are many interesting results on mean curvature flow on space-
like hypersurfaces in certain Lorentzian manifolds [10, 11, 12, 13]. For
higher codimension we refer to the previous work of the second author [19].

A special but important class of solutions to (1.1) are self-similar shrinking
solutions, whose profiles, space-like self-shrinkers, satisfy a system of
quasi-linear elliptic PDE of the second order

(1.2) $H=-\frac{X^{N}}{2}.$

Besides the Lagrangian space-like self-shrinkers [2, 14, 9], there is an
interesting paper on curves in the Minkowski plane [15]. The present paper is
devoted to general situation on space-like self-shrinker.

For a space-like $n-$submanifold $M$ in $\mathbb{R}^{m+n}_{m}$ we have the
Gauss map $\gamma:M\to\mathbb{G}_{n,m}^{m}$. The target manifold is a pseudo-
Grassmann manifold, dual space of the Grassmann manifold $\mathbb{G}_{n,m}$.
In the next section we will describe its geometric properties, which will be
used in the paper.

Choose a Lorentzian frame field $\\{e_{i},e_{\alpha}\\}$ in
$\mathbb{R}^{m+n}_{m}$ with space-like $\\{e_{i}\\}\in TM$ and time-like
$\\{e_{\alpha}\\}\in NM$ along the space-like submanifold
$F:M\to\mathbb{R}^{m+n}_{m}$. Define coordinate functions

$x^{i}=\left<F,e_{i}\right>,\;y^{\alpha}=-\left<F,e_{\alpha}\right>.$

We then have

$|F|^{2}=X^{2}-Y^{2},$

where $X=\sqrt{\sum_{i=1}^{n}(x^{i})^{2}},\quad
Y=\sqrt{\sum_{\alpha=n+1}^{m+n}(y^{\alpha})^{2}}.$ We call $|F|^{2}$ the
pseudo-distance function from the origin $0\in M$.

We always put the origin on $M$ in the paper. We see that $|F|^{2}$ is
invariant under the Lorentzian action up to the choice of the origin in
$\mathbb{R}^{m+n}_{m}$. Set $z=|F|^{2}$. It has been proved that $z$ is proper
provided $M$ is closed with the Euclidean topology (see [4] for $m=1$ and [16]
for any codimension $m$).

Following Colding and Minicozzi [6] we can also introduce the drift Laplacian,

(1.3) $\displaystyle\mathcal{L}=\Delta-\frac{1}{2}\langle
F,\nabla(\cdot)\rangle=e^{\frac{z}{4}}div(e^{-\frac{z}{4}}\nabla(\cdot)).$

It can be showed that $\mathcal{L}$ is self-adjoint with respect to the
weighted volume element $e^{-\frac{z}{4}}d\mu,$ where $d\mu$ is the volume
element of $M$ with respect to the induced metric from the ambient space
$\mathbb{R}^{m+n}_{m}$. In the present paper we carry out integrations with
respect to this measure. We denote $\rho=e^{-\frac{z}{4}}$ and the volume form
$d\mu$ might be omitted in the integrations for notational simplicity.

For a space-like submanifold in $\mathbb{R}^{m+n}_{m}$ there are several
geometric quantities. The squared norm of the second fundamental form
$|B|^{2}$, the squared norm of the mean curvature $|H|^{2}$ and the
$w-$function, which is related to the image of the Gauss map. In §3 we will
calculate drift Laplacian $\mathcal{L}$ of those quantities, see Proposition
3.1.

Corresponding to the weighted measure and drift Laplacian there is so-called
the Baker-Emery Ricci tensor. It is noted that in [3]
$\text{Ric}_{f}\geq\frac{z}{4}$ with $f=\frac{z}{4}$. Using the comparison
technique the weighted volume of the geodesic ball can be estimated from above
in terms of the distance function [18].

For a space-like $n-$submanifold $M$ in $\mathbb{R}^{m+n}_{m}$ there are 3
kind global conditions: Closed one with Euclidean topology; entire graph;
complete with induced Riemannian metric. A complete space-like one has to be
entire graph, but the converse claim is not always the case. Closed one with
Euclidean topologg is complete under the parallel mean curvature assumption
(see [4] for codimension one and [16] for higher codimension).

In our case of closed one with Euclidean topology, the pseudo-distance
function $z$ is always proper. It is natural to consider the volume growth in
$z$. For the proper self-shrinkers in Euclidean space Ding-Xin [8] gave the
volume estimates. It has been generalized in [5] for more general situation.
But, the present case does not satisfy the conditions in Theorem 1.1 in [5].
However, the idea in [8] is still applicable for space-like self-shrinkers. In
§4 we will give volume estimates for space-like self-shrinkers, in a similar
manner as in [8], see Theorem 4.1.

Finally, using integral method we can obtain rigidity results as follows.

######  Theorem 1.1.

Let $M$ be a space-like self-shrinker of dimension $n$ in $R^{n+m}_{m},$ which
is closed with respect to the Euclidean topology. If there is a constant
$\alpha<\frac{1}{8}$, such that $|H|^{2}\leq e^{\alpha z}$, then $M$ is an
affine $n-$plane.

######  Theorem 1.2.

Let $M$ be a complete space-like self-shrinker of dimension $n$ in
$R^{n+m}_{m}$. If there is a constant $\alpha<\frac{1}{2}$, such that $\ln
w\leq e^{\alpha d^{2}(p,x)}$ for certain $p\in M$, where $d(p,\cdot)$ is the
distance function from $p$, then $M$ is affine $n-$plane.

######  Remark 1.1.

In the special situation, for the Lagrangian space-like self-shrinkers, the
rigidity results hold without the growth condition (see [9]). Let ${\tenmsb
R}^{2n}_{n}$ be Euclidean space with null coordinates
$(x,y)=(x_{1},\cdots,x_{n};\,y_{1},\cdots,y_{n})$, which means that the
indefinite metric is defined by $ds^{2}=\sum_{i}dx_{i}dy_{i}.$ If
$M=\\{(x,Du(x))\big{|}\ x\in{\tenmsb R}^{n}\\}$ is a space-like submanifold in
${\tenmsb R}^{2n}_{n}$, then $u$ is convex and the induced metric on $M$ is
given by $ds^{2}=\sum_{i,j}u_{ij}dx_{i}dx_{j}$. $M$ is a space-like Lagrangian
submanifold in $\mathbb{R}^{2n}_{n}$. It is worthy to point out that if $M$ is
entire gradient graph the potential function $u$ is proper, as the following
consideration. On $\mathbb{R}^{n}$ set $\rho=|x|=\sqrt{\sum x_{i}^{2}}$. At
any direction $\theta\in S^{n-1}$

$u_{i}=u_{\rho}\frac{\partial\rho}{\partial x_{i}}=\frac{x_{i}}{\rho}u_{\rho}$

and the pseudo-distance

$z=x_{i}u_{i}=\rho u_{\rho},$

which is positive when the origin is on $M$, since it is space-like. It
implies that $u$ is increasing in $\rho$. Moreover,

$z_{\rho}=u_{\rho}+\rho u_{\rho\rho}>0,$

which means that $z$ is also increasing in $\rho$. Hence,

$u(\rho)-u(\epsilon)=\int_{\epsilon}^{\rho}u_{\rho}d\rho=\int_{\epsilon}^{\rho}\frac{z}{\rho}d\rho\geq
z(\epsilon)\int_{\epsilon}^{\rho}\frac{1}{\rho}d\rho\geq
z(\epsilon)\int_{\epsilon}^{\rho}\frac{1}{\rho}d\rho\to\infty$

as $\rho\to\infty$.

######  Remark 1.2.

Rigidity problem for space-like extremal submanifolds was raised by E. Calabi
[1], and solved by Cheng-Yau [4] for codimension $1$. Later, Jost-Xin
generalized the results to higher codimension [16]. The rigidity problem for
space-like submanifolds with parallel mean curvature was studied in [20][22]
and [16] (see also in Chap. 8 of [21]).

##  2\. Geometry of $G_{n,m}^{m}$

In $\mathbb{R}^{n+m}_{m}$ all space-like $n-$subspaces form the pseudo-
Grassmannian $G^{m}_{n,m}.$ It is a specific Cartan-Hadamard manifold which is
the noncompact dual space of the Grassmann manifold $G_{n,m}.$

Let $P$ and $A\in G_{n,m}^{m}$ be two space-like $n-$plane in $R_{m}^{m+n}.$
The angles between $P$ and $A$ are defined by the critical values of angel
$\theta$ between a nonzero vector $x$ in P and its orthogonal projection
$x^{*}$ in $A$ as $x$ runs through $P$.

Assume that $e_{1},\cdots,e_{n}$ are orthonormal vectors which span the space-
like $P$ and $a_{1},\cdots,a_{n}$ for space-like $A.$ For a nonzero vector in
$P$

$x=\sum_{i}x_{i}e_{i},$

its orthonormal projections in $A$ is

$x^{*}=\sum_{i}x_{i}^{*}a_{i}.$

Thus, for any $y\in A,$ we have

$\langle x-x^{*},y\rangle=0.$

Set

$W_{i\,j}=\langle e_{i},a_{j}\rangle,$

We then have

$x_{j}^{*}=\sum_{i}W_{i\,j}x_{i}.$

Since $x$ is a vector in a space-like $n-$plane and its projection $x^{*}$ in
$A$ is also a space-like vector. We then have a Minkowski plane $R_{1}^{2}$
spanned by $x$ and $x^{*}.$ Then angle $\theta$ between $x$ and $x^{*}$ is
defined by

$\cosh\theta=\frac{\langle x,x^{*}\rangle}{|x||x^{*}|}.$

Let

$W=(W_{i\,j})=\left(\begin{array}[]{lll}\langle
e_{1},a_{1}\rangle&\cdots&\langle e_{n},a_{1}\rangle\\\ \quad\vdots&\
\vdots&\quad\vdots\\\ \langle e_{n},a_{1}\rangle&\cdots&\langle
e_{n},a_{n}\rangle\end{array}\right)$

Now define the $w-$function as

$w=\langle e_{1}\wedge\cdots\wedge e_{n},a_{1}\wedge\cdots\wedge
a_{n}\rangle=\det W.$

$W^{T}W$ is symmetric, its eigenvalues are $\mu_{1}^{2},\cdots,\mu_{n}^{2},$
then there exist $e_{1},\cdots,e_{n}$ in $P$, such that

$W^{T}W=\left(\begin{array}[]{lll}\mu_{1}^{2}&&0\\\ &\ddots&\\\
0&&\mu_{n}^{2}\end{array}\right),$

in which $\mu_{i}\geq 1$ and $\mu_{i}=\cosh\theta_{i}.$ Then

(2.1)
$w=\prod_{i}\cosh\theta_{i}=\prod_{i}\frac{1}{\sqrt{1-\lambda_{i}^{2}}},\;\lambda_{i}=\tanh\theta_{i}.$

The distance between $P$ and $A$ in the canonical Riemannian metric on
$\mathbb{G}_{n,m}^{m}$ is (see [17] for example)

$d(P,A)=\sqrt{\sum_{i}\theta_{i}^{2}}.$

For the fixed $A\in G_{n,m}^{m},$ which is spanned by $\\{a_{i}\\}$, choose
time-like $\\{a_{n+s}\\}$ such that $\\{a_{i},a_{n+s}\\}$ form an orthonormal
Lorentzian bases of $R^{n+m}_{m}.$

Set

$\displaystyle e_{i}$
$\displaystyle=\cosh\theta_{i}a_{i}+\sinh\theta_{i}a_{n+i}$ $\displaystyle
e_{n+i}$
$\displaystyle=\sinh\theta_{i}a_{i}+\cosh\theta_{i}a_{n+i}\,(\;\text{and}\;e_{n+\alpha}=a_{n+\alpha}\;\text{if}\;m>n).$

Then $e_{i}\in T_{p}M,e_{n+i}\in N_{p}M.$ In this case

(2.2) $\displaystyle w_{i\,\alpha}$ $\displaystyle=$ $\displaystyle\langle
e_{1}\wedge\cdots\wedge e_{i-1}\wedge e_{\alpha}\wedge e_{i+1}\cdots\wedge
e_{n},a_{1}\wedge\cdots\wedge a_{n}\rangle$ (2.3) $\displaystyle=$
$\displaystyle\cosh\theta_{1}\cosh\theta_{i-1}\sinh\theta_{i}\cosh\theta_{i+1}\cosh\theta_{n}=\lambda_{i}w\delta_{n+i\
\alpha},$

which is obtained by replacing $e_{i}$ by $e_{\alpha}$ in $w$. We also have
$w_{i\alpha j\beta}$ by replacing $e_{j}$ by $e_{\beta}$ in $w_{i\,\alpha}.$
We obtain

(2.7) $\displaystyle w_{i\alpha
j\beta}=\left\\{\begin{array}[]{lll}\lambda_{i}\lambda_{j}w&&\alpha=n+i,\beta=n+j\\\
-\lambda_{i}\lambda_{j}w&&\alpha=n+j,\beta=n+i\\\
0&&otherwise.\end{array}\right.$

##  3\. Drift Laplacian of some geometric quantities

The second fundamental form $B$ can be viewed as a cross-section of the vector
bundle Hom($\odot^{2}TM,NM$) over $M.$ A connection on Hom($\odot^{2}TM,NM$)
can be induced from those of $TM$ and $NM$ naturally. There is a natural fiber
metric on Hom($\odot^{2}TM,NM$) induced from the ambient space and it becomes
a Riemannian vector bundle. There is the trace-Laplace operator $\nabla^{2}$
acting on any Riemannian vector bundle.

In [19] we already calculate $\nabla^{2}B$ for general space-like
$n-$submanifolds in $\mathbb{R}^{m+n}_{m}$.

Set

$B_{i\,j}=B_{e_{i}\,e_{j}}=h_{i\,j}^{\alpha}e_{\alpha},\;S_{\alpha\,\beta}=h_{i\,j}^{\alpha}h_{i\,j}^{\beta}.$

From Proposition 2.1 in [19] we have

(3.1)
$\displaystyle\langle\nabla^{2}B,B\rangle=\langle\nabla_{i}\nabla_{j}H,B_{i\,j}\rangle+\langle
B_{i\,k},H\rangle\langle
B_{i\,l},B_{k\,l}\rangle-|R^{\perp}|^{2}-\sum_{\alpha,\beta}S_{\alpha\,\beta}^{2},$

where $R^{\perp}$ denotes the curvature of the normal bundle and

$|R^{\perp}|^{2}=-\langle
R_{e_{i}\,e_{j}}\nu_{\alpha},R_{e_{i}\,e_{j}}\nu_{\alpha}\rangle.$

Then from the self-shrinker equation (1.2) we obtain

$\displaystyle\nabla_{i}F^{N}$ $\displaystyle=$
$\displaystyle[\bar{\nabla}_{i}(F-\langle F,e_{j}\rangle e_{j})]^{N}$
$\displaystyle=$ $\displaystyle[e_{i}-\bar{\nabla}_{i}\langle F,e_{j}\rangle
e_{j}-\langle F,e_{j}\rangle\bar{\nabla}_{e_{i}}e_{j}]^{N}$ $\displaystyle=$
$\displaystyle-\langle F,e_{j}\rangle B_{i\,j},$

and

$\displaystyle\nabla_{i}\nabla_{j}F^{N}$ $\displaystyle=$
$\displaystyle-\nabla_{i}[\langle F,e_{k}\rangle B_{k\,j}]$ $\displaystyle=$
$\displaystyle-\delta_{i}^{k}B_{k\,j}-\langle F^{N},B_{k\,i}\rangle
B_{k\,j}-\langle F,e_{k}\rangle\nabla_{i}B_{k\,j}$ $\displaystyle=$
$\displaystyle-B_{i\,j}-\langle F^{N},B_{k\,i}\rangle B_{k\,j}-\langle
F,e_{k}\rangle\nabla_{k}B_{i\,j}$ $\displaystyle=$ $\displaystyle-
B_{i\,j}+\langle 2H,B_{k\,i}\rangle B_{k\,j}-\langle
F,e_{k}\rangle\nabla_{k}B_{i\,j}.$

Set $P_{i\,j}=\langle B_{i\,j},H\rangle,$ then

(3.2)
$\displaystyle\nabla_{i}\nabla_{j}H=\frac{1}{2}B_{i\,j}-P_{k\,i}B_{k\,j}+\frac{1}{2}\langle
F,e_{k}\rangle\nabla_{k}B_{i\,j}.$

Substituting (3.2) into (3.1)we obtain

$\displaystyle\langle\nabla^{2}B,B\rangle$ $\displaystyle=$
$\displaystyle\langle\frac{1}{2}B_{i\,j},B_{i\,j}\rangle-\langle
H,B_{k\,i}\rangle\langle B_{k\,j},B_{i\,j}\rangle+\frac{1}{2}\langle
F,e_{k}\rangle\langle\nabla_{k}B_{i\,j},B_{i\,j}\rangle$
$\displaystyle+\langle B_{i\,k},H\rangle\langle
B_{i\,l},B_{k\,l}\rangle-|R^{\perp}|^{2}-\sum_{\alpha,\beta}S^{2}_{\alpha\,\beta}.$

This also means that

(3.3) $\displaystyle\langle\nabla^{2}B,B\rangle=\frac{1}{2}\langle
B,B\rangle+\frac{1}{4}\langle F^{T},\nabla\langle
B,B\rangle\rangle-|R^{\perp}|^{2}-\sum_{\alpha,\beta}S^{2}_{\alpha\,\beta}.$

Note that $\Delta\langle
B,B\rangle=2\langle\nabla^{2}B,B\rangle+2\langle\nabla B,\nabla B\rangle,$ so

(3.4) $\displaystyle\Delta\langle B,B\rangle$ $\displaystyle=\langle
B,B\rangle+\frac{1}{2}\langle F^{T},\nabla\langle
B,B\rangle\rangle-2|R^{\perp}|^{2}-2\sum_{\alpha,\beta}S^{2}_{\alpha\,\beta}$
$\displaystyle+2\langle\nabla B,\nabla B\rangle.$

We denote

$|B|^{2}=-\langle B,B\rangle=\sum_{i,j,\alpha}h_{\alpha ij}^{2},\;|\nabla
B|^{2}=-\langle\nabla B,\nabla B\rangle.$ $|H|^{2}=-\langle
H,H\rangle,\;|\nabla H|^{2}=-\langle\nabla H,\nabla H\rangle$

then

(3.5) $\displaystyle\Delta|B|^{2}=|B|^{2}+\frac{1}{2}\langle
F^{T},\nabla|B|^{2}\rangle+2|R^{\perp}|^{2}+2\sum_{\alpha,\beta}S^{2}_{\alpha\,\beta}+2|\nabla
B|^{2}$

From (3.2) we also obtain

$\displaystyle\nabla^{2}H=\frac{1}{2}H-P_{k\,i}B_{k\,j}+\frac{1}{2}\langle
F,e_{k}\rangle\nabla_{k}H.$

Since

$\displaystyle\Delta|H|^{2}=-\Delta\langle
H,H\rangle=-2\langle\nabla^{2}H,H\rangle-2\langle\nabla H,\nabla H\rangle,$

we obtain

(3.6) $\displaystyle\Delta|H|^{2}$
$\displaystyle=-2\langle\frac{1}{2}H-P_{k\,i}B_{k\,i}+\frac{1}{2}\langle
F,e_{k}\rangle\nabla_{k}H,H\rangle-2\langle\nabla H,\nabla H\rangle$
$\displaystyle=|H|^{2}+2|P|^{2}+\frac{1}{2}\langle
F^{T},\nabla|H|^{2}\rangle+2|\nabla H|^{2},$

where $|P|^{2}=\sum_{i,j}P_{ij}^{2}$.

In the pseudo-Grassmann manifold $\mathbb{G}_{n,m}^{m}$ there are
$w-$functions with respect to a fixed point $A\in\mathbb{G}_{n,m}^{m}$, as
shown in §2. For the space-like $n-$submanifold $M$ in $\mathbb{R}^{m+n}_{m}$
we define the Gauss map $\gamma:M\to\mathbb{G}_{n,m}^{m}$, which is obtained
by parallel translation of $T_{p}M$ for any $p\in M$ to the origin in
$\mathbb{R}^{m+n}_{m}$. Then, we have functions $w\circ\gamma$ on $M$, which
is still denoted by $w$ for notational simplicity.

For any point $p\in M$ around $p$ there is a local tangent frame field
$\\{e_{i}\\}$, and which is normal at $p$. We also have a local orthonormal
normal frame field $\\{e_{\alpha}\\}$, and which is normal at $p$. Define a
$w-$function by

$w=\left<e_{1}\wedge\cdots\wedge e_{n},a_{1}\wedge\cdots\wedge a_{n}\right>,$

where $\\{a_{i}\\}$ is a fixed orthonormal vectors which span a fixed space-
like $n-$plane $A$. Denote

$e_{i\,\alpha}=e_{1}\wedge\cdots\wedge e_{\alpha}\wedge\cdots\wedge e_{n},$

which is got by substituting $e_{\alpha}$ for $e_{i}$ in
$e_{1}\wedge\cdots\wedge e_{n}$ and $e_{i\alpha j\beta}$ is obtained by
substituting $e_{\beta}$ for $e_{j}$ in $e_{i\,\alpha}.$ Then

(3.7) $\displaystyle\nabla_{e_{j}}w$ $\displaystyle=\sum_{i=1}^{n}\langle
e_{1}\wedge\cdots\bar{\nabla}_{e_{j}}e_{i}\wedge\cdots\wedge
e_{n},a_{1}\wedge\cdots\wedge a_{n}\rangle$
$\displaystyle=\sum_{i=1}^{n}\langle e_{1}\wedge\cdots\wedge B_{i\,j}\cdots
e_{n},a_{1}\wedge\cdots\wedge a_{n}\rangle$
$\displaystyle=\sum_{i=1}^{n}h_{i\,j}^{\alpha}\langle e_{1}\cdots\wedge
e_{\alpha}\wedge\cdots\wedge e_{n},a_{1}\wedge\cdots\wedge a_{n}\rangle$
$\displaystyle=\sum_{i=1}^{n}h_{i\,j}^{\alpha}\langle
e_{i\,\alpha},a_{1}\wedge\cdots\wedge a_{n}\rangle.$

Furthermore,

(3.10) $\displaystyle\nabla_{e_{i}}\nabla_{e_{j}}w$ $\displaystyle=$
$\displaystyle\langle\bar{\nabla}_{e_{i}}\bar{\nabla}_{e_{j}}(e_{1}\wedge\cdots\wedge
e_{n}),a_{1}\wedge\cdots\wedge a_{n}\rangle$ $\displaystyle=$
$\displaystyle\sum_{k\neq l}\langle
e_{1}\wedge\cdots\wedge\bar{\nabla}_{e_{j}}e_{k}\wedge\cdots\wedge\bar{\nabla}_{e_{i}}e_{l}\wedge\cdots\wedge
e_{n},a_{1}\wedge\cdots\wedge a_{n}\rangle$ $\displaystyle+\sum_{k}\langle
e_{1}\wedge\cdots\bar{\nabla}_{e_{i}}\bar{\nabla}_{e_{j}}e_{k}\wedge\cdots\wedge
e_{n},a_{1}\wedge\cdots\wedge a_{n}\rangle$ $\displaystyle=$
$\displaystyle\sum_{k\neq l}\langle e_{1}\wedge\cdots\wedge
B_{j\,k}\wedge\cdots\wedge B_{il}\wedge\cdots e_{n},a_{1}\wedge\cdots\wedge
a_{n}\rangle$ $\displaystyle+\sum_{k}\langle
e_{1}\wedge\cdots\wedge(\bar{\nabla}_{i}\bar{\nabla}_{j}e_{k})^{T}\wedge\cdots\wedge
e_{n},a_{1}\wedge\cdots\wedge a_{n}\rangle$ $\displaystyle+\sum_{k}\langle
e_{1}\wedge\cdots\wedge(\bar{\nabla}_{i}\bar{\nabla}_{j}e_{k})^{N}\wedge\cdots\wedge
e_{n},a_{1}\wedge\cdots\wedge a_{n}\rangle$

Note that

$\displaystyle(\ref{3.13})$ $\displaystyle=$ $\displaystyle\sum_{k\neq
l}h_{j\,k}^{\alpha}h_{i\,l}^{\beta}\langle e_{\alpha k\beta
l},a_{1}\wedge\cdots\wedge a_{n}\rangle$ $\displaystyle(\ref{3.14})$
$\displaystyle=$
$\displaystyle\langle\bar{\nabla}_{i}\bar{\nabla}_{j}e_{k},e_{k}\rangle
w=-\langle\bar{\nabla}_{j}e_{k},\bar{\nabla}_{i}e_{k}\rangle w=-\langle
B_{j\,k},B_{i\,k}\rangle w=h_{j\,k}^{\alpha}h_{i\,k}^{\alpha}w$
$\displaystyle(\ref{3.15})$ $\displaystyle=$
$\displaystyle-\langle(\bar{\nabla}_{i}\bar{\nabla}_{j}e_{k})^{N},e_{\alpha}\rangle\langle
e_{\alpha\,k},a_{1}\wedge\cdots\wedge a_{n}\rangle$ $\displaystyle=$
$\displaystyle-\langle(\bar{\nabla}_{i}(B_{j\,k}+\nabla_{e_{j}}e_{k}))^{N},e_{\alpha}\rangle\langle
e_{\alpha\,k},a_{1}\wedge\cdots\wedge a_{n}\rangle$ $\displaystyle=$
$\displaystyle-\langle\nabla_{i}B_{j\,k},e_{\alpha}\rangle\langle
e_{\alpha\,k},a_{1}\wedge\cdots\wedge
a_{n}\rangle=-\langle\nabla_{k}B_{i\,j},e_{\alpha}\rangle\langle
e_{\alpha\,k},a_{1}\wedge\cdots\wedge a_{n}\rangle,$

where we use the Codazzi equation in the last step. Thus, we obtain

$\Delta w=\sum_{i,k\neq l}h_{i\,k}^{\alpha}h_{i\,l}^{\beta}\langle e_{k\beta
l}^{\alpha},a_{1}\wedge\cdots\wedge
a_{n}\rangle+|B|^{2}w-\langle\nabla_{k}H,e_{\alpha}\rangle\langle e_{\alpha
k},a_{1}\wedge\cdots\wedge a_{n}\rangle,$

Since

$\nabla_{i}F^{N}=-\langle F,e_{j}\rangle B_{i\,j},$

from (1.2), we obtain

(3.11) $\displaystyle\nabla_{i}H$ $\displaystyle=\frac{1}{2}\langle
F,e_{j}\rangle B_{i\,j}$ $\displaystyle\langle\nabla_{i}H,e_{\alpha}\rangle$
$\displaystyle=-\frac{1}{2}\langle F,e_{j}\rangle h_{i\,j}^{\alpha},$

so,

(3.12) $\displaystyle\Delta w$ $\displaystyle=|B|^{2}w+\sum_{i,k\neq
l}h_{i\,k}^{\alpha}h_{i\,l}^{\beta}\langle e_{\alpha k\beta
l},a_{1}\wedge\cdots\wedge a_{n}\rangle+\frac{1}{2}\langle F,e_{i}\rangle
h_{k\,i}^{\alpha}\langle e_{\alpha k},a_{1}\wedge\cdots\wedge a_{n}\rangle$
$\displaystyle=|B|^{2}w+\sum_{i,k\neq
l}h_{i\,k}^{\alpha}h_{i\,l}^{\beta}\langle e_{\alpha k\beta
l},a_{1}\wedge\cdots\wedge a_{n}\rangle+\frac{1}{2}\langle F,\nabla w\rangle,$

where (3.7) has been used in the last equality.

######  Proposition 3.1.

For a space-like self-shrinker $M$ of dimension $n$ in $\mathbb{R}^{m+n}_{m}$
we have

(3.13)
$\mathcal{L}|B|^{2}=|B|^{2}+2|R^{\perp}|^{2}+2\sum_{\alpha,\beta}S^{2}_{\alpha\,\beta}+2|\nabla
B|^{2},$ (3.14) $\mathcal{L}|H|^{2}=|H|^{2}+2|P|^{2}+2|\nabla H|^{2},$ (3.15)
$\mathcal{L}(\ln w)\geq\frac{|B|^{2}}{w^{2}}.$

###### Proof.

From (1.3,3.5,3.6), we can obtain (3.13) and (3.14) easily.

From (1.3,3.12) we have

(3.16) $\displaystyle\mathcal{L}w$ $\displaystyle=|B|^{2}w+\sum_{i,k\neq
l}h_{i\,k}^{\alpha}h_{i\,l}^{\beta}\langle e_{\alpha k\beta
l},a_{1}\wedge\cdots\wedge a_{n}\rangle=|B|^{2}w+\sum_{i,k\neq
l}h_{i\,k}^{\alpha}h_{i\,l}^{\beta}w_{\alpha k\beta l}$
$\displaystyle=|B|^{2}w+\sum_{i,k\neq
l}\lambda_{k}\lambda_{l}(h_{i\,k}^{n+k}h_{i\,l}^{n+l}-h_{i\,k}^{n+l}h_{i\,l}^{n+k})w,$

Furthermore, since

$\mathcal{L}(\ln w)=\frac{1}{w}\mathcal{L}w-\frac{|\nabla w|^{2}}{w^{2}},$

we obtain

$\mathcal{L}(\ln w)=|B|^{2}+\sum_{i,k\neq
l}\lambda_{k}\lambda_{l}(h_{i\,k}^{n+k}h_{i\,l}^{n+l}-h_{i\,k}^{n+l}h_{i\,l}^{n+k})-\frac{|\nabla
w|^{2}}{w^{2}}.$

From (3.7),we obtain

$\displaystyle|\nabla w|^{2}$
$\displaystyle=\sum_{j=1}^{n}|\nabla_{e_{j}}w|^{2}=\sum_{j=1}^{n}(\sum_{i=1}^{n}\sum_{\alpha}h_{ij}^{\alpha}w_{i\,\alpha})^{2}$
$\displaystyle=\sum_{j=1}^{n}(\sum_{i=1}^{n}h_{ij}^{n+i}\lambda_{i}w)^{2}=\sum_{i,j,k=1}^{n}\lambda_{i}\lambda_{k}w^{2}h_{i\,j}^{n+i}h_{k\,j}^{n+k}.$

in the case of $m\geq n$ we rewrite (otherwise, we treat the situation
similarly)

$|B|^{2}=\sum_{j,k,\alpha>n}(h^{n+\alpha}_{jk})^{2}+\sum_{i,j}(h_{ij}^{n+i})^{2}+\sum_{j}\sum_{k<i}(h_{ij}^{n+k})^{2}+\sum_{j}\sum_{i<k}(h_{ij}^{n+k})^{2}.$

So, we obtain

(3.17) $\displaystyle\mathcal{L}(\ln w)$
$\displaystyle=|B|^{2}+\sum_{i,j,k\neq
i}\lambda_{i}\lambda_{k}(h_{i\,j}^{n+i}h_{j\,k}^{n+k}-h_{i\,j}^{n+k}h_{j\,k}^{n+i})-\sum_{i,j,k=1}^{n}\lambda_{i}\lambda_{k}h_{ij}^{n+i}h_{j\,k}^{n+k}$
$\displaystyle=|B|^{2}+\sum_{i,j,k\neq
i}\lambda_{i}\lambda_{k}h_{i\,j}^{n+i}h_{j\,k}^{n+k}-\sum_{i,j,k\neq
i}\lambda_{i}\lambda_{k}h_{i\,j}^{n+k}h_{j\,k}^{n+i}-\sum_{i,j,k=1}^{n}\lambda_{i}\lambda_{k}h_{ij}^{n+i}h_{j\,k}^{n+k}$
$\displaystyle=\sum_{j,k,\alpha>n}(h^{n+\alpha}_{jk})^{2}+\sum_{i,j}(h_{i\,j}^{n+i})^{2}+\sum_{j}\sum_{k<i}(h_{ij}^{n+k})^{2}+\sum_{j}\sum_{i<k}(h_{ij}^{n+k})^{2}$
$\displaystyle\hskip
144.54pt-\sum_{i,j}\lambda_{i}^{2}(h_{ij}^{n+i})^{2}-\sum_{ij,k\neq
i}\lambda_{i}\lambda_{k}h_{ij}^{n+k}h_{jk}^{n+i}$
$\displaystyle=\sum_{j,k,\alpha>n}(h^{n+\alpha}_{jk})^{2}+\sum_{i,j}(1-\lambda_{i}^{2})(h_{ij}^{n+i})^{2}+\sum_{j}\sum_{k<i}(h_{ij}^{n+k})^{2}+\sum_{j}\sum_{i<k}(h_{ij}^{n+k})^{2}$
$\displaystyle\hskip
180.67499pt-2\sum_{j}\sum_{k<i}\lambda_{k}\lambda_{i}h_{jk}^{n+i}h_{ij}^{n+k}$
$\displaystyle\geq\sum_{j,k,\alpha>n}(h^{n+\alpha}_{jk})^{2}+\sum_{i,j}(1-\lambda_{i}^{2})(h_{ij}^{n+i})^{2}+\sum_{j}\sum_{k<i}(1-\lambda_{i}^{2})(h_{ij}^{n+k})^{2}$
$\displaystyle\hskip
231.26378pt+\sum_{j}\sum_{i<k}(1-\lambda_{i}^{2})(h_{ij}^{n+k})^{2}$
$\displaystyle=\sum_{j,k,\alpha>n}(h^{n+\alpha}_{jk})^{2}+\sum_{i,j,k}(1-\lambda_{i}^{2})(h_{ij}^{n+k})^{2}$
$\displaystyle\geq\sum_{j,k,\alpha>n}(h^{n+\alpha}_{jk})^{2}+\prod_{i}(1-\lambda_{i}^{2})\sum_{i,j,k}(h_{ij}^{n+k})^{2}.$

Noting (2.1) the inequality (3.15) has been proved.

######  Remark 3.1.

For a space-like graph $M=(x,f(x))$ with $f:\mathbb{R}^{n}\to\mathbb{R}^{m}$
its induced metric is
$ds^{2}=(\delta_{ij}-f^{\alpha}_{i}f^{\alpha}_{j})dx^{i}dx^{j}.$ Set
$g=\det(\delta_{ij}-f^{\alpha}_{i}f^{\alpha}_{j})$ then
$w=\frac{1}{\sqrt{g}}$.

######  Remark 3.2.

(3.15) is a generalization of a formula (5.8) for space-like graphical self-
shrinkers in [7] to more general situation.

∎

##  4\. Volume growth

To draw our results we intend to integrate those differential inequalities
obtained in the last section. We need to know the volume growth in the pseudo-
distance function $z$ on the space-like submanifolds. In [16] the following
property has been proved.

######  Proposition 4.1.

(Proposition 3.1 in [16]) Let $M$ be a space-like $n-$submanifold in
$\mathbb{R}^{m+n}_{m}$. If $M$ is closed with respect to the Euclidean
topology, then when $0\in M$, $z=\left<F,F\right>$ is a proper function on
$M$.

we also need a lemma from [8]:

######  Lemma 4.1.

If $f(r)$ is a monotonic increasing nonnegative function on $[0,+\infty)$
satisfying $f(r)\leq C_{1}r^{n}f(\frac{r}{2})$ on $[C_{2},+\infty)$ for some
positive constant $n,C_{1},C_{2}$, here $C_{2}>1$, then $f(r)\leq
C_{3}e^{2n(\log r)^{2}}$ on $[C_{2},+\infty)$ for some positive constant
$C_{3}$ depending only on $n,C_{1},C_{2},f(C_{2})$.

Using the similar method as in [8] we obtain the following volume growth
estimates.

######  Theorem 4.1.

Let $z=\langle F,F\rangle$ be the pseudo-distance of $R_{m}^{n+m},$ where
$F\in R_{m}^{n+m}$ is the position vector with respect to the origin $0\in M$.
Let $M$ be an $n-$dimensional space-like self-shrinker of $R^{n+m}_{m}$.
Assume that $M$ is closed with respect to the Euclidean topology, then for any
$\alpha>0$, $\int_{M}e^{-\alpha z}d\mu$ is finite, in particular $M$ has
finite weighted volume.

###### Proof.

We have

$z_{i}\mathop{=}\limits^{def.}e_{i}(z)=2\left<F,e_{i}\right>,$
$z_{ij}\mathop{=}\limits^{def.}Hess(z)(e_{i},e_{j})=2\left(\delta_{ij}-y^{\alpha}h_{ij}^{\alpha}\right),$
(4.1) $\Delta z=2n-2y^{\alpha}H^{\alpha}=2n+Y^{2},$

where the self-shrinker equation (1.2) has been used in third equality. For
our self-shrinker $M^{n}$ in ${\tenmsb R}^{n+m}_{m}$, we define a functional
$F_{t}$ on any set $\Omega\subset M$ by

$F_{t}(\Omega)=\frac{1}{(4\pi
t)^{n/2}}\int_{\Omega}e^{-\frac{z}{4t}}d\mu,\quad\mathrm{for}\quad t>0.$

Set $B_{r}=\\{p\in\mathbb{R}^{m+n}_{m},z(p)<r^{2}\\}$ and $D_{r}=B_{r}\bigcap
M$. We differential $F_{t}(D_{r})$ with respect to $t$,

$F_{t}^{\prime}(D_{r})=(4\pi)^{-\frac{n}{2}}t^{-(\frac{n}{2}+1)}\int_{D_{r}}(-\frac{n}{2}+\frac{z}{4t})e^{-\frac{z}{4t}}d\mu.$

Noting (4.1)

(4.2) $\displaystyle-e^{\frac{z}{4t}}\mathrm{div}(e^{-\frac{z}{4t}}\nabla z)$
$\displaystyle=-\Delta z+\frac{1}{4t}\nabla z\cdot\nabla z$
$\displaystyle=-2n-Y^{2}+\frac{X^{2}}{t}$
$\displaystyle\geq\frac{z}{t}-2n\quad(\ when\ 0<t\leq 1\ ).$

Since

$\nabla z=2F^{T}$

and the unit normal vector to $\partial D_{r}$ is $\frac{F^{T}}{X}$, then

(4.3) $\displaystyle F_{t}^{\prime}(D_{r})\leq$
$\displaystyle\pi^{-\frac{n}{2}}(4t)^{-(\frac{n}{2}+1)}\int_{D_{r}}-\mathrm{div}(e^{-\frac{z}{4t}}\nabla
z)d\mu$ $\displaystyle=$
$\displaystyle\pi^{-\frac{n}{2}}(4t)^{-(\frac{n}{2}+1)}\int_{\partial
D_{r}}-2Xe^{-\frac{z}{4t}}\leq 0.$

We integrate $F_{t}^{\prime}(D_{r})$ over $t$ from $\frac{1}{r}$ to $1,\,r\geq
1$, and get

$\int_{D_{r}}e^{-\frac{z}{4}}d\mu\leq
r^{\frac{n}{2}}\int_{D_{r}}e^{-\frac{zr}{4}}d\mu,$

Since

$\int_{D_{r}}e^{-\frac{z}{4}}d\mu\geq e^{-\frac{r^{2}}{4}}\int_{D_{r}}1d\mu$

and

$\displaystyle\int_{D_{r}}e^{-\frac{zr}{4}}d\mu$ $\displaystyle=$
$\displaystyle\int_{D_{r}\backslash
D_{\frac{r}{2}}}e^{-\frac{zr}{4}}d\mu+\int_{D_{\frac{r}{2}}}e^{-\frac{zr}{4}}d\mu$
$\displaystyle\leq$ $\displaystyle
e^{-\frac{r^{3}}{16}}\int_{D_{r}}1d\mu+\int_{D_{\frac{r}{2}}}1d\mu.$

Set $V(r)=\int_{D_{r}}1d\mu$. Then,

$(e^{-\frac{r^{2}}{4}}-e^{-\frac{r^{3}}{16}}r^{\frac{n}{2}})V(r)\leq
r^{\frac{n}{2}}V(\frac{r}{2}).$

Let $g(r)=e^{-\frac{r^{2}}{4}}-e^{-\frac{r^{3}}{16}}r^{\frac{n}{2}}$. $g(r)>0$
when $r$ sufficiently large (say $r\geq 8n$) Since

$\displaystyle g^{\prime}(r)$
$\displaystyle=-\frac{r}{2}e^{-\frac{r^{2}}{4}}-\frac{n}{2}r^{\frac{n}{2}-1}e^{-\frac{r^{3}}{16}}+\frac{3r^{2}}{16}r^{\frac{n}{2}}e^{-\frac{r^{3}}{16}}$
$\displaystyle>(-\frac{r}{2}-\frac{n}{2}r^{\frac{n}{2}-1}+\frac{3}{16}r^{\frac{n}{2}+2})e^{-\frac{r^{3}}{16}}>0,$

$g(r)$ is increasing in $r$ and $g^{-1}(r)$ is decreasing in $r$. Therefore,

$g^{-1}(r)\leq\frac{1}{e^{-16n^{2}}-e^{-32n^{3}}(8n)^{\frac{n}{2}}}=C_{1}$

We then have

$V(r)\leq C_{1}r^{n}V(\frac{r}{2})\quad\quad\text{for $r$ sufficiently large
(say, $r\geq 8n$)}.$

By Lemma 4.1, we have

$V(r)\leq C_{4}e^{2n(\log r)^{2}}\quad for\quad r\geq 8n,$

here $C_{4}$ is a constant depending only on $n,V(8n)$. Hence, for any
$\alpha>0$

$\displaystyle\int_{M}e^{-\alpha z}d\mu=$
$\displaystyle\sum_{j=0}^{\infty}\int_{D_{8n(j+1)}\setminus D_{8nj}}e^{-\alpha
z}d\mu\leq\sum_{j=0}^{\infty}e^{-\alpha(8nj)^{2}}V(8n(j+1))$
$\displaystyle\leq$ $\displaystyle
C_{4}\sum_{j=0}^{\infty}e^{-\alpha(8nj)^{2}}e^{2n(\log(8n)+\log(j+1))^{2}}\leq
C_{5},$

where $C_{5}$ is a constant depending only on $n,V(8n)$. So we obtain our
estimates. Certainly, $M$ has weighted finite volume. ∎

######  Corollary 4.1.

Any space-like self-shrinker $M$ of dimension $n$ in $\mathbb{R}^{m+n}_{m}$
with closed Euclidean topology has finite fundamental group.

From the Gauss equation we have

$\text{Ric}(e_{i},e_{i})=\left<H,B_{ii}\right>-\sum_{j}\left<B_{ij},B_{ij}\right>,$

and

$\text{Hess}(f)(e_{i},e_{i})=\frac{1}{4}\text{Hess}(z)(e_{i},e_{i})=\frac{1}{2}\delta_{ij}+\frac{1}{2}\left<F,B_{ij}\right>=\frac{1}{2}\delta_{ij}-\left<H,B_{ij}\right>,$

It follows that

$\text{Ric}_{f}(e_{i},e_{i})=\text{Ric}(e_{i},e_{i})+\text{Hess}(f)(e_{i},e_{i})\geq\frac{1}{2}.$

Set $B_{R}(p)\subset M$, a geodesic ball of radius $R$ and centered at $p\in
M$. From Theorem 3.1 in [18] we know that for any $r$ there are constant
$A,\;B$ and $C$ such that

(4.4) $\int_{B_{R}(p)}\rho\leq A+B\int_{r}^{R}e^{-\frac{1}{2}t^{2}+Ct}dt$

##  5\. Rigidity results

Now, we are in a position to prove rigidity results mentioned in the
introduction.

######  Theorem 5.1.

Let $M$ be a space-like self-shrinker of dimension $n$ in $R^{n+m}_{m},$ which
is closed with respect to the Euclidean topology. If there is a constant
$\alpha<\frac{1}{8}$, such that $|H|^{2}\leq e^{\alpha z}$, then $M$ is an
affine $n-$plane.

###### Proof.

Let $\eta$ be a smooth function with compact support in $M,$ then by (3.14) we
obtain

(5.1) $\displaystyle\int_{M}(\frac{1}{2}|H|^{2}+|P|^{2}+|\nabla
H|^{2})\eta^{2}\rho$
$\displaystyle=\frac{1}{2}\int_{M}(\mathcal{L}|H|^{2})\eta^{2}\rho=\frac{1}{2}\int_{M}\text{div}(\rho\nabla|H|^{2})\eta^{2}$
$\displaystyle=-\int_{M}\eta\rho\nabla|H|^{2}\cdot\nabla\eta$
$\displaystyle=2\int_{M}\eta\rho\langle\nabla_{i}H,H\rangle\cdot\nabla_{i}\eta$
$\displaystyle\leq\int_{M}|\nabla
H|^{2}\eta^{2}\rho+\int_{M}|H|^{2}|\nabla\eta|^{2}\rho.$

We then have

(5.2)
$\int_{M}\left(\frac{1}{2}|H|^{2}+|P|^{2}\right)\eta^{2}\rho\leq\int_{M}|H|^{2}|\nabla\eta|^{2}\rho.$

Let $\eta=\phi(\frac{|F|}{r})$ for any $r>0,$ where $\phi$ is a nonnegative
function on $[0,+\infty)$ satisfying

$\displaystyle\phi(x)=\left\\{\begin{array}[]{lllll}1&&&if&x\in[0,1)\\\
0&&&if&x\in[2,+\infty),\end{array}\right.$

and $|\phi^{\prime}|\leq C$ for some absolute constant. Since $\nabla
z=2F^{T}$,

$\nabla\eta=\frac{1}{r}\phi^{\prime}\nabla\sqrt{z}=\frac{1}{r}\phi^{\prime}\frac{F^{T}}{\sqrt{z}}.$

By (1.2) we have

$|\nabla\eta|^{2}\leq\frac{C^{2}}{r^{2}}\frac{|F^{T}|^{2}}{z}=\frac{1}{r^{2}z}C^{2}(z+4|H|^{2}).$

It follows that (5.2) becomes

(5.3)
$\int_{D_{r}}\left(\frac{1}{2}|H|^{2}+|P|^{2}\right)\rho\leq\frac{C^{2}}{r^{2}}\int_{D_{2r}\setminus
D_{r}}|H|^{2}(1+\frac{4|H|^{2}}{z})\rho.$

By Theorem 4.1 then under the condition on $|H|$, we obtain that the right
hand side of (5.3) approaches to zero as $r\rightarrow+\infty.$ This implies
that $H\equiv 0.$

According to Theorem 3.3 in [16] we see that $M$ is complete with respect to
the induced metric from $\mathbb{R}^{m+n}_{m}$. In a geodesic ball $B_{a}(x)$
of radius $a$ and centered at $x\in M$ we can make gradient estimates of
$|B|^{2}$ in terms of the mean curvature. From (2.9) in [16] we have

$|B|^{2}\leq k\frac{2m(n-4)a^{2}}{(a^{2}-r^{2})^{2}}.$

Since $M$ is complete we can fix $x$ and let $a$ go to infinity. Hence,
$|B|^{2}=0$ at any $x\in M$ and $M$ is an $n-$plane. ∎

######  Theorem 5.2.

Let $M$ be a complete space-like self-shrinker of dimension $n$ in
$R^{n+m}_{m}$. If there is a constant $\alpha<\frac{1}{2}$, such that $\ln
w\leq e^{\alpha d^{2}(p,x)}$ for certain $p\in M$, where $d(p,\cdot)$ is the
distance function from $p$, then $M$ is affine $n-$plane.

###### Proof.

(3.15) tells us

$\mathcal{L}(\ln w)\geq\frac{|B|^{2}}{w^{2}}\geq 0.$

As an application to (4.4) (Theorem 3.1 in [18]) the Corollary 4.2 in [18]
tells us that $\ln w$ is constant. This forces $|B|^{2}\equiv 0$. ∎

## References

  * [1] E. Calabi: Examples of Bernstein problems for some nonlinear equations. Proc. Symp. Global Analysis U. C. Berkeley. (1968).
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