# The Diabetic Buddy: A Diet Regulator and Tracking System for Diabetics

Muhammad Usman  Division of Information and Computing Technology
College of Science and Engineering
Hamad Bin Khalifa University
Doha, Qatar
<EMAIL_ADDRESS>Kashif Ahmad  Division of Information and Computing
Technology
College of Science and Engineering
Hamad Bin Khalifa University
Doha, Qatar
<EMAIL_ADDRESS>Amir Sohail  Department of Information Engineering
and Computer Science
University of Trento
Trento, Italy
<EMAIL_ADDRESS>Marwa Qaraqe  Division of Information and
Computing Technology
College of Science and Engineering
Hamad Bin Khalifa University
Doha, Qatar
<EMAIL_ADDRESS>
###### Abstract

The prevalence of Diabetes mellitus (DM) in the Middle East is exceptionally
high as compared to the rest of the world. In fact, the prevalence of diabetes
in the Middle East is 17-20%, which is well above the global average of 8-9%.
Research has shown that food intake has strong connections with the blood
glucose levels of a patient. In this regard, there is a need to build
automatic tools to monitor the blood glucose levels of diabetics and their
daily food intake. This paper presents an automatic way of tracking continuous
glucose and food intake of diabetics using off-the-shelf sensors and machine
learning, respectively. Our system not only helps diabetics to track their
daily food intake but also assists doctors to analyze the impact of the food
in-take on blood glucose in real-time. For food recognition, we collected a
large-scale Middle-Eastern food dataset and proposed a fusion-based framework
incorporating several existing pre-trained deep models for Middle-Eastern food
recognition.

###### Index Terms:

Diabetes management, food recognition, continuous glucose monitoring, middle-
eastern food

##  I Introduction

Diabetes mellitus (DM) is a disorder that arises from a dysfunction of the
glucoregulatory system. Over 463 million people worldwide are diabetic and
this number is projected to jump to 700 million by 2040 [1]. With its high
prevalence worldwide, diabetes is becoming one of the fastest-growing health
challenges of the 21st century and is costing the healthcare system and wider
global economy an estimated 825 billion US dollars annually [2].

Diabetes requires constant management and, in addition to medication, self-
management of diabetes is vital in preventing acute complications and
minimizing the risk of long-term complications. The entire paradigm of
diabetes management has been transformed in the past decade due to the
development of new technologies such as continuous glucose monitoring (CGM)
devices. Diabetes management has become an ecosystem wherein food is an
important element that directly affects blood glucose levels in the human
body. A careful meal plan has the potential to not only regulate the blood
glucose level but also to keep it under the desired range [3].

In fact, what a person eats, when s/he eats, and the amount s/he eats affect
the blood glucose levels in the body. The normal range of blood glucose in our
body has a direct relation with the aforementioned questions. A blood glucose
level considered high at eight hours of fasting may be considered normal just
after eating or even two hours after eating. In this way, the blood glucose
level strongly depends on the type of food and time it was eaten. Table I
highlights the relation between food intake and blood glucose levels for non-
diabetics, pre-diabetics, and diabetics.

The current diabetes management system requires patients to track their meals
via a paper-based method. Although the process of CGM can be automated using
off-the-shelf sensors, such as FreeStyle Libre sensors [4], the process of
automating food tracking still lacks in the literature. Even though there have
been some applications aimed at food tracking, none of these applications can
precisely correlate meal time and type with CGM. CGM application, such as the
one from FreeStyle Libre, allows patients to manually write the meal-type and
time on the applications but this is not an efficient way as a patient may
forget to write in time. On the other hand, food tracking applications have no
option for CGM integration. Moreover, these applications are mainly designed
for Western Food, not local Middle-Eastern food. So this paper tries to
address two important gaps in the literature, (i) Mapping of food in-take to
CGM graph for better understanding of meal impact on blood glucose levels and
(ii) development of deep learning algorithms to correctly recognize the
Middle-Easters cuisines.

This work presents continuous glucose and food tracker (CGFT) for diabetics in
the Middle-East region having the ability to track real-time blood glucose
levels along with the food intake of the patient. The main highlights of the
proposed system are given below.

  * •

Being an integral part of the proposed application, the application is
equipped with a novel fusion-based middle eastern meal recognizer covering
most of the local Arab cuisines.

  * •

An interface to collect data from the FreeStyle Libre Sensor is developed,
which connects the mobile application with the sensor over Bluetooth
connection.

  * •

A web-platform is developed for the doctors and the patients, which stores the
data collected from the mobile application.

  * •

A mapping of CGM and the food-intake is provided to see the real-time impact
of the food on blood glucose.

For the demonstration purposes, we will give a live demo of the system where
it will be demonstrated how our system is able to help diabetics by tracking
continuous glucose and food intake using off-the-shelf sensors and ML.

TABLE I: Effect of food on blood glucose levels of non-diabetics, pre-
diabetics and diabetics [5].

|  Fasting (mg/dl) |  After Eating (mg/dl) |  2 Hours After Eating (mg/dl)
---|---|---|---
Non-Diabetic |  80-100 |  170-200 |  120-140
Pre-Diabetic |  101-125 |  190-230 |  140-160
Diabetic |  126+ |  220-300 |  200+

##  II Proposed System

The architecture of the proposed system is presented in Figure 1. There are
two main data sources in the application. one is for continuous glucose
readings from the FreeStyle Libre sensor worn by the patient. The data is
stored in mobile applications and the web-platform and plotted as a CGM graph.
The second source of data is the smartphone’s camera, which a patient uses to
take the picture of his meal. This data is stored in the mobile application
and the web-platform. Using data analytics, the food item is identified and
mapped in the CGM graph.

Figure 1: Systems architecture of the proposed system.

The details of the proposed system in terms of data collection, storage, and
analytics are given below.

###  II-A CGM Data

For continuous glucose monitoring, we rely on FreeStyle Libre Sensors. These
are invasive sensors that take a reading every 15 minutes. An illustration of
these sensors is given in Figure 2. The sensor can store eight hours of data.
Intrinsically, the data from these sensors can only be read by Near Field
Communication (NFC) of a smartphone or the reader shown in Figure 2. However,
this is not convenient for the patient to tap the reader/smartphone time and
again. In order to solve this problem, we use an NFC-to-Bluetooth converter
named NightRider BluCon 111https://www.ambrosiasys.com to make this data
transmission seamless to the patient. A reading is sent to the smartphone
every five minutes, which is plotted in terms of a time graph (CGM plot). A
snapshot of this graph on the mobile application is shown in Figure 3.

Figure 2: FreeStyle Libre Sensor with reader. Figure 3: CGM plot on the mobile
application.

###  II-B Food Data

Before every meal, the patient has to take the picture of his meal. A food
recognition framework identifies the food item in the picture. The identified
food item is then plotted on the CGM graph. For instance, chicken mandi is
identified in Figure 3 that the patient ate at around mid-night. A mobile
snapshot of the meal recognizer is presented in Figure 4. The details of the
food detection algorithm and the collected dataset are provided in subsequent
sub-sections.

Figure 4: Meal recognizer in the mobile application.

###  II-C Dataset

For the proposed application, we collected a large-scale dataset covering
images from 38 different categories of Middle-Eastern food. One of the biggest
challenges was the availability of traditional Middle-Eastern food images. In
order to collect a sufficient number of images, we crawled online sources,
such as Google and Flicker search engines to collect 7530 images from 38
different categories. However, some of the food categories resembled a lot in
terms of their appearance and ingredients. For instance, mandi and kabsa are
not only visually similar but their ingredients are also similar. Indeed, it
is very difficult for a human observer to differentiate among them. Based on
this observation, we merged some of the food categories and added a drop-down
menu for patients to select among merged categories.

Figure 5: Block diagram of the Food Recognizer. Figure 6: Block diagram of mobile application TABLE II: Experimental results of the middle eastern meal recognizer in terms of precision, recall, F-score and accuracy. Method | Avg. Pre. | Avg. Rec. | Avg. F-Score | Acc.
---|---|---|---|---
PSO based Fusion | 70.89 | 79.71 | 75.04 | 80.82
GA based Fusion | 69.83 | 77.30 | 73.37 | 79.61
Early Fusion | 66.08 | 73.10 | 69.41 | 76.53
Equal weights | 70.74 | 76.74 | 73.61 | 78.89

(a) Home page

(b) Dashboard

(c) User profile

(d) CGM and meal recognizer

Figure 7: Few snapshots of the web-platform

###  II-D Middle-Eastern Meal Recognizer

Figure 5 provides block diagram of the methodology adopted for the food/meal
recognizer module. As can be seen, the proposed methodology is composed of
three main components, namely (i) feature extraction, (ii) classification and
(iii) fusion. As a first stept, features are extracted via several existing
pre-trained deep models including AlexNet [6], VggNet [7], GoogleNet [8] and
ResNet [9]. It is important to mention that these models are used as features
descriptors only, where features are extracted from the last fully connected
layers without any fine-tuning and retraining. After feature extraction, SVMs
are trained on the features extracted through each individual model.

The final part of the methodology is based on the early (simple concatenation
of feature vectors) and three different late fusion methods, namely Particle
Swarm based Optimization (PSO), Genetic Algorithms (GA), and simple averaging
of the scores obtained with the individual models. The fusion of multiple deep
models is motivated by the fact that the deep models/architectures respond
differently to an image by extracting diverse but complementary features.
Combining such diverse but complementary features could result in an
improvement in the performance of our middle eastern meal recognizer.
Moreover, based on our previous experience, we believe merit-based weights
should be assigned to each model during the fusion. The selection of PSO and
GA based weight optimization of the weights to be used in the fusion is based
on their proven performance in other application domains [10, 11].

Our fitness function for PSO and GA based fusion methods is based on the
accumulative error computed on a validation set using Equ. 1 where $A_{acc}$
is computed using Equ. 2. Here $p_{n}$ represents the probabilities obtained
through $n^{th}$ model on the validation set while $x(n)$ represents the value
(weight) to be used and optimized for the $n^{th}$ model.

$error_{acc}=1-A_{acc}$ (1) $A_{acc}=x(1)*p_{1}+x(2)*p_{2}+...+x(n)*p_{n}$ (2)

Table II provides the experimental results of the proposed middle eastern meal
recognizer in terms of precision, recall, F1-score, and accuracy. As can be
seen, merit-based fusion (PSO and GA) have a clear advantage over the simple
averaging and early fusion method.

###  II-E Mobile Application

One of the most important components of our solution is the mobile application
that collects data from CGM sensors over Bluetooth and plots it on the time
graph. We integrate the meal recognizer discussed in subsection II-D into the
application. The application recognizes the meal and maps it over the CGM plot
as shown in 6. Glucose readings are stored in a database (DB) 1, while every
image taken by the patient is stored in DB2. DB3 contains the collected
dataset on which the system is trained. The application has the ability to
alert the patient in case the blood glucose level is high and recommend some
physical activity. An alert can be sent to the doctor and immediate family
members if the glucose level is very high. In addition, the patient can be
recommended not to eat high carb food.

###  II-F Web-Platform

The system also provides a web-platform for the doctors, patients, and
immediate family members to closely monitor the CGM and food-intake of the
patient. All the information stored in the mobile application is reflected in
the web-platform as well. Few snapshots of the developed application are
presented in Figure 7. It is important to note that CGM values plotted in
Figure 7(d) exactly duplicate the ones plotted in Figure 3.

##  III Conclusion

This paper addresses two important shortcomings of the literature. Firstly, it
maps the food intake of a patient on his/er CGM graph. Secondly, the food
recognition algorithm is customized for local Middle-Eastern Cuisines. The
end-system is a combination of a mobile application and a web-platform that
graphically represents the CGM and food-intake for the patients and the
doctors, respectively. In the future, we aim to estimate the number of carbs
and calories in the meal.

## Acknowledgment

This work was supported by internal fund of HBKU.

## References

  * [1] D. Atlas, “International diabetes federation,” _IDF Diabetes Atlas, 7th edn. Brussels, Belgium: International Diabetes Federation_ , 2015.
  * [2] G. Fagherazzi and P. Ravaud, “Digital diabetes: Perspectives for diabetes prevention, management and research,” _Diabetes & metabolism_, vol. 45, no. 4, pp. 322–329, 2019.
  * [3] H. K. Seligman, A. B. Bindman, E. Vittinghoff, A. M. Kanaya, and M. B. Kushel, “Food insecurity is associated with diabetes mellitus: results from the national health examination and nutrition examination survey (nhanes) 1999–2002,” _Journal of general internal medicine_ , vol. 22, no. 7, pp. 1018–1023, 2007.
  * [4] D. Rodbard, “Continuous glucose monitoring: a review of successes, challenges, and opportunities,” _Diabetes technology & therapeutics_, vol. 18, no. S2, pp. S2–3, 2016.
  * [5] M. Clinic, “Diabetes Diagnosis and Treatment,” https://www.mayoclinic.org/diseases-conditions/diabetes/diagnosis-treatment/drc-20371451, 2018, [Online; accessed 26-November-2018].
  * [6] A. Krizhevsky, I. Sutskever, and G. E. Hinton, “Imagenet classification with deep convolutional neural networks,” in _NIPS_ , 2012, pp. 1097–1105.
  * [7] K. Simonyan and A. Zisserman, “Very deep convolutional networks for large-scale image recognition,” _arXiv preprint arXiv:1409.1556_ , 2014.
  * [8] C. Szegedy, W. Liu, Y. Jia, P. Sermanet, S. Reed, D. Anguelov, D. Erhan, V. Vanhoucke, A. Rabinovich _et al._ , “Going deeper with convolutions,” in _CVPR_. CVPR, 2015\.
  * [9] K. He, X. Zhang, S. Ren, and J. Sun, “Deep residual learning for image recognition,” in _Proc. of the IEEE CVPR_ , 2016, pp. 770–778.
  * [10] K. Sheema, A. Kashif, A. Tahir, and A. Nasir, “Food items detection and recognition via multiple deep models,” _Journal of Electronic Imaging_ , vol. 28, pp. 28 – 28 – 10, 2019. [Online]. Available: https://doi.org/10.1117/1.JEI.28.1.013020
  * [11] K. Ahmad, M. L. Mekhalfi, N. Conci, F. Melgani, and F. D. Natale, “Ensemble of deep models for event recognition,” _ACM TOMM_ , vol. 14, no. 2, p. 51, 2018\.


# Sphere-foliated minimal and constant mean curvature hypersurfaces in product
spaces

Keomkyo Seo

###### Abstract.

In this paper, we prove that minimal hypersurfaces when $n\geq 3$ and nonzero
constant mean curvature hypersurfaces when $n\geq 2$ foliated by spheres in
parallel horizontal hyperplanes in ${\mathbb{H}}^{n}\times\mathbb{R}$ must be
rotationally symmetric.

Mathematics Subject Classification(2000) : 53A10, 53C42.
Key words and phrases : foliation, constant mean curvature, rotationally
symmetric hypersurface, product space.

##  1\. Introduction

In $\mathbb{R}^{3}$, catenoids and Riemann’s examples are the only minimal
surfaces foliated by circles. However, in higher-dimensional Euclidean space,
there are no examples of non-rotationally symmetric minimal hypersurfaces such
as Riemann’s examples in $\mathbb{R}^{3}$. In 1991, Jagy [6] proved that if
$M^{n}$ is a minimal hypersurface in $\mathbb{R}^{n+1}$ $(n\geq 3)$ and
foliated by $(n-1)$-dimensional spheres in parallel hyperplanes, then $M^{n}$
is rotationally symmetric about the axis containing the centers of all the
spheres. This result has been generalized to other spaces forms: sphere, the
hyperbolic and Lorentz-Minkowski space.(See [7], [8], and [11].)

In $\mathbb{H}^{2}\times\mathbb{R}$, Nelli and Rosenberg [9] found a
rotationally symmetric minimal surface which is called a $catenoid$ in
$\mathbb{H}^{2}\times\mathbb{R}$. In [4], Hauswirth provided several examples
of minimal surfaces foliated by horizontal curves of constant curvature in
$\mathbb{H}^{2}\times\mathbb{R}$. In particular, he constructed a two-
parameter family of Riemann type surfaces. Recently, Bérard and Sa Earp [2]
obtained some results on total curvature and index of higher-dimensional
catenoids in ${\mathbb{H}}^{n}\times\mathbb{R}$. On the other hand, Nelli et
al. [10] described the geometric behavior of rotationally symmetric constant
mean curvature surfaces in ${\mathbb{H}}^{2}\times\mathbb{R}$. They showed
that for $|H|>1/2$, the properties of rotationally symmetric constant mean
curvature surfaces in ${\mathbb{H}}^{2}\times\mathbb{R}$ are analogous to
those of the Delaunay surfaces in $\mathbb{R}^{3}$. Rotationally symmetric
constant mean curvature surfaces in ${\mathbb{H}}^{2}\times\mathbb{R}$ have
been studied in [1, 3, 5, 12].

Throughout this paper, we consider the upper half-space model of hyperbolic
space

$\displaystyle\mathbb{H}^{n}=\\{(x_{1},\cdots,x_{n})\in\mathbb{R}^{n}:x_{n}>0\\}$

equipped with the metric
$\displaystyle{ds^{2}=\frac{(dx_{1})^{2}+\cdots+(dx_{n})^{2}}{{x_{n}}^{2}}}$.
For a product space $\mathbb{H}^{n}\times\mathbb{R}$, we fix the metric
$ds^{2}+\varepsilon dt^{2}\,(\varepsilon=\pm 1)$. This metric is called
$Riemannian$ if $\varepsilon=1$ and $Lorentzian$ if $\varepsilon=-1$.

In this paper, we study hypersurfaces foliated by $(n-1)$-dimensional spheres
lying in parallel hyperplanes in some Riemannian and Lorentzian product
spaces. In Section 2, we shall prove that minimal hypersurfaces $(n\geq 3)$
and non-zero constant mean curvature hypersurfaces $(n\geq 2)$ foliated by
$(n-1)$-dimensional spheres in parallel horizontal hyperplanes in the
Riemannian product ${\mathbb{H}}^{n}\times\mathbb{R}$ should be rotationally
symmetric.(Theorem 2.1) As a consquence, one can see that there is no Riemann
type minimal hypersurfaces foliated by $(n-1)$-dimensional spheres in
${\mathbb{H}}^{n}\times\mathbb{R}$ for $n\geq 3$. We shall use Jagy’s idea [6]
to prove this result. (See also [8].) A key ingredient of the proof is the
following. We describe a hypersurface $M$ in
${\mathbb{H}}^{n}\times\mathbb{R}$ locally as the level set for a smooth
function $f$. If we orient $M$ by the unit normal vector field
$\displaystyle{N=-\frac{\nabla f}{|\nabla f|}}$, then the mean curvature $H$
is given by

(1.1) $\displaystyle nH=-{\rm div}\frac{\nabla f}{|\nabla f|},$

where $\nabla$ and ${\rm div}$ denote the gradient and divergence in
${\mathbb{H}}^{n}\times\mathbb{R}$, respectively. A straightforward
computation using the fact that $M$ is foliated by spheres in parallel
horizontal hyperplanes gives us the conclusion. In Section 3, applying the
similar arguments as in Section 2, we prove an analogue in the Lorentzian
product ${\mathbb{H}}^{n}\times\mathbb{R}$.

The author would like to thank the referee for useful suggestions on improving
the presentation of this paper.

##  2\. Sphere-foliated hypersurfaces in the Riemannian product
$\mathbb{H}^{n}\times\mathbb{R}$

In $\mathbb{H}^{n}\times\mathbb{R}$, a one-parameter family of hyperplanes
$\mathbb{H}^{n}\times\\{t\\}$ for $t\in\mathbb{R}$ are called $parallel$
$horizontal$ hyperplanes. We will deal with hypersurfaces foliated by spheres
in parallel horizontal hyperplanes in the Riemannian product space
$\mathbb{H}^{n}\times\mathbb{R}$.

######  Theorem 2.1.

Let $M$ be an $n$-dimensional hypersurface with constant mean curvature $H$ in
the Riemannian product $\mathbb{H}^{n}\times\mathbb{R}$ and foliated by
spheres in parallel horizontal hyperplanes. If $H\neq 0$ or $H=0$ and $n\geq
3$, then $M$ is a rotationally symmetric hypersurface.

Before proving the above theorem, we need the following well-known fact.

######  Lemma 2.2 ([13], p.81-82).

If an $(n-1)$-dimensional sphere has Euclidean center
$(0,\ldots,0,k)\in\mathbb{R}^{n}_{+}:=\\{(x_{1},\cdots,x_{n}):x_{n}>0\\}$ and
a Euclidean radius $r$, then it has the hyperbolic center
$(0,\ldots,0,K)\in\mathbb{R}^{n}_{+}$ and the hyperbolic radius $R$, where

$\displaystyle K=\sqrt{k^{2}-r^{2}}\,\,\mbox{and
}\,\,R=\frac{1}{2}\ln\frac{k+r}{k-r}.$

Proof of Theorem 2.1. Let $P_{t_{1}}=\mathbb{H}^{n}\times\\{t_{1}\\}$ and
$P_{t_{2}}=\mathbb{H}^{n}\times\\{t_{2}\\}$ be two horizontal hyperplanes of
the foliation for $t_{1}<t_{2}$. Let $M^{*}$ be the piece of $M$ between
$P_{t_{1}}$ and $P_{t_{2}}$. The boundary $\partial M^{*}$ of $M^{*}$ consists
of two $(n-1)$-dimensional spheres $(M^{*}\cap P_{t_{1}})\cup(M^{*}\cap
P_{t_{2}})$. After an isometric transformation in
$\mathbb{H}^{n}\times\mathbb{R}$, we may assume that the hyperbolic centers of
the two boundary spheres are given by $(0,\cdots,0,k_{1},t_{1})$ and
$(0,\cdots,0,k_{2},t_{2})$ in $\mathbb{R}^{n}_{+}\times\mathbb{R}$ for some
constants $k_{1},k_{2}>0$, respectively. Note that these two boundary spheres
are symmetric to the hyperplanes $\\{x_{1}=0\\},\cdots,\\{x_{n-1}=0\\}$. The
well-known Aleksandrov reflection principle shows that $M^{*}$ inherits the
symmetries of its boundary $\partial M^{*}$. Therefore, for each $t_{1}\leq
t\leq t_{2}$, the hyperbolic center of each level $M\cap\\{x_{n+1}=t\\}$ is
symmetric to the hyperplanes $\\{x_{1}=0\\},\cdots,\\{x_{n-1}=0\\}$. Hence it
follows that the hyperbolic center of each level lies in the $2$-plane
$\\{x_{1}=\cdots=x_{n-1}=0\\}$.

Using Lemma 2.2, we parametrize the hyperbolic centers of the
$(n-1)$-dimensional spheres by
$t\longmapsto(0,\ldots,0,K(t),t)\in\mathbb{H}^{n}\times\mathbb{R}$ for
$t\in[t_{1},t_{2}]$, and hence the Euclidean centers of the spheres by
$t\longmapsto(0,\ldots,0,k(t),t)\in\mathbb{H}^{n}\times\mathbb{R}$. Then it
follows from Lemma 2.2 that

$\displaystyle K(t)=\sqrt{k(t)^{2}-r(t)^{2}},$

where $r(t)$ is the Euclidean radius of $M\cap\mathbb{H}^{n}\times\\{t\\}$.
Note that $K(t)>0$ for $t\in[t_{1},t_{2}]$. Moreover, we see that $M^{*}$ is
the level set $\\{f=0\\}$ of a function $f$ given by

(2.1) $\displaystyle
f(x_{1},\ldots,x_{n},t)=\sum_{i=1}^{n-1}x_{i}^{2}+(x_{n}-k(t))^{2}-r(t)^{2}.$

To prove the theorem, it is sufficient to show that

$\displaystyle\frac{dK(t)}{dt}=0,$

which means that $M$ is a rotationally symmetric hypersurface whose rotation
axis is the geodesic $\gamma(t)=\\{(0,\ldots,0,K,t)\\}$ for some constant $K$.
Note that the metric on $\mathbb{H}^{n}\times\mathbb{R}$ is given by

$\displaystyle\sum_{i,j}g_{ij}dx_{i}\otimes
dx_{j}=\frac{1}{x_{n}^{2}}dx_{1}^{2}+\cdots+\frac{1}{x_{n}^{2}}dx_{n}^{2}+dt^{2}.$

Since

$\displaystyle\nabla f$ $\displaystyle=$
$\displaystyle\sum_{i,j}g^{ij}\frac{\partial f}{\partial
x_{i}}\frac{\partial}{\partial x_{j}}$ $\displaystyle=$
$\displaystyle(2x_{n}^{2}x_{1},\cdots,2x_{n}^{2}x_{n-1},2x_{n}^{2}(x_{n}-k),-2(x_{n}-k)k^{\prime}-2rr^{\prime}),$

we have

$\displaystyle|\nabla
f|^{2}=4\Big{(}x_{n}^{2}x_{1}^{2}+\cdots+x_{n}^{2}x_{n-1}^{2}+x_{n}^{2}(x_{n}-k)^{2}+((x_{n}-k)k^{\prime}+rr^{\prime})^{2}\Big{)}.$

Now we compute the mean curvature of $M^{*}$ using the equation (1.1).

$\displaystyle-nH={\rm div}\frac{\nabla f}{|\nabla f|}$ $\displaystyle=$
$\displaystyle\sum_{i,j}\frac{1}{\sqrt{g}}\frac{\partial}{\partial
x_{j}}\Big{(}\sqrt{g}\frac{g^{ij}}{|\nabla f|}\frac{\partial f}{\partial
x_{i}}\Big{)}$ $\displaystyle=$ $\displaystyle\sum_{j}\frac{\partial
Z^{j}}{\partial
x_{j}}+\sum_{j}\frac{1}{\sqrt{g}}\Big{(}\frac{\partial\sqrt{g}}{\partial
x_{j}}\Big{)}Z^{j},$

where $\displaystyle{Z^{j}=\sum_{i}\frac{g^{ij}}{|\nabla f|}\frac{\partial
f}{\partial x_{i}}}$ and $g=\det(g_{ij})=x_{n}^{-2n}$. Then we have

$\displaystyle-nH$
$\displaystyle=\frac{x_{n}^{2}}{S}-\frac{(x_{n}^{2}x_{1})^{2}}{S^{3}}+\cdots+\frac{x_{n}^{2}}{S}-\frac{(x_{n}^{2}x_{n-1})^{2}}{S^{3}}$
$\displaystyle\quad+\frac{x_{n}^{2}+2x_{n}(x_{n}-k)}{S}-\frac{x_{n}^{2}(x_{n}-k)\\{x_{n}r^{2}+x_{n}^{2}(x_{n}-k)+Ak^{\prime}\\}}{S^{3}}$
$\displaystyle\quad+\frac{B}{S}-\frac{A\\{x_{n}^{2}(x_{n}-k)k^{\prime}+AB\\}}{S^{3}}$
$\displaystyle\quad+x_{n}^{n}(-nx_{n}^{-n-1})\frac{x_{n}^{2}(x_{n}-k)}{S},$

where $A=(x_{n}-k)k^{\prime}+rr^{\prime},$ $B=k^{\prime
2}-(x_{n}-k)k^{\prime\prime}-r^{\prime 2}-rr^{\prime\prime},$ and

$\displaystyle S=\frac{|\nabla f|}{2}$ $\displaystyle=$
$\displaystyle\sqrt{x_{n}^{2}x_{1}^{2}+\cdots+x_{n}^{2}x_{n-1}^{2}+x_{n}^{2}(x_{n}-k)^{2}+((x_{n}-k)k^{\prime}+rr^{\prime})^{2}}$
$\displaystyle=$ $\displaystyle\sqrt{x_{n}^{2}r^{2}+A^{2}}.$

Thus we have

$\displaystyle-
nHS^{3}=2A^{2}x_{n}^{2}+(n-1)kr^{2}x_{n}^{3}+(n-2)kx_{n}A^{2}+r^{2}x_{n}^{2}B-2x_{n}^{3}k^{\prime}A+2kk^{\prime}x_{n}^{2}A.$

Squaring the above equation, we obtain

$\displaystyle n^{2}H^{2}S^{6}$
$\displaystyle=n^{2}H^{2}\\{x_{n}^{2}r^{2}+(k^{\prime}x_{n}+rr^{\prime}-kk^{\prime})^{2}\\}^{3}$
(2.2) $\displaystyle=\Big{[}\\{2rr^{\prime}k^{\prime}+(n-2)kk^{\prime
2}+(n-1)kr^{2}-r^{2}k^{\prime\prime}\\}x_{n}^{3}$
$\displaystyle\quad+\\{(k^{\prime 2}+r^{\prime
2}-rr^{\prime\prime}+kk^{\prime\prime})r^{2}+2(n-3)kk^{\prime}rr^{\prime}-2(n-2)k^{\prime
2}k^{2}\\}x_{n}^{2}$
$\displaystyle\quad+k(n-2)(rr^{\prime}-kk^{\prime})^{2}x_{n}\Big{]}^{2}$

Suppose that $H\neq 0$. Let us fix a section $t$. Since $x_{n}$ is varied, we
regard (2) as an equation on $x_{n}$ where the coefficients are functions of
the independent variable $t$. Comparing the degree $0$-terms in both sides of
(2), we get

$\displaystyle n^{2}H^{2}(rr^{\prime}-kk^{\prime})^{6}=0.$

Therefore it follows that

$\displaystyle\frac{dK(t)}{dt}=\frac{d}{dt}\sqrt{k(t)^{2}-r(t)^{2}}=\frac{kk^{\prime}-rr^{\prime}}{\sqrt{k^{2}-r^{2}}}=0.$

Now suppose that $H=0$ and $n\geq 3$. Comparing the coefficients of the degree
$2$-terms in both sides of (2), we have

$\displaystyle k(n-2)(rr^{\prime}-kk^{\prime})^{2}=0.$

Therefore $rr^{\prime}-kk^{\prime}=0$, which also implies that

$\displaystyle\frac{dK(t)}{dt}=0.$

Hence we can conclude that $M$ is a rotationally symmetric hypersurface in
both cases.

∎

Remark. In $\mathbb{H}^{2}\times\mathbb{R}$, Hauswirth [4] constructed several
Riemann type minimal surfaces foliated by circles. However, as mentioned in
the introduction, it follows from the above theorem that there is no Riemann
type minimal hypersurface which is not rotationally symmetric and foliated by
$(n-1)$-dimensional spheres lying in parallel horizontal hyperplanes in
$\mathbb{H}^{n}\times\mathbb{R}$ when $n\geq 3$.

##  3\. Sphere-foliated hypersurfaces in the Lorentzian product
$\mathbb{H}^{n}\times\mathbb{R}$

An immersed hypersurface $M$ in the Lorentz product space
$\mathbb{H}^{n}\times\mathbb{R}$ endowed with the Lorentzian metric

$\displaystyle
ds^{2}=\frac{(dx_{1})^{2}+\cdots+(dx_{n})^{2}}{{x_{n}}^{2}}-(dt)^{2}$

is called $spacelike$ if the induced metric on $M$ is a Riemannian metric. If
the hypersurface is locally the level set of a smooth function $f$, the fact
that $M$ is spacelike means that $\nabla f$ is a $timelike$ vector:

$\displaystyle\langle\nabla f,\nabla f\rangle<0.$

If we orient $M$ by the unit normal vector field
$\displaystyle{N=-\frac{\nabla f}{|\nabla f|}}$, then the mean curvature $H$
is given by

(3.1) $\displaystyle nH=-{\rm div}\frac{\nabla f}{|\nabla f|},$

where $|\nabla f|=\sqrt{-\langle\nabla f,\nabla f\rangle}$ and ${\rm div}$
denotes the divergence with respect to the Lorentzian metric on the product
space $\mathbb{H}^{n}\times\mathbb{R}$.

As in the proof of Theorem 2.1, consider two horizontal hyperplanes of the
foliation $P_{t_{1}}=\mathbb{H}^{n}\times\\{t_{1}\\}$ and
$P_{t_{2}}=\mathbb{H}^{n}\times\\{t_{2}\\}$ for $t_{1}<t_{2}$. Applying the
Aleksandrov reflection principle in ${\mathbb{H}}^{n}\times\mathbb{R}$, we see
that the piece $M^{*}$ between $P_{t_{1}}$ and $P_{t_{2}}$ has the symmetries
of its boundary $\partial M^{*}=(M^{*}\cap P_{t_{1}})\cup(M^{*}\cap
P_{t_{2}})$. Therefore, for each $t_{1}\leq t\leq t_{2}$, the hyperbolic
center of each level $M\cap\\{x_{n+1}=t\\}$ lies in the same $2$-plane. After
a translation in ${\mathbb{H}}^{n}\times\mathbb{R}$, we may assume that this
$2$-plane is defined by $x_{1}=\cdots=x_{n-1}=0$.

Using Lemma 2.2 again, we can parametrize the hyperbolic centers of the
$(n-1)$-dimensional spheres by
$t\longmapsto(0,\ldots,0,K(t),t)\in\mathbb{H}^{n}\times\mathbb{R}$ for
$t\in[t_{1},t_{2}]$, and the Euclidean centers of the spheres by
$t\longmapsto(0,\ldots,0,k(t),t)\in\mathbb{H}^{n}\times\mathbb{R}$, where
$K(t)=\sqrt{k(t)^{2}-r(t)^{2}}$ and $r(t)$ is the Euclidean radius of
$M\cap\mathbb{H}^{n}\times\\{t\\}$. Then $M^{*}$ is the level set $\\{f=0\\}$
of a function $f$ defined as in (2.1). Note that the metric on the Lorentzian
product $\mathbb{H}^{n}\times\mathbb{R}$ is given by

$\displaystyle\sum_{i,j}g_{ij}dx_{i}\otimes
dx_{j}=\frac{1}{x_{n}^{2}}dx_{1}^{2}+\cdots+\frac{1}{x_{n}^{2}}dx_{n}^{2}-dt^{2}.$

Since

$\displaystyle\nabla f$ $\displaystyle=$
$\displaystyle\sum_{i,j}g^{ij}\frac{\partial f}{\partial
x_{i}}\frac{\partial}{\partial x_{j}}$ $\displaystyle=$
$\displaystyle\Big{(}2x_{n}^{2}x_{1},\cdots,2x_{n}^{2}x_{n-1},2x_{n}^{2}(x_{n}-k),2((x_{n}-k)k^{\prime}+rr^{\prime})\Big{)},$

we get

$\displaystyle-\langle\nabla f,\nabla
f\rangle=4\Big{(}-x_{n}^{2}x_{1}^{2}-\cdots-
x_{n}^{2}x_{n-1}^{2}-x_{n}^{2}(x_{n}-k)^{2}+((x_{n}-k)k^{\prime}+rr^{\prime})^{2}\Big{)}.$

Using the mean curvature equation (3.1), we have

$\displaystyle-nH={\rm div}\frac{\nabla f}{|\nabla f|}$ $\displaystyle=$
$\displaystyle\sum_{i,j}\frac{1}{\sqrt{|g|}}\frac{\partial}{\partial
x_{j}}\Big{(}\sqrt{|g|}\frac{g^{ij}}{|\nabla f|}\frac{\partial f}{\partial
x_{i}}\Big{)},$

where $|\nabla f|=\sqrt{-\langle\nabla f,\nabla f\rangle}$ and
$|g|=|\det(g_{ij})|=x_{n}^{-2n}$.

A similar computation as in the proof of Theorem 2.1 shows that

$\displaystyle n^{2}H^{2}$
$\displaystyle\\{-x_{n}^{2}r^{2}+(k^{\prime}x_{n}+rr^{\prime}-kk^{\prime})^{2}\\}^{3}$
(3.2) $\displaystyle=\Big{[}\\{2rr^{\prime}k^{\prime}-(n-2)kk^{\prime
2}-(n-1)kr^{2}+r^{2}k^{\prime\prime}\\}x_{n}^{3}$
$\displaystyle\quad+\\{(k^{\prime 2}+r^{\prime
2}-rr^{\prime\prime}+kk^{\prime\prime})r^{2}-2(n-1)kk^{\prime}rr^{\prime}+2(n-2)k^{\prime
2}k^{2}\\}x_{n}^{2}$
$\displaystyle\quad-k(n-2)(rr^{\prime}-kk^{\prime})^{2}x_{n}\Big{]}^{2}$

Suppose that $H\neq 0$. Comparing the degree $0$-terms in both sides of the
equation (3) of variable $x_{n}$, we obtain

$\displaystyle n^{2}H^{2}(rr^{\prime}-kk^{\prime})^{6}=0.$

Therefore it follows that

(3.3)
$\displaystyle\frac{dK(t)}{dt}=\frac{d}{dt}\sqrt{k(t)^{2}-r(t)^{2}}=\frac{kk^{\prime}-rr^{\prime}}{\sqrt{k^{2}-r^{2}}}=0.$

Now suppose that $H=0$ and $n\geq 3$. Comparing the coefficients of the degree
$2$-terms in both sides of (3), we have

$\displaystyle k(n-2)(rr^{\prime}-kk^{\prime})^{2}=0.$

So we have $rr^{\prime}-kk^{\prime}=0$, which also implies that

(3.4) $\displaystyle\frac{dK(t)}{dt}=0.$

From (3.3) and (3.4), it follows that the hyperbolic center of each
hypersphere in parallel horizontal hyperplane lies in a vertical geodesic line
of the Lorentzian product $\mathbb{H}^{n}\times\mathbb{R}$. Therefore we
obtain the following.

######  Theorem 3.1.

Let $M$ be an $n$-dimensional spacelike hypersurface with constant mean
curvature $H$ in the Lorentzian product $\mathbb{H}^{n}\times\mathbb{R}$ and
foliated by spheres in parallel horizontal hyperplanes. If $H\neq 0$ or $H=0$
and $n\geq 3$, then $M$ is rotationally symmetric.

## References

  * [1] U. Abresch and H. Rosenberg, A Hopf differential for constant mean curvature surfaces in $\mathbb{S}^{2}\times\mathbb{R}$ and $\mathbb{H}^{2}\times\mathbb{R}$, Acta Math. 193 (2004), no. 2, 141-174.
  * [2] P. Bérard and R. Sa Earp, Minimal hypersurfaces in ${\mathbb{H}}^{n}\times\mathbb{R}$, total curvature and index, arXiv: 0808.3838v1.
  * [3] M. Cavalcante and J. de Lira, Examples and structure of CMC surfaces in some Riemannian and Lorentzian homogeneous spaces, Michigan Math. J. 55 (2007), no. 1, 163-181.
  * [4] L. Hauswirth, Minimal surfaces of Riemann type in three-dimensional product manifolds, Pacific J. Math. 224 (2006), no. 1, 91-117.
  * [5] W.-T. Hsiang and W.-Y. Hsiang, On the uniqueness of isoperimetric solutions and imbedded soap bubbles in noncompact symmetric spaces. I, Invent. Math. 98 (1989), no. 1, 39-58.
  * [6] W. Jagy, Minimal hypersurfaces foliated by spheres, Michigan Math. J. 38 (1991), no. 2, 255-270.
  * [7] W. Jagy, Sphere-foliated constant mean curvature submanifolds, Rocky Mountain J. Math. 28 (1998), no. 3, 983-1015.
  * [8] R. López, Constant mean curvature hypersurfaces foliated by spheres, Differential Geom. Appl. 11 (1999), no. 3, 245-256.
  * [9] B. Nelli and H. Rosenberg, Minimal surfaces in ${\mathbb{H}}^{2}\times\mathbb{R}$, Bull. Braz. Math. Soc. (N.S.) 33 (2002), no. 2, 263-292.
  * [10] B. Nelli, R. Sa Earp, W. Santos, and E. Toubiana, Uniqueness of $H$-surfaces in ${\mathbb{H}}^{2}\times\mathbb{R},|H|\leq 1/2$, with boundary one or two parallel horizontal circles, Ann. Global Anal. Geom. 33 (2008), no. 4, 307-321.
  * [11] S.-H. Park, Sphere-foliated minimal and constant mean curvature hypersurfaces in space forms and Lorentz-Minkowski space, Rocky Mountain J. Math. 32 (2002), no. 3, 1019-1044.
  * [12] R. Sa Earp and E. Toubiana, Screw motion surfaces in $\mathbb{H}^{2}\times\mathbb{R}$ and $\mathbb{S}^{2}\times\mathbb{R}$, Illinois J. Math. 49 (2005), no. 4, 1323-1362.
  * [13] S. Stahl, The Poincare half-plane. A gateway to modern geometry. Jones and Bartlett Publishers, Boston, MA, 1993.

Department of Mathematics
Sookmyung Women’s University
Hyochangwongil 52, Yongsan-ku
Seoul, 140-742, Korea
e-mail<EMAIL_ADDRESS>