# PMCV hypersurfaces in non-flat pseudo-Riemannian space forms

Chao Yang1, Jiancheng Liu1,∗ and Li Du2

1\. College of Mathematics and Statistics, Northwest Normal University,
Lanzhou, 730070, China
2\. School of Science, Chongqing University of Technology, Chongqing, 400054,
China

> Abstract: In this paper, we prove that PMCV (i.e. $\Delta\vec{H}$ is
> proportional to $\vec{H}$) hypersurface $M^{n}_{r}$ of a non-flat pseudo-
> Riemannian space form $N^{n+1}_{s}(c)$ with at most two distinct principal
> curvatures is minimal or locally isoparametric, and compute the mean
> curvature for the isoparametric ones. As an application, we give full
> classification results of such non-minimal Lorentzian hypersurfaces of non-
> flat Lorentz space forms.
>
> Keywords: proper mean curvature vector field; pseudo-Riemannian space form;
> isoparameric hypersurface; mean curvature; Lorentz space form
>
> Mathematics Subject Classification (2020) 53C50

000This work is supported by National Natural Science Foundation of China (No.
12161078), Science and Technology Project of Gansu Province (No. 20JR5RA515),
Innovation Ability Enhancement Project of Gansu Higher Education Institution
(2023B-063). 000E-mail address<EMAIL_ADDRESS>
##  1 Introduction

Let $N^{n+1}_{s}(c)$ be the ($n+1$)-dimensional pseudo-Riemannian space form
of constant curvature $c$ with index $s\geq 0$, and $x:\\!M^{n}_{r}\to
N^{n+1}_{s}(c)$ be an isometric immersion of a pseudo-Riemannian hypersurface
$M^{n}_{r}$ into $N^{n+1}_{s}(c)$. Denote by $\vec{H}$ and $\Delta$ the mean
curvature vector field and the Laplace operator of $M^{n}_{r}$ with respect to
the induced metric, respectively.

If the hypersurface $M^{n}_{r}$ satisfies the equation

$\Delta\vec{H}=\lambda\vec{H},$

for some real constant $\lambda$, then it is said to have proper mean
curvature vector field (abbreviated as PMCV). Specially, when $\lambda=nc$,
the hypersurface $M^{n}_{r}$ is known as biharmonic one.

The following geometric question was proposed by B.-Y. Chen in [1, 2]:

_Determine all PMCV submanifolds of pseudo-Euclidean space
$\mathbb{E}^{n+1}_{s}$. In particular, classify PMCV hypersurfaces in
$\mathbb{E}^{n+1}_{s}$._

In 1988, B.-Y. Chen [1] initiated the study of this problem for $s=0$, and
proved that the PMCV surface $M^{2}$ in $\mathbb{E}^{3}$ is minimal, or
locally congruent to a circular cylinder. In 1991, A. Ferrández and P. Lucus
showed in [7] that the PMCV hypersurface $M^{n}$ of $\mathbb{E}^{n+1}$ with at
most two distinct principal curvatures is minimal, or locally congruent to
$\mathbb{E}^{l}\times\mathbb{S}^{n-l}(\frac{\lambda}{n-l})\ (0\leq l\leq
n-1)$. Afterwards, B.-Y. Chen and O. J. Garay [3] derived that
$\delta(2)$-ideal PMCV hypersurfaces in $\mathbb{E}^{n+1}$ is minimal, or
locally congruent to
$\mathbb{E}^{1}\times\mathbb{S}^{n-1}(\frac{\lambda}{n-1})$.

For the case $s=1$, A. Ferrández and P. Lucus [8] classified PMCV surfaces,
and generalized the result in [9] to $n$-dimensional hypersurfaces whose
minimal polynomial of the shape operator is at most of degree two. In [12], C.
Yang and J.-C. Liu weaken this condition to the number of distinct principal
curvatures be not larger than two and obtained classification result. For
general index $s$, L. Du [4] classified hypersuraces with diagonalizable shape
operator and at most two distinct principal curvatures.

Naturally, it is interesting to consider the classification problem of PMCV
hypersurfaces in non-flat pseudo-Riemannian space forms $N^{n+1}_{s}(c)$.
Under the assumption that the shape operator is diagonalizable, L. Du [4]
classified such hypersurfaces of $N^{n+1}_{s}(c)\ (c\neq 0)$ with at most two
distinct principal curvatures. In this paper, we show that such PMCV
hypersurface $M^{n}_{r}$ of $N^{n+1}_{s}(c)\ (c\neq 0)$ is minimal or locally
isoparametric and give the value or range of the mean curvature, without the
restriction that the shape operator is diagonalizable. As an application, we
classify completely such non-minimal Lorentzian PMCV hypersurfaces of
$N^{n+1}_{1}(c)\ (c\neq 0)$ by using classification theory of isoparametric
hypersurfaces.

##  2 Preliminaries

Let $\mathbb{E}^{n}_{s}$ $(0\leq s\leq n)$ be the $n$-dimensional pseudo-
Euclidean space equipped with the canonical pseudo-Euclidean metric of index
$s$

$g_{0}=-\sum_{i=1}^{s}\text{d}x_{i}^{2}+\sum_{j=s+1}^{n}\text{d}x_{j}^{2},$

where $(x_{1},x_{2},\cdots,x_{n})$ is a rectangular coordinate system of
$\mathbb{E}^{n}_{s}$. Set

$\displaystyle\mathbb{S}^{n}_{s}(c)=\left\\{x\in\mathbb{E}^{n+1}_{s}\Big{|}-\sum_{i=1}^{s}x_{i}^{2}+\sum_{j=s+1}^{n+1}x_{j}^{2}=\frac{1}{c}>0\right\\},$
$\displaystyle\mathbb{H}^{n}_{s}(c)=\left\\{x\in\mathbb{E}^{n+1}_{s+1}\Big{|}-\sum_{i=1}^{s+1}x_{i}^{2}+\sum_{j=s+2}^{n+1}x_{j}^{2}=\frac{1}{c}<0\right\\},$

where $c$ is a nonzero real constant. $\mathbb{S}^{n}_{s}(c)$ and
$\mathbb{H}^{n}_{s}(c)$ are pseudo-Riemannian manifolds of curvature $c$ with
index $s$, which together with $\mathbb{E}^{n}_{s}$ are called _pseudo-
Riemannian space forms_ , denoted by $N^{n}_{s}(c)$. In particular, for $s=1$,
$N^{n}_{1}(c)$ are called _Lorentz space forms_ , $\mathbb{S}^{n}_{1}(c)$ and
$\mathbb{H}^{n}_{1}(c)$ known as _de Sitter space_ and _anti-de Sitter space_
, respectively. A non-zero vector $X$ in $N^{n}_{s}(c)$ is said to be _time-
like_ , _space-like_ or _light-like_ , according to whether $\langle
X,X\rangle$ is negative, positive or zero.

Let $M^{n}_{r}$ be a nondegenerate hypersurface in $N^{n+1}_{s}(c)$,
$\vec{\xi}$ denote a unit normal vector field to $M^{n}_{r}$, then
$\varepsilon=\langle\vec{\xi},\vec{\xi}\rangle=\pm 1$. Set $\nabla$ and
$\tilde{\nabla}$ the Levi-Civita connections of $M^{n}_{r}$ and
$N^{n+1}_{s}(c)$, respectively. For any vector fields $X,Y$ tangent to
$M^{n}_{r}$, it follows that

$\widetilde{\nabla}_{X}Y=\nabla_{X}Y+h(X,Y)\vec{\xi},$

where $h$ is the scalar-valued second fundamental form. Denote by $A$,
$\vec{H}$ and $H$ the shape operator of $M^{n}_{r}$ associated to $\vec{\xi}$,
the mean curvature vector field and the mean curvature, then
$\vec{H}=H\vec{\xi}$ and $H=\frac{1}{n}\varepsilon\text{tr}A$. For any vector
fields $X,Y,Z$ tangent to $M^{n}_{r}$, the Codazzi and the Gauss equations are
given by

$\langle(\nabla_{X}A)Y,Z\rangle=\langle(\nabla_{Y}A)X,Z\rangle,$ (2.1)
$R(X,Y)Z=c(\langle Y,Z\rangle X-\langle X,Z\rangle Y)+\varepsilon\langle
A(Y),Z\rangle A(X)-\varepsilon\langle A(X),Z\rangle A(Y),$

where $R(X,Y)Z=\nabla_{X}\nabla_{Y}Z-\nabla_{Y}\nabla_{X}Z-\nabla_{[X,Y]}Z$.

According to [5], $M^{n}_{r}$ is a PMCV hypersurface if and only if

$A(\nabla H)=-\frac{n}{2}\varepsilon H(\nabla H),$ (2.2) $\Delta H+\varepsilon
H\text{tr}A^{2}=\lambda H.$ (2.3)

Suppose the eigenpolynomial $\text{det}(\lambda I-A)$ of $A$ can be expressed
as
$(\lambda-\lambda_{1})^{n_{1}}(\lambda-\lambda_{2})^{n_{2}}\cdots(\lambda-\lambda_{k})^{n_{k}}$,
then $n_{i}$ is called the _algebraic multiply_ of $\lambda_{i}$. The
_geometric multiply_ of $\lambda_{i}$ refers to the dimension of the
eigensubspace $\text{ker}(\lambda_{i}I-A):=\\{X|AX=\lambda_{i}X\\}$. A
hypersurface whose shape operator has constant eigenpolynomial is said to be
_isoparametric_ (cf. [10]), which is equivalent to the principal curvatures
and their algebriac multiplicities being constants. In this paper, without
special statement, multiplicity refers to algebraic multiplicity.

##  3 PMCV hypersurfaces in non-flat pseudo-Riemannian space forms

In this section, we study PMCV hypersurface $M^{n}_{r}$ of $N^{n+1}_{s}(c)\
(c\neq 0)$ with at most two distinct principal curvatures, and prove that it
is minimal or locally isoparametric (see Theorem 3.1). Furthermore, we
estimate or compute the value of the mean curvature (see Theorems 3.3 and
3.4).

First of all, we recall from [2, 12] that the shape operator $A$ of the
hypersurface $M^{n}_{r}$ can be expressed as an almost diagonal matrix

$A=\text{diag}\\{A_{1},\cdots,A_{t},\bar{A}_{t+1},\cdots,\bar{A}_{m}\\},\
\text{for some}\ t,\ 0\leq t\leq m,$

where

$\displaystyle A_{i}\\!=\\!\left(\begin{array}[]{ccccc}\lambda_{i}&&&&\\\
1&\lambda_{i}&&&\\\ &\ddots&\ddots&&\\\ &&1&\lambda_{i}&\\\
&&&1&\lambda_{i}\\\ \end{array}\right)\\!,\
\bar{A}_{j}\\!=\\!\left(\begin{array}[]{ccccccc}\gamma_{j}&\tau_{j}&&&&\\\
-\tau_{j}&\gamma_{j}&0&&&\\\ 1&0&\gamma_{j}&\tau_{j}&&\\\
&\ddots&\ddots&\ddots&\ddots&\\\ &&1&0&\gamma_{j}&\tau_{j}\\\
&&&1&-\tau_{j}&\gamma_{j}\\\ \end{array}\right),\tau_{j}\neq 0,$

for $1\leq i\leq t$ and $t+1\leq j\leq m$, relative to the basis
$\mathfrak{B}\\!=\\!\\{u_{i_{1}},u_{i_{2}},\cdots,u_{i_{\alpha_{i}}},u_{\bar{j}_{1}},$
$u_{\tilde{j}_{1}},u_{\bar{j}_{2}},u_{\tilde{j}_{2}},\cdots,u_{\bar{j}_{\beta_{j}}},u_{\tilde{j}_{\beta_{j}}}|1\leq
i\leq t,\ t+1\leq j\leq m\\}$ with all scalar products zero except $\langle
u_{i_{a}},u_{i_{b}}\rangle=\varepsilon_{i}=\pm 1$ and $\langle
u_{\bar{j}_{c}},u_{\bar{j}_{d}}\rangle=1=-\langle
u_{\tilde{j}_{c}},u_{\tilde{j}_{d}}\rangle$ if $a+b=\alpha_{i}+1$. Here $1\leq
i\leq t,\ c+d=\beta_{j}+1,\ t+1\leq j\leq m$ and
$\sum_{i=1}^{t}\alpha_{i}+2\sum_{j=t+1}^{m}\beta_{j}=n.$

It follows from the form of $A$ that $M^{n}_{r}$ has real principal curvatures
$\lambda_{1},\lambda_{2},\cdots,\lambda_{t}$ and imaginary eigenvalues
$\gamma_{t+1}+\tau_{t+1}\sqrt{-1},\gamma_{t+1}-\tau_{t+1}\sqrt{-1},\cdots,\gamma_{m}+\tau_{m}\sqrt{-1},\gamma_{m}-\tau_{m}\sqrt{-1}.$
Notice that the imaginary principal curvatures appear as conjugate pairs.

Theorem 3.1  _Let $M^{n}_{r}$ be a PMCV hypersurface of $N^{n+1}_{s}(c)\
(c\neq 0)$. Suppose that $M^{n}_{r}$ has at most two distinct principal
curvatures, then $M^{n}_{r}$ has constant mean curvature. Furthermore, it is
minimal or locally isoparametric._

Proof Firstly, we prove that if the mean curvature $H$ of $M^{n}_{r}$ is a
constant, then $M^{n}_{r}$ is minimal or locally isoparametric. Let $x\in
M^{n}_{r}$, there exists a neighborhood $U_{x}$ such that the multiplicities
of distinct principal curvatures are constants on $U_{x}$. We work on $U_{x}$
and show that $M^{n}_{r}$ is isoparametric.

Suppose $\mu$ and $\nu$ are distinct principal curvatures of $M^{n}_{r}$ with
multiplicities $l$ $(1\leq l\leq n)$ and $n-l$, then

$n\varepsilon H=\text{tr}A=l\mu+(n-l)\nu,$ (3.1)

and $\text{tr}A^{2}=l\mu^{2}+(n-l)\nu^{2}$. If $H$ is a nonzero constant, then
(2.3) implies $\text{tr}A^{2}=\varepsilon\lambda$. So,

$l\mu^{2}+(n-l)\nu^{2}=\varepsilon\lambda.$ (3.2)

It follows from (3.1) and (3.2) that $\mu$ and $\nu$ are constants. Hence,
$M^{n}_{r}$ is isoparametric on $U_{x}$.

In the following, we show that $M^{n}_{r}$ has constant mean curvature.

When the principal curvatures are imaginary, $-\frac{n}{2}\varepsilon H$ (a
real function) is not an eigenvalue of $A$. It follows from (2.2) that $\nabla
H\equiv 0$, i.e. $H=\text{const.}$ We need only treat the situation that the
principal curvatures are all real.

Assume that $H$ is not a constant, then there exists a neighborhood $U_{x}$ of
$x\in M^{n}_{r}$ such that $H\neq 0$ and $\nabla H\neq 0$. From (2.2), we get
that $-\frac{n}{2}\varepsilon H$ is an eigenvalue of $A$, and $\nabla H$ is a
corresponding eigenvector. When the principal curvatures of $M^{n}_{r}$ are
all the same, denoted by $\mu$, then $\mu=\frac{1}{n}\text{tr}A=\varepsilon
H$. Thus, $-\frac{n}{2}\varepsilon H\ (\neq\mu)$ is not an eigenvalue of $A$,
a contradiction.

Now, we assume that the number of distinct principal curvatures is two, i.e.
$A=\text{diag}\\{A_{1},\cdots,A_{m}\\}$ and there are two distinct values
among $\\{\lambda_{1},\cdots,\lambda_{m}\\}$. Observe the form of $A$, $\nabla
H$ is in one of the directions $u_{i_{\alpha_{i}}},1\leq i\leq m$. Without
loss of generality, we suppose $\nabla H$ is in the direction of
$u_{1_{\alpha_{1}}}$, which may or may not be a light-like vector. We follow
different processes to lead contradictions for those two cases.

At the begining, we provide some important equations. We remark that, if no
otherwise specified, the ranges of $i$ and $a$ in the index $i_{a}$ are as:
$1\leq i\leq m$ and $1\leq a\leq\alpha_{i}$. And when $a=\alpha_{i}$ (or
$a=1$), the connection coefficients with the index $i_{a+1}$ (or $i_{a-1}$)
disappear.

Let $[i]=\\{j|\ \lambda_{j}=\lambda_{i}\\}$ for $1\leq i\leq m$ and
$\nabla_{u_{i_{a}}}u_{j_{b}}=\sum_{k=1}^{m}\sum_{d=1}^{\alpha_{k}}\Gamma^{k_{d}}_{i_{a}j_{b}}u_{k_{d}}$.
In view of the assumption that $\nabla H$ is in the direction of
$u_{1_{\alpha_{1}}}$, as well as the compatibility and symmetry of the
connection, we obtain

$\begin{cases}u_{1_{1}}(H)\neq 0,\ u_{B}(H)=0,\
\Gamma^{1_{1}}_{BC}=\Gamma^{1_{1}}_{CB},\\\
\Gamma^{i_{\alpha_{i}-a+1}}_{Di_{a}}=0,\
\Gamma^{i_{\alpha_{i}-b+1}}_{Di_{a}}=-\Gamma^{i_{\alpha_{i}-a+1}}_{Di_{b}},\
\Gamma^{j_{\alpha_{j}-d+1}}_{Di_{a}}=-\varepsilon_{i}\varepsilon_{j}\Gamma^{i_{\alpha_{i}-a+1}}_{Dj_{d}},\end{cases}$
(3.3)

for $B,C,D\in\\{k_{e},1\leq k\leq m,1\leq e\leq\alpha_{k}\\}$ and $B,C\neq
1_{1}$.

Putting $X=u_{i_{a}},Y=u_{j_{b}},Z=u_{k_{\alpha_{k}-d+1}}$ into (2.1), it
gives

$\displaystyle
u_{i_{a}}\\!(\lambda_{j})\delta_{j_{b}}^{k_{d}}\\!+\\!(\lambda_{j}-$
$\displaystyle\lambda_{k})\Gamma_{i_{a}j_{b}}^{k_{d}}\\!+\\!\Gamma_{i_{a}j_{b+1}}^{k_{d}}\\!-\\!\Gamma_{i_{a}j_{b}}^{k_{d-1}}\\!=\\!u_{j_{b}}\\!(\lambda_{i})\delta_{i_{a}}^{k_{d}}\\!+\\!(\lambda_{i}-\lambda_{k})\Gamma_{j_{b}i_{a}}^{k_{d}}\\!+\\!\Gamma_{j_{b}i_{a+1}}^{k_{d}}\\!-\\!\Gamma_{j_{b}i_{a}}^{k_{d-1}},$
(3.4)

where $\delta_{j_{b}}^{i_{a}}=1$ if $i_{a}=j_{b}$, and
$\delta_{j_{b}}^{i_{a}}=0$ if $i_{a}\neq j_{b}$. Using (3.3), we deduce from
(3.4) that

$\displaystyle\text{for}\
i=j=k=d=1\Rightarrow\Gamma_{1_{\alpha_{1}-1}1_{\alpha_{1}}}^{1_{\alpha_{1}-1}}\\!=\cdots=\\!\Gamma_{1_{1}1_{\alpha_{1}}}^{1_{1}}\\!=0;$
(3.5) $\displaystyle\text{for}\ i=k=d=1,j\in[1],j\neq
1\Rightarrow\Gamma_{1_{a+1}j_{\alpha_{j}}}^{1_{1}}\\!=0;$
$\displaystyle\text{for}\
i=k=d=1,j\notin[1]\Rightarrow\Gamma_{1_{a}j_{b}}^{1_{1}}\\!=0;$
$\displaystyle\text{for}\
i\notin[1],j=1,b=\alpha_{1},k=1\Rightarrow\Gamma_{1_{\alpha_{1}}i_{a}}^{1_{d}}\\!=0;$
$\displaystyle\text{for}\ i,j\neq
1,j\notin[i],k=d=1\Rightarrow\Gamma_{i_{a}j_{b}}^{1_{1}}\\!=0;$
$\displaystyle\text{for}\
i,j\\!\neq\\!1,j\\!\in\\![i],k=\\!d\\!=\\!1\\!\Rightarrow\\!\Gamma_{i_{a}1_{\alpha_{1}}}^{j_{b}}\\!=\\!\Gamma_{i_{a+1}1_{\alpha_{1}}}^{j_{b+1}}\\!,\
\Gamma_{i_{\alpha_{i}}j_{b+1}}^{1_{1}}\\!=\\!\Gamma_{i_{a}\\!1_{\alpha_{1}}\\!}^{j_{e}}\\!=\\!0,e<a;$
$\displaystyle\text{for}\
\\!i\\!=\\!k(\\!\neq\\!1)\\!,d\\!=\\!a,j\\!=\\!1,b\\!=\\!\alpha_{1}\\!\Rightarrow\\!u_{1_{\alpha_{1}}}\\!(\lambda_{i})\\!=\\!(-\frac{n}{2}\varepsilon
H-\lambda_{i})\Gamma_{i_{\alpha_{i}}1_{\alpha_{1}}}^{i_{\alpha_{i}}}\\!,\
\Gamma_{1_{\alpha_{1}}i_{a+1}}^{i_{a}}\\!=\\!0;\\!$ $\displaystyle\text{for}\
i,a\neq
1,j=d=1,b=\alpha_{1},k=i\Rightarrow\Gamma_{1_{\alpha_{1}}i_{a}}^{i_{1}}\\!=0.$

Let

$\begin{cases}A_{i,0,j_{b}}=\Gamma_{i_{1}j_{b}}^{i_{1}}+\Gamma_{i_{2}j_{b}}^{i_{2}}+\cdots+\Gamma_{i_{\alpha_{i}}j_{b}}^{i_{\alpha_{i}}},\\\
A_{i,1,j_{b}}=\Gamma_{i_{2}j_{b}}^{i_{1}}+\Gamma_{i_{3}j_{b}}^{i_{2}}+\cdots+\Gamma_{i_{\alpha_{i}}j_{b}}^{i_{\alpha_{i}-1}},\\\
\quad\quad\quad\quad\quad\vdots\\\
A_{i,\alpha_{i}-1,j_{b}}=\Gamma_{i_{\alpha_{i}}j_{b}}^{i_{1}},\end{cases}\ \
\text{for}\ j\in[1],\ i\notin[1],$

then (3.4) for $k=i$ implies

$\begin{cases}(-\frac{n}{2}\varepsilon
H-\lambda_{i})A_{i,0,j_{b}}+A_{i,0,j_{b+1}}-A_{i,1,j_{b}}=\alpha_{i}u_{j_{b}}(\lambda_{i}),\\\
(-\frac{n}{2}\varepsilon
H-\lambda_{i})A_{i,r,j_{b}}+A_{i,r,j_{b+1}}-A_{i,r+1,j_{b}}=0,\\\
(-\frac{n}{2}\varepsilon
H-\lambda_{i})A_{i,\alpha_{i}-1,j_{b}}+A_{i,\alpha_{i}-1,j_{b+1}}=0,\end{cases}$

$1\leq r\leq\alpha_{i}-2$. It follows that

$A_{i,s,j_{b}}=0,\ 1\leq s\leq\alpha_{i}-1,\\\ $ (3.6)
$(-\frac{n}{2}\varepsilon
H-\lambda_{i})A_{i,0,j_{b}}+A_{i,0,j_{b+1}}=\alpha_{i}u_{j_{b}}(\lambda_{i}).$
(3.7)

Set

$\begin{cases}B_{i,k,0,j_{b}}=\Gamma_{i_{1}j_{b}}^{k_{1}}+\Gamma_{i_{2}j_{b}}^{k_{2}}+\cdots+\Gamma_{i_{y_{0}}j_{b}}^{k_{y_{0}}},\\\
B_{i,k,1,j_{b}}=\Gamma_{i_{2}j_{b}}^{k_{1}}+\Gamma_{i_{3}j_{b}}^{k_{2}}+\cdots+\Gamma_{i_{y_{1}}j_{b}}^{k_{y_{1}-1}},\\\
\quad\quad\quad\quad\quad\vdots\\\
B_{i,k,\alpha_{i}-1,j_{b}}=\Gamma_{i_{\alpha_{i}}j_{b}}^{k_{1}},\end{cases}\\!$

for $j\in[1]$, $i,k\notin[1]$, $i\neq k$ and
$y_{r}=\min\\{\alpha_{i},\alpha_{k}+r\\}$, $0\leq r\leq\alpha_{i}-1$, then we
obtain from (3.4) that

$\begin{cases}(-\frac{n}{2}\varepsilon
H-\lambda_{k})B_{i,k,s,j_{b}}+B_{i,k,s,j_{b+1}}-B_{i,k,s+1,j_{b}}=0,\\\
(-\frac{n}{2}\varepsilon
H-\lambda_{k})B_{i,k,\alpha_{i}-1,j_{b}}+B_{i,k,\alpha_{i}-1,j_{b+1}}=0,\end{cases}$

which implies

$B_{i,k,r,j_{b}}=0.$ (3.8)

We treat the two cases that $\nabla H$ is not light-like or light-like
respectively and deduce contradictions.

_Case 1: $\nabla H$ is not light-like._

Since $\nabla H$ is in the direction $u_{1_{\alpha_{1}}}$, we have $\langle
u_{1_{\alpha_{1}}},u_{1_{\alpha_{1}}}\rangle\neq 0$, which implies
$2\alpha_{1}=\alpha_{1}+1$, i.e. $\alpha_{1}=1$. We find from the seventh
equation of (3.5) that if $\lambda_{i}=-\frac{n}{2}\varepsilon H$ for some
$i\neq 1$, then $u_{1_{1}}(H)=0$, which contradicts to (3.3). Hence,
$\lambda_{i}\neq-\frac{n}{2}\varepsilon H$ for $2\leq i\leq m$. Combining
$\lambda_{1}=-\frac{n}{2}\varepsilon H$ and $\text{tr}A=n\varepsilon H$, we
know $\lambda_{2}=\cdots=\lambda_{m}=\frac{3n\varepsilon H}{2(n-1)}$. It
follows from the seventh equation of (3.5) that

$W:=\Gamma^{i_{a}}_{i_{a}1_{1}}\\!=-\frac{3u_{1_{1}}(H)}{(n+2)H},\ 2\leq i\leq
m.$ (3.9)

Lemma 3.2  _We have_

$u_{1_{1}}(W)+W^{2}=\frac{3\varepsilon\varepsilon_{1}n^{2}H^{2}}{4(n-1)}-c\varepsilon_{1}.$
(3.10)

Proof Let $j=b=1,d=1$ in (3.4), then

$-\frac{n(n+2)\varepsilon
H}{2(n-1)}\Gamma_{i_{a}1_{1}}^{k_{1}}\\!=\Gamma_{1_{1}i_{a+1}}^{k_{1}},\ 2\leq
i,k\leq m,\ i\neq k.$ (3.11)

Combining the sixth equation of (3.5), it leads to

$\Gamma_{1_{1}i_{a+1}}^{k_{1}}\\!=0,\ 2\leq a\leq\alpha_{i}-1.$ (3.12)

Calculate $\langle R(u_{1_{1}},u_{i_{\alpha_{i}}})u_{1_{1}},u_{i_{1}}\rangle$
by using the Gauss equation, combining (3.5) and (3.12), we have

$u_{1_{1}}(W)+W^{2}-\frac{3\varepsilon\varepsilon_{1}n^{2}H^{2}}{4(n-1)}+c\varepsilon_{1}=P_{i},\
2\leq i\leq m,$ (3.13)

where

$P_{i}=\sum_{2\leq k\leq m,k\neq
i}\\!\left(\Gamma_{k_{\alpha_{k}}1_{1}}^{i_{\alpha_{i}}}\Gamma_{1_{1}i_{\alpha_{i}}}^{k_{\alpha_{k}}}\\!-\Gamma_{i_{\alpha_{i}}1_{1}}^{k_{\alpha_{k}}}\Gamma_{1_{1}k_{\alpha_{k}}}^{i_{\alpha_{i}}}\\!+\Gamma_{k_{\alpha_{k}-1}1_{1}}^{i_{\alpha_{i}}}\Gamma_{1_{1}i_{\alpha_{i}}}^{k_{\alpha_{k}-1}}\\!-\Gamma_{i_{\alpha_{i}}1_{1}}^{k_{\alpha_{k}}}\Gamma_{k_{\alpha_{k}}1_{1}}^{i_{\alpha_{i}}}\right).$

From the sixth equation of (3.5), we obtain that
$\Gamma^{i_{1}}_{k_{1}1_{1}}\\!=\Gamma^{i_{\alpha_{k}}}_{k_{\alpha_{k}}1_{1}}\\!=0$
if $\alpha_{k}<\alpha_{i}\ (i,k\neq 1)$. And if $\alpha_{k}=\alpha_{i}\
(i,k\neq 1)$, (3.8) tells us that
$\Gamma^{i_{1}}_{k_{1}1_{1}}\\!+\Gamma^{i_{2}}_{k_{2}1_{1}}\\!+\cdots+\Gamma^{i_{\alpha_{i}}}_{k_{\alpha_{k}}1_{1}}\\!=0,$
which together with the sixth equation of (3.5) implies
$\Gamma^{i_{1}}_{k_{1}1_{1}}\\!=0$. If $\alpha_{k}>\alpha_{i}\ (i,k\neq 1)$,
taking $j=b=1,\ d=a+1$ in (3.4) and combining
$\Gamma^{k_{1}}_{i_{1}1_{1}}\\!=0$, we have

$-\frac{n(n+2)\varepsilon
H}{2(n-1)}\Gamma^{k_{a+1}}_{i_{a}1_{1}}=\Gamma^{k_{a+1}}_{1_{1}i_{a+1}}-\Gamma^{k_{a}}_{1_{1}i_{a}}.$
(3.14)

When $\alpha_{k}>\alpha_{i}+1$, combining (3.3), the sixth equation of (3.5)
implies that

$\Gamma^{k_{2}}_{i_{1}1_{1}}\\!=\cdots=\Gamma^{k_{\alpha_{i}+1}}_{i_{\alpha_{i}}1_{1}}\\!=0.$
(3.15)

(3.14) together with (3.15) yields

$\Gamma^{k_{1}}_{1_{1}i_{1}}\\!=0,\ \text{if}\ \alpha_{k}>\alpha_{i}+1.$
(3.16)

When $\alpha_{k}=\alpha_{i}+1$, we deduce from (3.3), the sixth equation of
(3.5) and (3.14) that

$\displaystyle\Gamma^{k_{2}}_{i_{1}1_{1}}\\!=\\!\Gamma^{k_{\alpha_{k}}}_{i_{\alpha_{i}}1_{1}}\\!=\\!\varepsilon_{i}\varepsilon_{k}\Gamma^{i_{1}}_{k_{1}1_{1}}\\!,\
\Gamma^{k_{1}}_{1_{1}i_{1}}\\!=\\!\alpha_{i}\Gamma^{k_{\alpha_{i}}}_{1_{1}i_{\alpha_{i}}}\\!=\\!\varepsilon_{i}\varepsilon_{k}\frac{n(n+2)\varepsilon
H}{2(n-1)}\alpha_{i}\Gamma^{i_{1}}_{k_{1}1_{1}}\\!.$ (3.17)

By (3.3), we know
$\Gamma_{i_{\alpha_{i}}1_{1}}^{j_{\alpha_{j}}}=\varepsilon_{i}\varepsilon_{j}\Gamma_{j_{1}1_{1}}^{i_{1}}$
and
$\Gamma_{j_{\alpha_{j}-1}1_{1}}^{i_{\alpha_{i}}}=\varepsilon_{i}\varepsilon_{j}\Gamma_{i_{1}1_{1}}^{j_{2}}$.
Using the sixth equation of (3.5), (3.11), (3.15), (3.16) and (3.17), $P_{i}$
can be rewritten as

$\displaystyle P_{i}=$ $\displaystyle\\!\sum_{k\neq
i}\\!\left(-\Gamma_{i_{1}1_{1}}^{k_{1}}\Gamma_{1_{1}k_{1}}^{i_{1}}\\!-\Gamma_{i_{1}1_{1}}^{k_{2}}\Gamma_{1_{1}k_{2}}^{i_{1}}\\!+\Gamma_{k_{1}1_{1}}^{i_{1}}\Gamma_{1_{1}i_{1}}^{k_{1}}\\!-\Gamma_{i_{1}1_{1}}^{k_{1}}\Gamma_{k_{1}1_{1}}^{i_{1}}\right)$
(3.18) $\displaystyle=$
$\displaystyle\\!-\\!\sum_{\alpha_{k}=\alpha_{i}-1}\\!\Gamma_{i_{1}1_{1}}^{k_{1}}\Gamma_{1_{1}k_{1}}^{i_{1}}\\!-\\!\sum_{\alpha_{k}=\alpha_{i}+1}\\!\Gamma_{i_{1}1_{1}}^{k_{2}}\Gamma_{1_{1}k_{2}}^{i_{1}}\\!+\\!\sum_{\alpha_{k}=\alpha_{i}+1}\\!\Gamma_{k_{1}1_{1}}^{i_{1}}\Gamma_{1_{1}i_{1}}^{k_{1}}$
$\displaystyle=$ $\displaystyle\frac{n(n+2)\varepsilon
H}{2(n-1)}\varepsilon_{i}\left[\sum_{\alpha_{k}=\alpha_{i}+1}\\!\varepsilon_{k}(\alpha_{i}+1)(\Gamma^{i_{1}}_{k_{1}1_{1}})^{2}-\\!\sum_{\alpha_{k}=\alpha_{i}-1}\\!\varepsilon_{k}\alpha_{k}(\Gamma^{k_{1}}_{i_{1}1_{1}})^{2}\right].$

Choose
$\\{\alpha_{1_{1}},\alpha_{1_{2}},\cdots,\alpha_{1_{\xi_{1}}},\alpha_{2_{1}},\alpha_{2_{2}},\cdots,\alpha_{2_{\xi_{2}}},\cdots,\alpha_{h_{1}},\alpha_{h_{2}},\cdots,\alpha_{h_{\xi_{h}}}\\}$
in $\\{\alpha_{2},\cdots,\alpha_{m}\\}$
such that
$\tilde{\alpha}_{t}:=\alpha_{t_{1}}=\alpha_{t_{2}}=\cdots=\alpha_{t_{\xi_{t}}}$
for $1\leq t\leq h$, $\tilde{\alpha}_{k}+1=\tilde{\alpha}_{k+1}$ for $1\leq
k\leq h-1$, and
$\tilde{\alpha}_{1}-1,\tilde{\alpha}_{h}+1\notin\\{\alpha_{2},\cdots,\alpha_{m}\\}$.
Since $P_{i}=P_{k}$ for distinct $i,k$, we can denote $P_{i}$ by $P$. Puting
$i=t_{1},t_{2},\cdots,t_{\xi_{t}}$ in (3.18) and taking the sum of those
equations gives

$\xi_{t}P=(\tilde{\alpha}_{t}+1)Q_{t}-\tilde{\alpha}_{t-1}Q_{t-1},\ 1\leq
t\leq h,$ (3.19)

where

$Q_{t}=\\!\sum_{{\\!\tiny\\!\begin{array}[]{c}\\!1\\!\leq\\!p\\!\leq\\!\xi_{t}\\!\\\
\\!1\\!\leq\\!q\\!\leq\\!\xi_{t+1}\\!\end{array}\\!}\\!}\\!\frac{n(n+2)\varepsilon
H}{2(n-1)}\varepsilon_{\rho}\varepsilon_{\sigma}(\Gamma^{\rho_{1}}_{\sigma_{1}1_{\alpha_{1}}})^{2},\
\ \rho=t_{p},\ \sigma=(t+1)_{q}.$

(3.19) implies $Q_{t}=0$, $1\leq t\leq h$ and $P=0$. Thus, (3.10) follows from
(3.13). $\square$

Now, we continue the proof of Theorem 3.1 for case 1.

Following the process of the proof of [12, Lemma 3.4], we derive

$\displaystyle
u_{1_{1}}u_{1_{1}}(H)+(n-1)Wu_{1_{1}}(H)-\varepsilon_{1}\varepsilon\frac{(n+8)n^{2}H^{3}}{4(n-1)}+\lambda\varepsilon_{1}H=0.$
(3.20)

We obtain from (3.9) and (3.10) that

$\displaystyle
u_{1_{1}}u_{1_{1}}(H)=\frac{(n+2)(n+5)H}{9}W^{2}-\frac{(n+2)n^{2}\varepsilon\varepsilon_{1}H^{3}}{4(n-1)}+\frac{1}{3}(n+2)c\varepsilon_{1}H.$

Substituting into (3.20) leads to

$\displaystyle\frac{2(n-4)(n+2)}{9}W^{2}+\frac{n^{2}(n+5)\varepsilon\varepsilon_{1}H^{2}}{2(n-1)}-\frac{1}{3}(n+2)c\varepsilon_{1}-\varepsilon_{1}\lambda=0.$
(3.21)

Differentiating (3.21) along $u_{1_{1}}$, and applying (3.9) and (3.10), we
deduce

$\displaystyle-4(n-4)W^{2}+\frac{3n^{2}}{n-1}\varepsilon\varepsilon_{1}H^{2}-4(n-4)c\varepsilon_{1}=0,$

which together with (3.21) gives a polynomial equation of $H$, which tells us
that $H$ must be a constant, a contradiction.

_Case 2: $\nabla H$ is light-like._

We know $\alpha_{1}\geq 2$ in this case. Suppose
$\lambda_{1}=\cdots=\lambda_{p}=-\frac{n}{2}\varepsilon H$ and
$\lambda_{p+1}=\cdots=\lambda_{m}(\neq\\!-\frac{n}{2}\varepsilon H)$. Denote
$l=\alpha_{1}+\cdots+\alpha_{p}$, then $\alpha_{p+1}+\cdots+\alpha_{m}=n-l$
since $\alpha_{1}+\cdots+\alpha_{m}=n$. Combining
$\sum_{i=1}^{m}\alpha_{i}\lambda_{i}=\text{tr}A=n\varepsilon H$, we get
$\lambda_{m}=\frac{(2+l)n\varepsilon H}{2(n-l)}$. It follows from the seventh
equation of (3.5) and (3.7) that

$\displaystyle\Gamma_{i_{1}1_{\alpha_{1}}}^{i_{1}}\\!=0,\ \ \
\sum_{a=1}^{\alpha_{i}}\left(\frac{-n(n+2)\varepsilon
H}{2(n-l)}\Gamma_{i_{a}k_{d}}^{i_{a}}+\Gamma_{i_{a}k_{d+1}}^{i_{a}}\right)=0,$
(3.22)

for $2\leq i\leq m$, $1\leq k\leq p$ and $k_{d}\neq 1_{1}$. As
$\Gamma_{i_{a}1_{\alpha_{1}}}^{j_{b}}\\!=\Gamma_{i_{a+1}1_{\alpha_{1}}}^{j_{b+1}}$
for $j\in[i]$, $i,j\neq 1$ (see (3.5)), we derive from (3.6) and (3.8) that

$\displaystyle\sum_{a=2}^{\alpha_{i}}\sum_{1\leq
b<a}\Gamma_{i_{a}1_{d}}^{i_{b}}\Gamma_{i_{b}1_{\alpha_{1}}}^{i_{a}}\\!=\Gamma_{i_{1}1_{\alpha_{1}}}^{i_{2}}A_{i,1,1_{d}}+\Gamma_{i_{1}1_{\alpha_{1}}}^{i_{3}}A_{i,2,1_{d}}+\cdots+\Gamma_{i_{1}1_{\alpha_{1}}}^{i_{\alpha_{i}}}A_{i,\alpha_{i}-1,1_{d}}=0,$
$\displaystyle\sum_{a=1}^{\alpha_{i}}\\!\sum_{\tiny\begin{array}[]{c}\\!1\\!\leq\\!b\\!\leq\\!\alpha_{k}\\!\\\
\\!b\leq
a\\!\end{array}}\\!\Gamma_{i_{a}1_{d}}^{k_{b}}\Gamma_{k_{b}1_{\alpha_{1}}}^{i_{a}}\\!=\\!\Gamma_{k_{1}1_{\alpha_{1}}}^{i_{1}}\\!B_{i,k,0,1_{d}}\\!+\\!\Gamma_{k_{1}1_{\alpha_{1}}}^{i_{2}}\\!B_{i,k,1,1_{d}}\\!+\\!\cdots+\\!\Gamma_{k_{1}1_{\alpha_{1}}}^{i_{\alpha_{i}}}\\!B_{i,k,\alpha_{i}-1,1_{d}}\\!=\\!0,$

for $p+1\leq i,k\leq m$ and $i\neq k$.

Using the Gauss equation for $\langle
R(u_{1_{2}},u_{i_{a}})u_{1_{\alpha_{1}}}\\!,u_{i_{\alpha_{i}-a+1}}\rangle$ and
$\langle
R(u_{1_{1}},u_{i_{a}})u_{1_{\alpha_{1}}}\\!,u_{i_{\alpha_{i}-a+1}}\rangle$ for
$p+1\leq i\leq m$, and taking sum of the results over $i$ and $a$, as well as
combining (3.3), (3.5), (3.22) and the above two equations, we deduce

$\displaystyle\sum_{i=p+1}^{m}\left(\sum_{d=3}^{\alpha_{1}-1}\Gamma_{1_{2}1_{\alpha_{1}}}^{1_{d}}\\!\sum_{a=1}^{\alpha_{i}}\\!\Gamma_{i_{a}1_{d}}^{i_{a}}+\sum_{k=2}^{p}\sum_{d=2}^{\alpha_{k}}\Gamma_{1_{2}1_{\alpha_{1}}}^{k_{d}}\\!\sum_{a=1}^{\alpha_{i}}\\!\Gamma_{i_{a}k_{d}}^{i_{a}}\right)=0,$
(3.23)
$\displaystyle\sum_{i=p+1}^{m}\\!\left(\sum_{d=2}^{\alpha_{1}-1}\\!\Gamma_{1_{1}1_{\alpha_{1}}}^{1_{d}}\\!\sum_{a=1}^{\alpha_{i}}\\!\Gamma_{i_{a}1_{d}}^{i_{a}}\\!+\\!\sum_{k=2}^{p}\sum_{d=1}^{\alpha_{k}}\\!\Gamma_{1_{1}1_{\alpha_{1}}}^{k_{d}}\\!\sum_{a=1}^{\alpha_{i}}\\!\Gamma_{i_{a}k_{d}}^{i_{a}}\right)\\!=\\!(n-l)c\varepsilon_{1}\\!-\frac{1}{4}(l+2)n^{2}\varepsilon\varepsilon_{1}H^{2}.$
(3.24)

Multiplying both sides of (3.24) by $\frac{n(n+2)\varepsilon H}{2(n-l)}$, and
combining (3.22), it gives

$\displaystyle\sum_{i=p+1}^{m}\left(\sum_{d=2}^{\alpha_{1}-2}\Gamma_{1_{1}1_{\alpha_{1}}}^{1_{d}}\sum_{a=1}^{\alpha_{i}}\Gamma_{i_{a}1_{d+1}}^{i_{a}}+\sum_{k=2}^{p}\sum_{d=1}^{\alpha_{k}-1}\Gamma_{1_{1}1_{\alpha_{1}}}^{k_{d}}\sum_{a=1}^{\alpha_{i}}\Gamma_{i_{a}k_{d+1}}^{i_{a}}\right)$
(3.25) $\displaystyle=\frac{n(n+2)\varepsilon
H}{2}\left(c\varepsilon_{1}-\frac{(l+2)n^{2}\varepsilon\varepsilon_{1}H^{2}}{4(n-l)}\right).$

By (3.23), equation (3.25) reduces to

$4(n-l)c\varepsilon_{1}-(2+l)n^{2}\varepsilon\varepsilon_{1}H^{2}=0.$

Therefore, $H$ is a constant, a contradiction. $\hfill\square$

Based on the results of Theorem 3.1, in the following, we estimate the range
of that constant mean curvature (see Theorem 3.3), and compute that constant
for the special case that the the algebraic and geometric multiplicities of
some principal curvature coincide (see Theorem 3.4).

We illustrate that under the assumption that the number of distinct principal
curvatures is not larger than two, the principal curvatures are all real or
imaginary, since imaginary principal curvatures are appear as conjugate pairs.
As before, $\varepsilon$ is the inner product of the normal vector field $\xi$
with itself.

Theorem 3.3  _Let $M^{n}_{r}$ be a non-minimal PMCV hypersurface of
$N^{n+1}_{s}(c)(c\neq 0)$, with PMCV constant $\lambda$._

  * •

_When_ $M^{n}_{r}$ has one or two distinct real principal curvatures, we have
$H^{2}\leq\frac{\varepsilon\lambda}{n}$ with $\varepsilon\lambda>0$. The
equality holds if and only if the principal curvatures are all equal;

  * •

_When_ $M^{n}_{r}$ has a pair of imaginary principal curvatures, we have
$H^{2}>\frac{\varepsilon\lambda}{n}$.

Proof Suppose $\mu$ and $\nu$ are distinct principal curvatures of
$M^{n}_{r}$, with multiplicities $l$ $(1\leq l\leq n)$ and $n-l$.

When $\mu$ and $\nu$ are real, we get from (3.1) and (3.2) that
$n^{2}H^{2}=(l\mu+(n-l)\nu)^{2}\leq
n(l\mu^{2}+(n-l)\nu^{2})=n\varepsilon\lambda$. So, we have
$\varepsilon\lambda>0$ and $H^{2}\leq\frac{\varepsilon\lambda}{n}$, with the
equality holding if and only if $l=n$.

When $\mu$ and $\nu$ are a pair of imaginary numbers with multiply
$l=\frac{n}{2}$, we suppose $\mu=\gamma+\tau\sqrt{-1}$ and
$\nu=\gamma-\tau\sqrt{-1}$, with $\tau\neq 0$. Then (3.1) and (3.2) reduces to
$\gamma=\varepsilon H$ and $n(\gamma^{2}-\tau^{2})=\varepsilon\lambda$. It
follows that
$nH^{2}>nH^{2}-n\tau^{2}=n(\gamma^{2}-\tau^{2})=\varepsilon\lambda$, i.e.
$H^{2}>\frac{\varepsilon\lambda}{n}$. $\hfill\square$

Remark 3.4 We point out that the case that $M^{n}_{r}$ has a pair of imaginary
principal curvatures in Theorem 3.3 does not exist when $n$ is an odd number.

In particular, when the algebraic and geometric multiplicities of some
principal curvature coincide, we can compute the value of $H$.

Theorem 3.5  _Let $M^{n}_{r}$ be a non-minimal PMCV hypersurface of
$N^{n+1}_{s}(c)(c\neq 0)$, with PMCV constant $\lambda$. Suppose that
$M^{n}_{r}$ has two distinct principal curvatures $\mu$ (multiply $l$) and
$\nu$, and the algebraic and geometric multiplicities of $\mu$ or $\nu$
coincide, then one of the following holds._

  * •

_When_ $\mu$ and $\nu$ are real, we have

$H^{2}=\frac{1}{2n^{2}}\left[n\varepsilon\lambda-4l(n-l)c\varepsilon\pm(2l-n)\sqrt{\lambda^{2}-4l(n-l)c^{2}}\right],$
$\mu^{2}=\frac{\varepsilon\lambda\pm\sqrt{\lambda^{2}-4l(n-l)c^{2}}}{2l},\
\nu^{2}=\frac{\varepsilon\lambda\mp\sqrt{\lambda^{2}-4l(n-l)c^{2}}}{2(n-l)},$

_with_ $\varepsilon\lambda\geq 2\sqrt{l(n-l)}|c|$. In particular, if
$\varepsilon\lambda=n|c|$, then $H^{2}=-\frac{4l(n-l)c\varepsilon}{n^{2}}$,
$\mu^{2}=-\frac{(n-l)c\varepsilon}{l}$, $\nu^{2}=-\frac{lc\varepsilon}{n-l}$
for $c\varepsilon<0$, or $H^{2}=\frac{(2l-n)^{2}c\varepsilon}{n^{2}}$,
$\mu^{2}=\nu^{2}=c\varepsilon$ for $c\varepsilon>0$.

  * •

_When_ $\mu$ and $\nu$ are imaginary, we have
$H^{2}=\frac{1}{2n}(\varepsilon\lambda-nc\varepsilon)$ with $c\varepsilon<0$
and $|\varepsilon\lambda|<-nc\varepsilon$.

Proof According to Theorem 3.1, $M^{n}_{r}$ is locally isoparametric. Then, by
the basic identity of Cartan in [10, Theorem 2.9], we obtain
$c+\varepsilon\mu\nu=0$.

(i) When $\mu$ and $\nu$ are real, by applying (3.1), (3.2) and
$c+\varepsilon\mu\nu=0$, we have

$\begin{cases}\varepsilon\lambda+2\sqrt{l(n-l)}c\varepsilon=(\sqrt{l}\mu-\sqrt{n-l}\nu)^{2}\geq
0,\\\
\varepsilon\lambda-2\sqrt{l(n-l)}c\varepsilon=(\sqrt{l}\mu+\sqrt{n-l}\nu)^{2}\geq
0,\end{cases}$

which implies $\varepsilon\lambda\geq 2\sqrt{l(n-l)}|c|$. Calculate $\mu^{2}$,
$\nu^{2}$ and $H^{2}$ from (3.1), (3.2) and $c+\varepsilon\mu\nu=0$, we derive

$\mu^{2}=\frac{\varepsilon\lambda\pm\sqrt{\lambda^{2}-4l(n-l)c^{2}}}{2l},\ \
\nu^{2}=\frac{\varepsilon\lambda\mp\sqrt{\lambda^{2}-4l(n-l)c^{2}}}{2(n-l)},$
$H^{2}=\frac{1}{2n^{2}}\left[n\varepsilon\lambda-4l(n-l)c\varepsilon\pm(2l-n)\sqrt{\lambda^{2}-4l(n-l)c^{2}}\right].$
(3.26)

Specially, for the case $\varepsilon\lambda=n|c|$, we check the values in the
above equations:

  * •

when $c\varepsilon>0$,

    * $\bullet$

$\mu^{2}=\nu^{2}=c\varepsilon$ and
$H^{2}=\frac{(2l-n)^{2}c\varepsilon}{n^{2}}$;

    * $\bullet$

$\mu^{2}=\frac{n-l}{l}c\varepsilon$, $\nu^{2}=\frac{l}{n-l}c\varepsilon$ and
$H^{2}=0$, a contradiction.

  * •

when $c\varepsilon<0$, we have

    * $\bullet$

$\mu^{2}=-\frac{n-l}{l}c\varepsilon$, $\nu^{2}=-\frac{l}{n-l}c\varepsilon$ and
$H^{2}=-\frac{4l(n-l)}{n^{2}}c\varepsilon$;

    * $\bullet$

$\mu^{2}=\nu^{2}=H^{2}=-c\varepsilon$, which contradicts to $\mu\neq\nu$.

By the way, we verify that the right-hand side of (3.26) for the case
$\varepsilon\lambda\neq n|c|$ is larger than $0$ and less than
$\frac{\varepsilon\lambda}{n}$. In other words, the values of $H^{2}$ are all
being in the range given by Theorem 3.3. Indeed, by using
$n\varepsilon\lambda\geq 2n\sqrt{l(n-l)}|c|\geq 4l(n-l)|c|$ and

$\frac{[n\varepsilon\lambda\pm
4l(n-l)c\varepsilon]^{2}}{(n-2l)^{2}[\lambda^{2}-4l(n-l)c^{2}]}=1+\frac{4l(n-l)(\lambda\pm
nc)^{2}}{(n-2l)^{2}[\lambda^{2}-4l(n-l)c^{2}]}>1,$

we easily derive $n\varepsilon\lambda\pm
4l(n-l)c\varepsilon>\pm(2l-n)\sqrt{\lambda^{2}-4l(4-l)c^{2}},$ which leads to

$0<\frac{1}{2n^{2}}[n\varepsilon\lambda-4l(n-l)c\varepsilon\pm(2l-n)\sqrt{\lambda^{2}-4l(4-l)c^{2}}]<\frac{\varepsilon\lambda}{n}.$

(ii) When $\mu$ and $\nu$ are imaginary, we suppose
$\mu=\gamma+\tau\sqrt{-1}$ and $\nu=\gamma-\tau\sqrt{-1}$. (3.1), (3.2) and
$c+\varepsilon\mu\nu=0$ be rewritten as

$\gamma=\varepsilon H,\ \varepsilon\lambda=n(\gamma^{2}-\tau^{2}),\
-c\varepsilon=\gamma^{2}+\tau^{2}>0.$

Then, $-nc\varepsilon-\varepsilon\lambda=2n\tau^{2}>0\ \text{and}\
\varepsilon\lambda-nc\varepsilon=2n\gamma^{2}>0$ follows. Therefore,
$|\varepsilon\lambda|<-nc\varepsilon$ and
$H^{2}=\gamma^{2}=\frac{\varepsilon\lambda-nc\varepsilon}{2n}.$
$\hfill\square$

##  4 Classification of non-minimal Lorentzian PMCV hypersurfaces in
$\mathbb{H}^{n+1}_{1}$ and $\mathbb{S}^{n+1}_{1}$

In this section, applying the results in Section 3 and the classification
theory of isoparametric hypersurfaces, we classify non-minimal Lorentzian PMCV
hypersurfaces of $\mathbb{H}^{n+1}_{1}(-1)$ and $\mathbb{S}^{n+1}_{1}(1)\
(n\geq 3)$ with at most two distinct principal curvatures. The classification
results for $n=2$ have been provided in [6].

According to [13], for Lorentzian hypersurface $M^{n}_{1}$ in
$N_{1}^{n+1}(c)$, the shape operator $A$ and the metric matrix $G$ have the
following four possible forms:

$\begin{array}[]{|c|c|c|c|}\hline\cr\\!\textbf{Form I}\\!&\\!\textbf{Form
I\\!I}\\!&\\!\textbf{Form I\\!I\\!I}\\!&\\!\textbf{Form I\\!V}\\\
\hline\cr\\!A=\\!\left(\\!\begin{array}[]{ccc}\\!\lambda_{1}\\!&\\!&\\!\\\
\\!\\!&\\!\ddots\\!&\\!\\\
\\!&\\!&\\!\lambda_{n}\\!\end{array}\\!\right)\\!&\\!A=\\!\left(\\!\begin{array}[]{ccc}\\!\lambda_{1}\\!&\\!&\\!\\\
\\!1\\!&\\!\lambda_{1}\\!&\\!\\\
\\!&\\!&\\!D_{n-2}\\!\end{array}\\!\right)\\!&\\!A=\\!\left(\\!\begin{array}[]{cccc}\\!\lambda_{1}\\!&\\!&\\!&\\!\\\
\\!1\\!&\\!\lambda_{1}\\!&\\!&\\!\\\ \\!&\\!1\\!&\\!\lambda_{1}\\!&\\!\\\
\\!&\\!&\\!&\\!D_{n-3}\\!\end{array}\\!\right)\\!&\\!A=\\!\left(\\!\begin{array}[]{ccc}\\!D_{n-2}\\!&\\!&\\!\\\
\\!&\\!\gamma\\!&\\!\tau\\!\\\
\\!&\\!-\tau\\!&\\!\gamma\\!\end{array}\\!\right)\\!\\\
\\!G\\!=\\!\left(\\!\begin{array}[]{cc}\\!I_{n-1}\\!&\\!\\\
\\!&\\!-1\\!\end{array}\\!\right)\\!&\\!G\\!=\\!\left(\\!\begin{array}[]{ccc}\\!0\\!&\\!1\\!&\\!\\\
\\!1\\!&\\!0\\!&\\!\\\
\\!&\\!&\\!I_{n-2}\\!\end{array}\\!\right)\\!&\\!G\\!=\\!\left(\\!\begin{array}[]{cccc}\\!0\\!&\\!0\\!&\\!1\\!&\\!\\\
\\!0\\!&\\!1\\!&\\!0\\!&\\!\\\ \\!1\\!&\\!0\\!&\\!0\\!&\\!\\\
\\!&\\!&\\!&\\!I_{n-3}\\!\end{array}\\!\right)\\!&\\!G\\!=\\!\left(\\!\begin{array}[]{ccc}\\!I_{n-2}\\!&\\!&\\!\\\
\\!&\\!1\\!&\\!0\\!\\\ \\!&\\!0\\!&-1\\!\end{array}\\!\right)\\!\\\
\hline\cr\end{array},$

where $D_{n-2}\\!=\\!\text{diag}\\{\lambda_{2},\cdots,\lambda_{n-1}\\}$,
$D_{n-3}\\!=\\!\text{diag}\\{\lambda_{2},\cdots,\lambda_{n-2}\\}$ and $I$ the
identity matrix. For simplicity, a hypersurface whose shape operator has the
form I (II, III or IV) is called the type I (II, III or IV)
hypersurface.

Observe the form IV, we find $A$ has at least three distinct eigenvalues when
$n\geq 3$. For the form I, the classification result of non-minimal PMCV
hypersurfaces with at most two distinct principal curvatures has been obtained
in [4]. So, we only need to discuss the type II and III hypersurfaces.

From Theorem 3.1, we know that PMCV hypersurfaces of
$\mathbb{H}^{n+1}_{1}(-1)$ and $\mathbb{S}^{n+1}_{1}(1)$ with at most two
distinct principal curvatures are isoparametric. In turn, we can show that the
following type II and III isoparametric hypersurfaces in
$\mathbb{H}^{n+1}_{1}(1)$ and $\mathbb{S}^{n+1}_{1}(1)$ (cf. [11, 14]) are
also PMCV hypersurfaces with at most two distinct principal curvatures.

Example 4.1 (Type II hypersurfaces in $\mathbb{H}^{n+1}_{1}(-1)$, cf. [14])

Let $\\{E_{1}(t),\ E_{2}(t),\ \cdots,\ E_{n+2}(t)\\}$ be a single parameter
frame of $\mathbb{E}^{n+2}_{2}$ defined on the interval $I$ with all scalar
products zero except $\langle E_{1},E_{2}\rangle=\langle E_{i},E_{i}\rangle=1\
(3\leq i\leq n+1)$ and $\langle E_{n+2},E_{n+2}\rangle=-1$, and satisfies

$\displaystyle E_{1}=E_{n+2}^{{}^{\prime}},\ E_{n+1}^{{}^{\prime}}=\mu
E_{1}+BE_{2},$ $\displaystyle
E_{\alpha}^{{}^{\prime}}\in\text{span}\\{E_{2},E_{p+1},\cdots,E_{n}\\},\
p+1\leq\alpha\leq n,$

where $2\leq p\leq n$, $\mu$ is a nonzero real constant and $B$ is a nonzero
function of variable $t$. The parametrized hypersurfaces in
$\mathbb{H}^{n+1}_{1}(-1)$ given by

$\displaystyle x(t,y)=$
$\displaystyle\left(1+\frac{1}{2}\sum_{i=3}^{n}y_{i}^{2}\right)E_{n+2}-\frac{1}{2}\sum_{i=3}^{n}y_{i}^{2}E_{n+1}+\sum_{j=2}^{n}y_{j}E_{j},\
\text{when}\ \mu^{2}=1;$ $\displaystyle x(t,y)=$
$\displaystyle\text{sgn}(1-\mu^{2})\left[\frac{1}{(\mu^{2}-1)^{2}}-\sum_{i=3}^{p}\frac{y_{i}^{2}}{\mu^{2}-1}\right]^{\frac{1}{2}}(E_{n+2}-\mu
E_{n+1})+\sum_{j=2}^{n}y_{j}E_{j}$
$\displaystyle+\text{sgn}(\mu(1-\mu^{2}))\left[\frac{\mu^{2}}{(\mu^{2}-1)^{2}}+\sum_{\alpha=p+1}^{n}\frac{y_{\alpha}^{2}}{\mu^{2}-1}\right]^{\frac{1}{2}}(E_{n+1}-\mu
E_{n+2}),\ \text{when}\ \mu^{2}\neq 1,$

for $t\in I$ and $y=(y_{2},\cdots,y_{n})\in\mathbb{E}^{n-1}$, have at most two
distinct principal curvatures $\mu$ (with multiply $p$) and $\frac{1}{\mu}$,
and the shape operator $A$ with the form

$\left(\begin{array}[]{cccc}\mu&&&\\\ 1&\mu&&\\\ &&\mu I_{p-2}&\\\
&&&\frac{1}{\mu}I_{n-p}\end{array}\right).$

It follows that $H=\frac{1}{n}\text{tr}A=\frac{p\mu}{n}+\frac{n-p}{n\mu}$ is a
real constant and $\text{tr}A^{2}=p\mu^{2}+(n-p)\frac{1}{\mu^{2}}$. Then,
(2.2) and (2.3) hold for $\lambda=p\mu^{2}+(n-p)\frac{1}{\mu^{2}}$, i.e. the
type II parametrized Lorentzian hypersurfaces in $\mathbb{H}^{n+1}_{1}(-1)$
have proper mean curvature vector field.

Example 4.2 (Type III hypersurfaces in $\mathbb{H}^{n+1}_{1}(-1)$, cf. [14])

Let $\\{E_{1}(t),\ E_{2}(t),\ \cdots,\ E_{n+2}(t)\\}$ be a single parameter
pseudo-orthonormal frame of $\mathbb{E}^{n+2}_{2}$ defined on the interval $I$
with all scalar products zero except $\langle E_{1},E_{3}\rangle=\langle
E_{2},E_{2}\rangle=\langle E_{i},E_{i}\rangle=1\ (3\leq i\leq n+1)$ and
$\langle E_{n+1},E_{n+1}\rangle=-1$, and satisfies

$\displaystyle E_{1}=E_{n+2}^{{}^{\prime}},\
E_{\alpha}^{{}^{\prime}}\in\text{span}\\{E_{3},E_{p+1},\cdots,E_{n}\\},\
p+1\leq\alpha\leq n\ \text{and}\ E_{n+1}^{{}^{\prime}}=\mu E_{1}+BE_{2},$

where $3\leq p\leq n$, $\mu$ is a nonzero real constant and $B$ is a nonzero
function of variable $t$. The parametrized hypersurfaces in
$\mathbb{H}^{n+1}_{1}(-1)$ given by

$\displaystyle x(t,y)=$
$\displaystyle\left(1+\frac{1}{2}y_{2}^{2}+\frac{1}{2}\sum_{i=4}^{n}y_{i}^{2}\right)E_{n+2}-\frac{1}{2}(y_{2}^{2}+\sum_{i=4}^{n}y_{i}^{2})E_{n+1}+\sum_{j=2}^{n}y_{j}E_{j},\
\text{when}\ \mu^{2}=1;$ $\displaystyle x(t,y)=$
$\displaystyle\text{sgn}(1-\mu^{2})\left[\frac{1}{(\mu^{2}-1)^{2}}-\frac{y_{2}^{2}}{\mu^{2}-1}-\sum_{i=4}^{p}\frac{y_{i}^{2}}{\mu^{2}-1}\right]^{\frac{1}{2}}(E_{n+2}-\mu
E_{n+1})+\sum_{j=2}^{n}y_{j}E_{j}$
$\displaystyle+\text{sgn}(\mu(1-\mu^{2}))\left[\frac{\mu^{2}}{(\mu^{2}-1)^{2}}+\sum_{\alpha=p+1}^{n}\frac{y_{\alpha}^{2}}{\mu^{2}-1}\right]^{\frac{1}{2}}(E_{n+1}-\mu
E_{n+2}),\ \text{when}\ \mu^{2}\neq 1,$

for $t\in I$ and $y=(y_{2},\cdots,y_{n})\in\mathbb{E}^{n-1}$, have at most two
distinct principal curvatures, and the shape operator takes the form

$\left(\begin{array}[]{ccccc}\mu&&&&\\\ 1&\mu&&&\\\ &1&\mu&&\\\ &&&\mu
I_{p-3}&\\\ &&&&\frac{1}{\mu}I_{n-p}\end{array}\right).$

After following the process of Example 4.1, we can also verify that the non-
minimal type III parametrized hypersurfaces in $\mathbb{H}^{n+1}_{1}(-1)$
are PMCV hypersurfaces, with the PMCV constant
$\lambda=p\mu^{2}+(n-p)\frac{1}{\mu^{2}}$.

Example 4.3 (Type II hypersurfaces in $\mathbb{S}^{n+1}_{1}$(1), cf. [11])

Let $\\{E_{1}(t),\ E_{2}(t),\ \cdots,\ E_{n+2}(t)\\}$ be a single parameter
frame of $\mathbb{E}^{n+2}_{1}$ defined on the interval $I$ with all scalar
products zero except $\langle E_{1},E_{2}\rangle=\langle E_{i},E_{i}\rangle=1\
(3\leq i\leq n+2)$, and satisfies

$\begin{cases}E_{1}^{{}^{\prime}}=\sum_{i=3}^{n+2}B_{i-2}E_{i},\
E_{2}^{{}^{\prime}}=-E_{3},\ E_{3}^{{}^{\prime}}=E_{1}+B_{1}E_{2},\\\
E_{4}^{{}^{\prime}}=B_{2}E_{2},\ \cdots,\ E_{n+1}^{{}^{\prime}}=B_{n-1}E_{2},\
E_{n+2}^{{}^{\prime}}=B_{n}E_{2},\end{cases}$

where $B_{1},B_{2},\cdots,B_{n}$ are smooth functions on $I$ and
$\sum_{i=p}^{n}B_{i}^{2}>0$. For some $2\leq p\leq n$ and $\theta\in[0,2\pi]$,
$\theta\neq\frac{k\pi}{4}\ (k=1,3,5,7)$, the parametrized hypersurface in
$\mathbb{S}^{n+1}_{1}(1)$ given by

$x(t,v,y,z)=\cos(\theta-\frac{\pi}{4})\left(vE_{2}+\sum_{i=3}^{p+1}y_{i}E_{i}\right)-\sin(\theta-\frac{\pi}{4})\sum_{\alpha=p+2}^{n+2}z_{\alpha}E_{\alpha},$

for $t\in I$, $v\in\mathbb{R}$,
$y=(y_{3},\cdots,y_{p+1})\in\mathbb{S}^{p-2}_{+}\subset\mathbb{E}^{p-1}$ and
$z=(z_{p+2},\cdots,z_{n+2})\in\mathbb{S}^{n-p}\subset\mathbb{E}^{n-p+1}$, has
principal curvatures $\cot(\theta+\frac{\pi}{4})$ (with multiply $p$) and
$-\tan(\theta+\frac{\pi}{4})$, and the shape operator with the form

$\left(\begin{array}[]{cccc}\cot(\theta+\frac{\pi}{4})&&&\\\
1&\cot(\theta+\frac{\pi}{4})&&\\\ &&\cot(\theta+\frac{\pi}{4})I_{p-2}&\\\
&&&-\tan(\theta+\frac{\pi}{4})I_{n-p}\\\ \end{array}\right),\ 2\leq p\leq n.$

It follows from $nH=\text{tr}A$ that the mean curvature $H$ is a constant.
Combining
$\text{tr}A^{2}=p\cot^{2}(\theta+\frac{\pi}{4})+(n-p)\tan^{2}(\theta+\frac{\pi}{4})$,
it is easy to check that (2.2) and (2.3) hold for
$\lambda=p\cot^{2}(\theta+\frac{\pi}{4})+(n-p)\tan^{2}(\theta+\frac{\pi}{4})$.
Hence, the non-minimal type II parametrized hypersurface in
$\mathbb{S}^{n+1}_{1}(1)$ has proper mean curvature vector field.

Example 4.4 (Type III hypersurfaces in $\mathbb{S}^{n+1}_{1}(1)$, cf. [11])

Let $\\{E_{1},E_{2},\cdots,E_{n+1}\\}$ be a parameter frame of
$\mathbb{E}^{n+2}_{1}$ defined on the interval $I$ with all scalar products
zero except $\langle E_{1},E_{3}\rangle=\langle E_{2},E_{2}\rangle=\langle
E_{i},E_{i}\rangle=1\ (3\leq i\leq n+2)$, and satisfies

$\begin{cases}E_{1}^{{}^{\prime}}=B_{1}E_{2}+\sum_{i=4}^{n+2}B_{i-1}E_{i},\
E_{3}^{{}^{\prime}}=E_{2},\ E_{2}^{{}^{\prime}}=-E_{1}+B_{1}E_{3},\\\
E_{r}^{{}^{\prime}}=B_{r-1}E_{3}+\sum_{\alpha=p+2}^{n+2}C_{r\alpha}E_{\alpha},\
r=4,\cdots,p+1,\\\
E_{\beta}^{{}^{\prime}}=B_{\beta-1}E_{3}-\sum_{s=4}^{p+1}C_{s\beta}E_{s},\
\beta=p+2,\cdots,n+2,\\\ \end{cases}$

where $3\leq p\leq n$, $B_{i}\ (i=1,3,\cdots,n),C_{r\alpha}$
($r=4,\cdots,p+1$, $\alpha=p+2,\ \cdots,n+2$) are smooth functions on $I$ and
the matrix $(C_{r\alpha})\neq 0$. For some $\theta\in[0,2\pi]$,
$\theta\neq\frac{k\pi}{4}\ (k=1,3,5,7)$, the parametrized hypersurface in
$\mathbb{S}^{n+1}_{1}(1)$ given by

$\displaystyle x(t,v,y,z)=$
$\displaystyle\cos(\theta-\frac{\pi}{4})\left(vE_{3}+y_{3}E_{2}+\sum_{r=4}^{p+1}y_{r}E_{r}\right)-\sin(\theta-\frac{\pi}{4})\sum_{\alpha=p+2}^{n+2}z_{\alpha}E_{\alpha}$
$\displaystyle-\sqrt{2}\frac{\Psi}{y_{3}}\sin\theta E_{3},\
\Psi=\sum_{r=4}^{p+1}\sum_{\alpha=p+2}^{n+2}y_{r}z_{\alpha}C_{r\alpha},$

for $t\in I$, $v\in\mathbb{R}$,
$y=(y_{3},\cdots,y_{p+1})\in\mathbb{S}^{p-2}\subset\mathbb{E}^{p-1}$ and
$z=(z_{p+2},\cdots,z_{n+2})\in\mathbb{S}^{n-p}\subset\mathbb{E}^{n-p+1}$, has
the shape operator of the form

$\left(\begin{array}[]{ccccc}\cot(\theta+\frac{\pi}{4})&&&&\\\
1&\cot(\theta+\frac{\pi}{4})&&&\\\ &1&\cot(\theta+\frac{\pi}{4})&&\\\
&&&\cot(\theta+\frac{\pi}{4})I_{p-3}&\\\
&&&&-\tan(\theta+\frac{\pi}{4})I_{n-p}\\\ \end{array}\right).$

Similar to Example 4.3, we conclude that the non-minimal type III
parametrized hypersurface in $\mathbb{S}^{n+1}_{1}(1)$ is a PMCV one with the
PMCV constant
$\lambda=p\cot^{2}(\theta+\frac{\pi}{4})+(n-p)\tan^{2}(\theta+\frac{\pi}{4})$.

In the following, we give classification results of non-minimal type II and
III PMCV hypersurfaces with at most two distinct principal curvatures. In
view of the forms II and III, we know the multiply $l$ of some principal
curvature is equal or greater than two or three, respectively. Hence, we can
suppose $2\leq l\leq n$ for type II and $3\leq l\leq n$ for type III.

Theorem 4.5  _Let $M^{n}_{1}$ $(n\geq 3)$ be a non-minimal type II (or III)
PMCV hypersurface of $\mathbb{H}^{n+1}_{1}(-1)$ with the PMCV constant
$\lambda$. Suppose that $M^{n}_{1}$ has at most two distinct principal
curvatures with multiplicities $l$ and $n-l$, $2\leq l\leq n$ (or $3\leq l\leq
n$), then $\lambda>0$ for $l=n$, and $\lambda\geq 2\sqrt{l(n-l)}$ for $l<n$.
Furthermore, $M^{n}_{1}$ is locally congruent to that of Example 4.1 (or
Example 4.2) with $p=n$, $\mu^{2}=\frac{\lambda}{n}$ if $l=n$, or $p=l$,
$\mu^{2}=\frac{1}{2l}[\lambda\pm\sqrt{\lambda^{2}-4l(n-l)}]$ if $l>n$._

Proof We only treat type II hypersurfaces, the other ones are similar. Denote
by $\tilde{\mu}$ and $\tilde{\nu}$ the principal curvatures of $M^{n}_{1}$
with multiply $l$ and $n-l$.

When $l=n$, i.e. $M^{n}_{1}$ has a simple principal curvature, we obtain from
Theorem 3.3 that $\lambda>0$ and $H^{2}=\frac{\lambda}{n}$. Combining
$n\tilde{\mu}=\text{tr}A=n\varepsilon H$, we get
$\tilde{\mu}^{2}=\frac{\lambda}{n}$. Thus, $M^{n}_{1}$ is a totally umbilical
isoparametric hypersurface with the square of the principal curvature being
equal to $\frac{\lambda}{n}$. According to [14], it is locally congruent to
that of Example 4.1 with $p=n$, $\mu^{2}=\frac{\lambda}{n}$.

When $2\leq l<n$, we easily find from the form II that the geometric
multiplicity of $\tilde{\mu}$ is equal to the algebraic multiplicity $n-l$.
Applying Theorem 3.5, we derive that $\lambda\geq 2\sqrt{l(n-l)}$ and
$\tilde{\mu}^{2}=\frac{1}{2l}[\lambda\pm\sqrt{\lambda^{2}-4l(n-l)}]$. So,
$M^{n}_{1}$ is the isoparametric hypersurface with two distinct principal
curvatures (of multiplicities $l$ and $n-l$), and the square of one principal
curvature (with multiply $l$) being equal to
$\frac{1}{2l}[\lambda\pm\sqrt{\lambda^{2}-4l(n-l)}]$. The result follows from
[14, Theorem 4.2]. $\square$

Similar to the proof of Theorem 4.5, we obtain the following classification
result of non-minimal type II and III PMCV hypersurfaces in
$\mathbb{S}^{n+1}_{1}(1)$.

Theorem 4.6  _Let $M^{n}_{1}$ $(n\geq 3)$ be a non-minimal type II (or III)
PMCV hypersurface of $\mathbb{S}^{n+1}_{1}(1)$ with the PMCV constant
$\lambda$. Suppose that $M^{n}_{1}$ has at most two distinct principal
curvatures with multiplicities $l$ and $n-l$, $2\leq l\leq n$ (or $3\leq l\leq
n$), then $\lambda>0$ for $l=n$, and $\lambda\geq 2\sqrt{l(n-l)}$ for $l<n$.
Furthermore, $M^{n}_{1}$ is locally congruent to that of Example 4.3 (or
Example 4.4) with $p=n$, $\cot^{2}(\theta+\frac{\pi}{4})=\frac{\lambda}{n}$ if
$l=n$, or $p=l$,
$\cot^{2}(\theta+\frac{\pi}{4})=\frac{1}{2l}[\lambda\pm\sqrt{\lambda^{2}-4l(n-l)}]$
if $l>n$._

## References

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[1]Vishwa S. Parekh

1]University of Maryland Medical Intelligent Imaging (UM2ii) Center,
University of Maryland School of Medicine, Baltimore, MD

# One Copy Is All You Need: Resource-Efficient Streaming of Medical Imaging
Data at Scale

Pranav Kulkarni<EMAIL_ADDRESS>Adway Kanhere
<EMAIL_ADDRESS>Eliot Siegel<EMAIL_ADDRESS>Paul H. Yi
<EMAIL_ADDRESS><EMAIL_ADDRESS>[

###### Abstract

Large-scale medical imaging datasets have accelerated development of
artificial intelligence tools for clinical decision support. However, the
large size of these datasets is a bottleneck for users with limited storage
and bandwidth. Many users may not even require such large datasets as AI
models are often trained on lower resolution images. If users could directly
download at their desired resolution, storage and bandwidth requirements would
significantly decrease. However, it is impossible to anticipate every users’
requirements and impractical to store the data at multiple resolutions. What
if we could store images at a single resolution but send them at different
ones? We propose MIST, an open-source framework to operationalize progressive
resolution for streaming medical images at multiple resolutions from a single
high-resolution copy. We demonstrate that MIST can dramatically reduce imaging
infrastructure inefficiencies for hosting and streaming medical images by
$>$90%, while maintaining diagnostic quality for deep learning applications.

## Main

Figure 1: a, Illustration of the conventional approach to medical imaging data
curation. In this scenario, four users download a cloud-hosted liver CT
dataset for different use-cases. Each use case requires a different sized
image, yet the user must download the data at the baseline resolution before
processing them into their desired resolutions for further analysis, resulting
in inefficient use of bandwidth, storage, and compute. b, How can progressive
encoding address the computational, network and storage inefficiencies
associated with current computation infrastructures developed to host medical
imaging datasets? Progressively encoding the liver CT dataset produces a
storage saving of 4x for the host organization while reducing the total data
transferred from the host organization by 9x. Similarly, savings in the data
storage requirements for the users range from 4x–64x.

The curation of large-scale, open-source medical imaging databases has greatly
accelerated the development of artificial intelligence (AI) tools for clinical
decision support. The Cancer Imaging Archive (TCIA) is one of the largest
cancer imaging databases with 196 open-source datasets accessed by more than
768,000 users from 216 countries with monthly downloads exceeding 140 TB [1,
2]. The large size of these datasets is a major bottleneck for users with
limited computer storage capacity and internet connections (e.g., those in
resource-limited areas or working remotely) [3]. For example, consider a
hypothetical liver CT dataset with images of size 512x512xN hosted in the
NifTI format spanning a total storage space of 180 GB (Figure 1a). Each user
working with this dataset would be required to download the entire 180 GB data
to their machines for further processing. This would not only take more than
10 hours to download on a 100 Mbps internet connection, but also require a
user to have 180 GB free storage space available. These large image storage
sizes can quickly exhaust users’ resources, especially when working with
numerous datasets. Furthermore, many users may not even require such large
datasets, as medical images are frequently downsampled to resolutions much
lower than the original images (e.g., 224x224 for deep transfer learning);
such downsampling for the referenced liver CT dataset would reduce total
dataset size by 5x to 35 GB. If large datasets hosted by groups like the TCIA
could be downloaded directly at a desired resolution, storage and bandwidth
requirements for hosting and streaming medical images would dramatically
decrease, and the computational resources for extra image processing would be
eliminated.

These inefficiencies unfortunately scale with the number of users accessing a
database. For the example in Figure 1a, there would be an unnecessary data
transfer of 450 GB in addition to increased storage requirements and image
processing operations for data curation – all for just four clients in the
setup. When considering that the TCIA has more than 768,000 users, the
potential storage and bandwidth savings are enormous, in addition to reduced
carbon footprint and impact on the environment. The ability to store and send
images to users at their desired resolutions could significantly reduce
current imaging infrastructure inefficiencies by optimizing the distribution
of medical images on the cloud. However, it is impossible to anticipate the
requirements of every user accessing these datasets or store the data at
multiple resolutions as it will result in increased cloud storage costs and
unnecessary data duplication. But what if we could store the images at a
single resolution but send them at different resolutions?

Progressive resolution (PR) is a promising technology capable of delivering
images at multiple resolutions, even if they are stored at a single resolution
[4, 5]. PR has revolutionized the way we consume media content over the
internet, addressing similar infrastructural inefficiencies associated with
hosting and streaming consumer media. The core concept behind PR has enabled
streaming platforms like Netflix to optimize delivery of media at scale, by
streaming content at different resolutions depending on individual users’
devices (e.g., iPhone vs. 4K TV) and network speeds from a single high-
resolution copy [6]. However, the computational overhead of PR codecs has held
back its adoption in medical imaging beyond niche web-based applications [4].
PR has received recent interest within the medical imaging community with the
proposed integration of High-Throughput JPEG 2000 (HTJ2K) [7, 8] – a state-of-
the-art lossless PR codec that addresses its predecessors’ drawbacks – into
DICOMWeb, the industry standard for web-based medical imaging [9].
Furthermore, HTJ2K has been adopted by the recently launched Amazon Web
Services’ HealthLake Imaging as part of their toolkit for encoding medical
images for rapid retrieval [10]. Briefly, HTJ2K encodes images into a series
of progressive decompositions, each with an increasing level of detail, and
enables them to be decoded into different resolutions by selecting different
subset of bytes (i.e., partial bytestreams). In contrast, vast majority of
image codecs (e.g., JPEG) sequentially encode an image from the top-left to
the bottom-right pixels; this sequential encoding prevents images from being
decoded from a partial bytestream in the same way that HTJ2K allows (Figure
2).

Figure 2: a, A chest X-Ray image encoded using JPEG (sequential) and HTJ2K
(progressive) encoders. b, Decoding the chest X-ray image using partial
bytestreams from both the JPEG and HTJ2K encodings. The JPEG decoded image was
a partially visible image (left) whereas the HTJ2K decoded image was visible,
albeit lower in resolution (right).

Despite widespread and growing enthusiasm for the integration of HTJ2K in
medical imaging data standards, there is a critical unmet need to
operationalize PR for streaming medical imaging, with current software
difficult to use and lacking specificity for medical imaging data management.
To that end, we propose the Medical Image Streaming Toolkit (MIST), an open-
source framework that operationalizes PR for medical imaging data by
progressively encoding and streaming medical imaging datasets. MIST is built
on the scientific premise that PR can 1) dramatically reduce medical image
data infrastructure costs by reducing storage and data transfer bandwidth
requirements and 2) improve image data useability by making it more practical
for users to download this data, even with low storage or slow internet
connections (Figure 1b). In this work, we aim to demonstrate that MIST can
dramatically reduce medical image data infrastructure requirements for hosting
and streaming datasets while maintaining diagnostic image quality and
information for deep learning analyses and applications.

## Results

We developed MIST as an end-to-end toolkit with the capability to
progressively encode, decode, and stream medical imaging data over the
internet and consisting of two major components: 1) progressive encoder-
decoder framework and 2) progressive streaming framework, as shown in Figure
3.

### Progressive Encoder-Decoder Framework

The progressive encoding technology provides the basis for storing images at a
single resolution with the ability to send the images at different resolutions
as requested by the user. The MIST framework utilizes the HTJ2K codec to
enable progressive encode-decode capabilities, achieve a high compression
ratio, and enable rapid access to lower resolution versions of the original
high-resolution copy. MIST’s progressive encoder-decoder framework’s builds on
the capabilities of HTJ2K through two important capabilities: 1) multi-format
encode-decode capability to enable users to progressively encode both 2D and
3D medical imaging data acquired in different imaging formats (e.g., DICOM,
NifTI, PNG, etc.) across various imaging modalities (CT, MRI, X-ray) and 2)
stream optimization map to enable users to download images at their desired
resolutions directly from the high-resolution copy.

Figure 3: Illustration of the Medical Image Streaming Toolkit (MIST). a, The progressive encoder-decoder framework encodes the medical images (b) into progressively encoded HTJ2K images (c) and generates a stream optimization map (d) between different byte subsets and image resolutions. e, The progressive streaming framework enables the users (f-g) to request images at their desired resolutions by determining the appropriate partial bytestream and decoding them into desired imaging formats for further analysis. Using MIST, the left user (f) was able to download the entire liver data in 1.58 GB (as opposed to 18.43 GB) and the right user (g) was able to download the entire brain mpMRI data in 7.63 GB (as opposed to 133.73 GB) resulting in 15x bandwidth and storage savings. Table 1: Summary of the quantitative evaluation of MIST’s encoding, decoding, and streaming efficiency, compared to the conventional approach. For the host, MIST’s encoding efficiency and lossless decoding efficiency is measured. For the client, MIST’s streaming and lossy decoding efficiency is measured at the user’s requested resolution. Quantitative metrics reported as Mean $\pm$ SD. Dataset | NIH Chest X-Ray | MSD Liver | UPENN-GBM
---|---|---|---
Modality | X-ray | CT | mpMRI
Format | PNG | NifTI | DICOM
Compression | Lossless | Lossless | -
Storage Size (GB) |  |  |
Original | 42.39 | 18.34 | 133.37
HTJ2K | 40.72 (-3.95%) | 4.53 (-75.28%) | 34.10 (-74.43%)
Quantitative Metrics |  |  |
SSIM | $1.000\pm 0.0000$ | $0.999\pm 0.0002$ | $1.000\pm 0.0001$
MAD | $0.000\pm 0.0000$ | $0.001\pm 0.0001$ | $0.001\pm 0.0002$
Requested Resolution | 224x224 | 256x256 | 128x128
Data Transmitted (GB) |  |  |
Original | 42.39 | 18.34 | 133.37
HTJ2K | 3.53 (-91.68%) | 1.58 (-91.38%) | 7.63 (-94.28 %)
Quantitative Metrics |  |  |
SSIM |  0$.971\pm 0.0087$ | $0.959\pm 0.0150$ | $0.968\pm 0.0590$
MAD | $0.011\pm 0.0022$ | $0.010\pm 0.0038$ | $0.011\pm 0.0137$
|  |  |

We evaluated the progressive encoder-decoder framework using three datasets
sourced from different repositories. The first dataset is a multiparametric
brain MRI dataset sourced from the UPENN-GBM collection on TCIA [2, 11] and
comprising of 3,301 mpMRI studies in the DICOM format, encompassing a total of
133.37 GB of storage space. The second dataset is a liver CT dataset, sourced
from the MSD archive, consisting of 131 3D CT volumes in the NifTI format,
encompassing a total space of 18.34 GB [12]. The third dataset is the NIH
Chest X-Ray dataset [13] with 112,120 frontal-view chest X-rays in the PNG
format from 30,805 unique patients, encompassing a total of 42.39 GB of
storage space. All three datasets were first encoded using MIST’s progressive
encoder, followed by lossless decoding of each of the datasets back to their
original resolution. We measured the framework’s encoding efficiency across
all three imaging formats by comparing the storage space required to host the
original dataset and the progressively encoded dataset. Furthermore, we
quantitatively evaluated MIST’s lossless decoding capability by measuring the
structural similarity index measure (SSIM) and mean absolute deviation (MAD)
between the reconstructed images against the original images across all three
datasets. As shown in Table 1, the progressively encoded datasets produced
excellent encoding efficiency, compressing the brain mpMRI dataset by 74.43%
(3.91x) from 133.37 GB down to 34.10 GB, liver CT dataset by 75.28% (4.07x)
from 18.43 GB down to 4.53 GB, and the NIH Chest X-Ray dataset by 3.95%
(1.04x) from 42.39 GB to 40.72 GB. Furthermore, the images reconstructed back
to original resolution using the progressive decoder produced a mean SSIM of
1.00 and MAD of 0.00 across all datasets.

### Progressive Streaming Framework

The progressive streaming framework provides open-source tools and APIs that
allow users to interface with MIST’s progressively encoded imaging database
and download datasets at desired resolutions, consisting of a host API and a
client API, to facilitate this interaction. The host API interfaces with the
progressively encoded imaging database and the stream optimization map
database to splice and stream partial bytestreams that are sufficient to
reconstruct the images to resolution requested by the user. If the requested
resolution is not available, MIST automatically transmits the partial
bytestream corresponding to the closest sub-resolution available. This ability
to splice and stream partial bytestreams is vital component behind MIST’s data
transmission efficiency as currently available software offerings that utilize
PR are not equipped with the capability to splice, transmit, and decode
partial bytestreams. On the other hand, the client API will enable the user to
request and reconstruct the partial bytestreams directly into imaging formats
like NifTI, which can be seamlessly utilized by various open-source tools
including MONAI and nnUNet [14, 15, 16] for subsequent AI modeling and
analysis. In this study, we evaluate 1) the ability of the progressive
streaming framework to reduce the bandwidth and storage requirements for
streaming large-scale datasets over the network and 2) compare the performance
of AI models developed using progressively streamed data at the desired
resolution with AI models developed using conventionally downsampled data at
the same resolution.

Table 2: Summary of the performance of deep learning classification and segmentation models trained on conventionally downsampled and progressively streamed medical imaging datasets. Mean AUC scores reported with 95% C.I. in brackets. Dice scores reported as Mean $\pm$ SD. | Internal Test Set | External Test Set
---|---|---
Classification
Dataset | NIH Chest X-Ray | MIMIC-CXR-JPG
Format | PNG | JPEG
Sample Size, N | 22,330 | 377,110
Conventionally Downsampled Model |  |
Loss | 0.1502 | 0.3697
Mean AUC | 0.8135 [0.8063, 0.8199] | 0.7400 [0.7386, 0.7414]
p-value | - | -
Progressively Streamed Model |  |
Loss | 0.1498 | 0.3836
Mean AUC | 0.8166 [0.8090, 0.8242] | 0.7391 [0.7375, 0.7406]
p-value | 0.37 | 0.22
Segmentation |  |
Dataset | MSD Liver | BTCV
Format | NifTI | NifTI
Sample Size, N | 25 | 30
Conventionally Downsampled Model |  |
Dice Score |  0.9451 $\pm$ 0.0171 |  0.8705 $\pm$ 0.1106
p-value | - | -
Progressively Streamed Model |  |
Dice Score |  0.9541 $\pm$ 0.0093 |  0.8447 $\pm$ 0.1610
p-value |  $<$0.001 | 0.06
|  |

We used the progressively encoded NIH Chest X-ray, MSD liver CT, and UPENN-GBM
brain mpMRI datasets from the previous experiment to request and reconstruct
sub-resolution images from the partial bytestreams for each dataset.
Furthermore, we repeated the quantitative analysis to measure the SSIM and MAD
comparing the reconstructed sub-resolution images to the conventional approach
(i.e., downsampling the original dataset). In short, the NIH Chest X-ray
dataset was requested at a resolution of 224x224. Since this resolution was
unavailable in the stream optimization map, MIST automatically streamed
256x256 images, i.e., the closest sub-resolution available. The liver CT
dataset was requested, streamed, and decoded at a resolution of 256x256 since
the requested resolution was available. Similarly, the brain mpMRI dataset was
requested, streamed, and decoded at a resolution of 128x128. As shown in Table
1, the NIH Chest X-ray, liver CT, and brain mpMRI datasets were streamed using
3.53 GB, 1.58 GB, and 7.63 GB of data, i.e., 91.68%, 91.38%, and 94.28% less
data transmitted compared to the original dataset, respectively. In aggregate,
MIST dramatically reduced the amount of data transmitted over the network by
93.44% (15.24x) from 194.10 GB to 12.74 GB. Furthermore, the mean SSIM between
the conventionally downsampled and progressively streamed datasets was 0.97,
0.96, and 0.97 for the NIH Chest X-Ray, liver CT, and brain mpMRI dataset
respectively with a MAD of 0.01 across all three datasets.

The primary goal of developing the progressive streaming framework is to
reduce computational infrastructure and network bandwidth costs for
researchers training and validating AI models on large datasets. Thus, it is
important to evaluate the performance of AI models trained on progressively
streamed datasets in comparison to AI models trained using original datasets.
We compared the performance of deep learning classification and segmentation
models trained using progressively streamed NIH Chest X-ray and liver CT
datasets with models trained using conventionally downsampled datasets.

#### Deep Learning Classification

We trained two deep learning classifications models using transfer learning on
the NIH Chest X-Ray dataset. The first model was trained on the original PNG
images and the second model was trained on the progressively streamed images.
Both models were then evaluated on the original internal test set ($n=22,330$)
and externally tested on the MIMIC-CXR-JPG dataset ($n=377,110$) [17, 18]. The
performance of both models was measured by the binary cross-entropy (BCE) loss
and the mean area under the receiver operating characteristic curve (AUC)
across all observed disease labels. As shown in Table 2, we observe that the
classification model trained using progressively streamed images demonstrated
similar performance across the internal and external test sets with a mean AUC
of 0.82 ($p=0.37$) and 0.74 ($p=0.22$) respectively, when compared to the
model trained using conventionally downsampled images.

#### Deep Learning Segmentation

We trained two deep learning 3D segmentation models on the MSD liver CT
dataset. The first model was trained on the original NifTI volumes and the
second model was trained on progressively streamed volumes. Both models were
similarly evaluated on the original internal test set ($n=25$) with external
testing on the BTCV dataset ($n=30$) [19]. The performance of both models was
measured by the mean dice score. As shown in Table 2, we observe that the
segmentation model trained using progressively streamed volumes demonstrated
similar, if not better, performance across the internal test set with a mean
dice score of 0.95 ($p<0.001$). On the external BTCV test set, we observed a
slight, albeit non-significant, degradation to the performance of the model
trained using progressively streamed volumes with a mean dice score of 0.85
($p=0.06$).

## Discussion

Our results demonstrate that the scientific premise behind MIST holds true
with concrete results from three large-scale real-world medical imaging
datasets – i.e., MIST can efficiently compress and encode, stream, and decode
medical imaging data across different formats, modalities, and body parts,
while maintaining diagnostic quality for deep learning applications. We show
that MIST has the potential to dramatically reduce medical image data
infrastructure costs by reducing storage and bandwidth requirements to host
and stream data over the network, while making large medical imaging datasets
more accessible to users with limited storage and slow internet connections.
More importantly, by reducing the network and computational overhead of data
transmission and image preprocessing (e.g., downsampling) without affecting
further downstream applications for medical images, MIST has the potential to
significantly reduce the carbon footprint and environmental impact of large-
scale medical image data infrastructure.

In stark contrast to currently available software offerings and toolkits, MIST
provides two significant improvements – owing to its high-throughput and
resource-efficiency. First, MIST’s progressive encoder enables users to
progressively encode both 2D and 3D medical imaging data in different imaging
formats (PNG, DICOM, NifTI, etc.) across various modalities (MRI, CT, X-ray,
etc.). In contrast, current progressive encoding solutions only support
encoding of raw DICOM images [10] – a major limiting factor due to the
prominence of other imaging formats such as NifTI in further downstream
analyses and applications [12, 15, 14]. Finally, MIST has the capability to
splice, stream, and decode partial bytestreams using the stream optimization
map. Since the progressive encoding technology was not developed to transmit
partial bytestream but to deliver the complete bytestream with the goal of
enhancing the user experience in case of network interruptions (e.g., for
Netflix movie streaming), current tools lack the specificity to decode partial
bytestreams nor do they generate or store an encoding dictionary that maps
different byte subsets to different image resolutions, which is a key
technical requirement for reconstructing images from partial bytestreams.
Furthermore, MIST has the capability to decode the partial bytestream directly
into imaging formats such as NifTI for further downstream applications – a
missing link between current software offerings and successfully
operationalizing PR for medical imaging.

Unfortunately, at this stage of development, MIST is not devoid of its own
limitations that are an impediment to its deployment in current large-scale
medical image data infrastructures. Due to current progressive encoder-decoder
tools lacking the specificity required for medical images, imaging data stored
with high-precision floating point intensities suffer from a minimal loss of
quality during the encoding process – 0.001 MAD for the liver CT and brain
mpMRI datasets. While our results show that this loss of quality does not
negatively impact further downstream deep learning applications, it is crucial
to address this limitation prior to MIST’s deployment. Promisingly, the HTJ2K
codec supports encoding floating point data despite the limitations of current
encoding frameworks [8, 4]. On the other hand, 3D medical imaging data (e.g.,
NifTI) poses a challenge to integrating MIST in current cloud-based solutions.
Since the HTJ2K codec only supports encoding 2D data, there is an additional
overhead to correctly process 3D data for progressive encoding-decoding.
However, in our prior work, we have showed that even with this additional
overhead, encoding, streaming, and decoding of 3D imaging data using HTJ2K is
significantly faster and more resource-efficient than NifTI encoded volumes
[20]. Finally, future work is warranted to extensively validate the MIST
framework for not just deep learning applications, but also for further
downstream analyses (e.g. radiomics) and clinical use-cases through in-depth
diagnostic equivalence reader studies with board-certified radiologists to
evaluate.

Despite current limitations, our results demonstrate that MIST provides
substantial improvements compared to current software solutions and has the
potential to overhaul large-scale medical imaging data infrastructures.
Returning back to our hypothetical four-node liver CT set up in Figure 1a, if
we were to use MIST to progressively encode the entire dataset and then only
stream the partial bytestream required by each user for their particular use-
case, we would dramatically reduce the bandwidth, storage, and compute
utilization across all users without additional computational overhead. In
addition, the lossless compression provided by HTJ2K would also reduce the
storage requirement for the hosting organization (e.g., TCIA). As shown in
Figure 1b, the host organization would reduce their overall storage by 4x from
180 GB to 45 GB and their total data transmission by 9x from 720 GB to 77.8 GB
using MIST. Similarly, all users would observe tremendous savings in terms of
storage space required and bandwidth used to download the data (savings
between 4x–64x), eliminating additional computational overhead required to
downsample the imaging data. As cloud-based solutions for medical imaging data
infrastructure gain popularity, MIST presents an open-source, resource-
efficient, and scalable framework to meet the critically unmet need to
dramatically reduce infrastructural inefficiencies associated with hosting and
streaming medical images over the Internet.

## Methods

### High-Throughput JPEG 2000 (HTJ2K)

High-Throughput JPEG 2000 (HTJ2K) is an extension to the existing JPEG 2000
(J2K) codec, that – unlike the original JPEG standard – enables faster
decoding, higher quality, lossless encoding with high scalability, and
progressive decoding of images [4, 8]. The HTJ2K codec utilizes new
lightweight and efficient block-coder algorithm that yields up to 10-30x
increase in throughput for lossless coding and a 6-10% increase in efficiency
when compared to J2K [7]. Furthermore, HTJ2K’s lower computational complexity
and support for parallelism results in a throughput on par, or even better,
than the heavily optimized original JPEG standard in single and multi-threaded
applications [8].

Figure 4: Illustration of the ability of the HTJ2K codec for encoding a chest
X-ray dataset (a) and subsequently decoding the dataset at different
resolutions by selecting different subsets of the bytestream i.e., partial
bytestreams (b-f).

In short, imaging data is progressively encoded using lossless HTJ2K as $N$
decompositions, i.e., subsets of the bytestream, each with a native resolution
as a factor of $2^{N}$ of the full image resolution. The native resolution of
a decomposition refers to the proportion of pixels of the full resolution
image transmitted prior to any interpolation. Unlike the sequential streams of
the original JPEG standard, HTJ2K streams are transmitted as a sequence of
these $N$ decompositions. For example, a 2D chest X-ray with a resolution of
1024x1024 can be progressively encoded as 5 decompositions with native
resolutions 64x64, 128x128, 256x256, 512x512, and 1024x1024 (visualized in
Figure 4). This allows HTJ2K to both achieve a high compression ratio and
enable rapid access to lower resolution versions of the original high-
resolution image, thus minimizing data duplication.

### Medical Image Streaming Toolkit (MIST)

The Medical Imaging Streaming Toolkit (MIST) is an end-to-end toolkit with the
capability to progressively encode, decode, and stream medical imaging data
over the network, comprising of two major components: 1) progressive encoder-
decoder framework, and 2) progressive streaming framework, as previously shown
in Figure 3. Our implementation of MIST is open-source and available on GitHub
[21], with source code to replicate our study in a separate repository [22].

#### Progressive Encoder-Decoder Framework

The progressive encoder-decoder framework provides two important capabilities:
1) multi-format encode-decode and 2) stream optimization map. MIST’s multi-
format encode-decode capability enables users to progressively encode both 2D
and 3D medical imaging data acquired in different imaging formats (e.g.,
DICOM, NifTI, PNG, JPEG, etc.) across any imaging modalities (CT, MRI, X-Rays,
etc.) as a series of $N$ decompositions, as shown in Figure 4. We integrate
OpenJPH, an open-source implementation of the HTJ2K codec [23], within MIST to
interface with HTJ2K capabilities.

Correctly encoding imaging data across multiple formats and accounting for
variations within each format is critical to operationalizing PR for medical
imaging. In general, all imaging data is encoded by OpenJPH as lossless with
64x64 block size, $N$ decompositions (determined by Equation 1), and tile-part
markers to identify location of each decomposition (i.e. sub-resolution)
within the complete bytestream. While encoding 2D imaging data is
straightforward, there is an additional overhead to correctly encode 3D
imaging data. For example, the NifTI format encodes the entire 3D volume
within a single file. When encoding NifTI data, MIST splits and progressively
encodes each slice separately as a 2D image. Not only does this require
additional preprocessing during the encoding process, but also during the
decoding process as individual slices must be tracked, reconstructed, and
restitched to yield the final 3D volume. Fortunately, this process is
relatively straightforward as MIST strips DICOM and NifTI metadata prior to
encoding and tracks all relevant information within a manifest file, and
enables MIST to achieve a higher compression ratio than NifTI.

$\small
N=\left\lfloor\log_{2}\left(\frac{\min(X,Y)}{\alpha}\right)\right\rfloor$ (1)

where $N$ is the number of decompositions to encode, $(X,Y)$ is the resolution
of an image, and $\alpha$ is lower bound for the resolution of the first
decomposition (i.e., the smallest sub-resolution) such that it exceeds
$\alpha$. In our study, we arbitrarily set $\alpha=32$.

Due to limitations in current open source HTJ2K encode-decode frameworks,
there is a minor loss of image fidelity during progressive encoding. For
example, the OpenJPH library is written in C++ and only supports encoding and
decoding at 16-bit precision for unsigned integers (i.e. pixel values ranging
from 0 to 65,535) [23]. However, due to limitations in integrating OpenJPH
with Python, we were only able to maintain 8-bit precision (i.e. pixel values
ranging for 0 to 255) [24]. Consequently, prior to encoding, all imaging data
is resampled to fit within the [0, 255] range and original pixel value range
is stored in the manifest file. During decoding, the reconstructed image is
subsequently resampled back to the original data’s pixel value range. While
this severely limits the fidelity of our current setup, for future work, we
have a two-fold plan of action: 1) Extend precision to 16-bit in current
setup, and 2) Implement an end-to-end integration of OpenJPH in Python with
direct encode-decode capabilities to leverage HTJ2K’s 16-bit floating point
precision (i.e. values ranging from -65,504 to 65,504). Our implementation of
OpenJPH’s Python integration is open-source and available on GitHub [25].

After imaging data is encoded, a stream optimization map is generated. In
short, the stream optimization map enables rapid sub-resolution access by
mapping the sub-resolution to the partial bytestream necessary to decode the
imaging data at the sub-resolution (e.g., reconstructing image at 256x256
resolution requires the first 12 KB of the complete 354 KB bytestream). This
map is generated by locating each tile-part marker in the bytestream
(identified by bytes 0xFF90 and 0xFFD9 [26]), emphasizing the significance of
including tile-part markers during the encoding process. Similarly, each sub-
resolution is automatically calculated as a factor of $2^{N}$, where $N$ is
the number of decompositions (e.g., a 1024x1024 image consisting of 5
decompositions has sub-resolutions 64x64, 128x128, etc.).

#### Progressive Streaming Framework

The progressive streaming framework provides two interfaces to interact with
the progressive imaging database generated by the progressive encoder-decoder
framework: 1) host API, and 2) client API. In short, both APIs function
together to enable users to request imaging data at their desired resolutions.
The host API interfaces with the imaging database to splice (i.e., truncate)
and stream partial bytestreams to the user using the stream optimization map.
The client API uses these partial bytestream to decode and reconstruct medical
imaging data at the user’s requested resolution. If the user’s requested
resolution is not available in the stream optimization map, the client API
requests the closest sub-resolution. This enables MIST to account for all
possible applications, while providing the substantial improvements of
streaming partial bytestreams. Furthermore, the client API handles all
necessary conversions based on the user’s request into formats such as PNG,
JPEG, DICOM, NifTI, etc. For example, the client API can automatically
reconstruct and restitch imaging data into NifTI (including adapting all
affine transformations to ensure pixel coordinates map correctly to real-world
coordinates – a crucial step to maintain scale of anatomical parts for
clinical use-cases).

### Datasets

#### NIH Chest X-Ray 14

The NIH Chest X-Ray dataset [13] consists of 14 disease labels with
$n=112,120$ frontal-view chest x-rays from 30,805 unique patients,
encompassing a total size of 42.39 GB. All images within the dataset have a
fixed 1024x1024 resolution and are encoded in PNG (lossless) format.

#### MSD Liver

The Medical Segmentation Decathlon (MSD) dataset is a collection of 2,633 CT
and MRI studies spanning 10 body parts [12]. We selected the liver
segmentation subset of the MSD dataset, consisting of $n=131$ CT studies in
the NifTI format with a fixed resolution of 512x512xN (i.e., N is the varying
number of slices in each volume), encompassing a total size of 18.34 GB.

#### UPENN-GBM

The UPENN-GBM dataset is an extensive collection hosted on TCIA focusing on
glioblastoma patients and comprising of multiparametric MRI (mpMRI) studies,
patient demographics, clinical outcomes, and more [11, 1]. For our study, we
selected the brain mpMRI subset of the collection, containing $n=3,301$
studies from 630 unique patients in the raw DICOM format (i.e., implicit VR
endian) consisting of T1-pre, T1-post, FLAIR, and T2-weighted sequences
acquired at different resolutions ranging from 128x128 to 512x512,
encompassing a total of 133.7 GB of storage space.

#### MIMIC-CXR-JPG

The MIMIC-CXR-JPG dataset [18, 17] consists of 14 disease labels, seven of
which are shared with the NIH Chest X-Ray dataset, with $n=377,110$ frontal
and lateral chest x-rays from 65,379 unique patients. All images within the
dataset are encoded in JPEG format with resolutions ranging from 1470x1461 to
4280x3520. For our study, only the seven disease labels shared with the NIH
Chest X-Ray dataset – i.e., atelectasis, cardiomegaly, consolidation, edema,
pleural effusion, pneumonia, and pneumothorax – were considered. To handle
uncertain labels in the dataset, we chose the U-Zeros approach i.e. all
uncertain labels were treated as negatives.

#### BTCV

The Beyond The Cranial Vault (BTCV) dataset [19] is a multi-organ abdominal CT
dataset consisting of $n=30$ studies in the NifTI format with a fixed
resolution of 512x512xN and 13 segmentation labels for each volume. For our
study, only the liver segmentation masks for each volume were used.

### Experimental Design

In our study, we evaluate MIST across each of its capabilities, from
progressive encoding-decoding to its downstream applications in deep learning,
across different modalities and body parts, using three datasets: 1) NIH Chest
X-Ray dataset, 2) MSD Liver CT dataset, and 3) UPENN-GBM Brain mpMRI dataset.

#### Progressive Encoding-Decoding

We validate MIST’s progressive encoder-decoder framework by progressively
encoding all three datasets, followed by losslessly decoding each of the
datasets back to their original resolution. We measured the framework’s
encoding efficiency across all three datasets by comparing the storage space
required to host the original dataset and the progressively encoded dataset.
Furthermore, we quantitively evaluate MIST’s lossless decoding capability by
measuring the structural similarity index measure (SSIM), as described in Wang
et al. [27], and the mean absolute deviation (MAD), as described in Equation
2, between the reconstructed images and the original images across all three
datasets.

$\small\text{MAD}=\frac{1}{XY}\sum_{i=1}^{X}\sum_{j=1}^{Y}|M_{ij}-M^{\prime}_{ij}|$
(2)

where $M,M^{\prime}$ are two 2D images with resolution $(X,Y)$.

#### Progressive Streaming

We validate MIST’s progressive streaming framework by analyzing its effect on
deep learning classification and segmentation – two foundational downstream
use-cases for medical imaging data. In our study, for both tasks, we train two
distinct deep learning models using fixed architecture and hyperparameters
with the first model trained on progressively streamed images, while the
second model is trained on conventionally downsampled images. Both models are
subsequently evaluated on a held-out internal test set and an external test
set (each test set’s original format is used for testing).

Deep Learning Classification: We used a multi-label DenseNet121 model
architecture [28], initialized with ImageNet weights, and used transfer
learning to train the model on the NIH Chest X-Ray dataset. The dataset was
randomly divided into training (70%, $n=78,571$), validation (10%,
$n=11,219$), and testing (20%, $n=22,330$) splits, while ensuring no patient
appears in more than one split. For preprocessing, all images were downsampled
to 224x224 resolution, normalized between 0 and 1, and data augmentation
(e.g., random crop, rotation, contrast, etc.) was used. Both models were
trained, using a batch size of 64, in a two-fold manner: 1) The new
classification heads were warmed up, while the rest of the model weights
remained frozen, using transfer learning for 15 epochs with a learning rate of
5e-2. 2) The best performing model from the first round was chosen and fine-
tuned after unfreezing all model weights for 50 epochs with a learning rate of
5e-4 and early stopping after 10 epochs with no improvement. The best models
were chosen by the minimum BCE loss on the validation set. After the training
phase, both models were then evaluated on the held-out internal NIH Chest
X-Ray test set ($n=22,330$) and the external MIMIC-CXR-JPG ($n=377,110$) to
measure the BCE loss and mean AUC scores. In a multi-label scenario, the mean
AUC score is the macro average of the AUC for each disease label. For the NIH
Chest X-Ray test set, both metrics were measured on the complete set of 14
disease labels, while for the MIMIC-CXR-JPG dataset, only the seven common
disease labels were used. We statistically compared the mean AUC from each
test set across both models using bootstrapping and t-test. Statistical
significance was defined as $p<0.05$.

Deep Learning Segmentation: We used a 3D-UNet [29] using the MONAI framework
[14] for the segmentation of liver using the MSD Liver CT dataset. The model
architecture consisted of five two-convolutional residual units with down/up-
sampling by a factor of 2 at each unit. The dataset was randomly divided into
training ($n=106$) and validation/testing ($n=25$). For this study, all liver
tumor annotations were discarded, maintaining annotations only for the organ.
For preprocesing, all volumes were resized to 256x256x128 with constant voxel
spacing of $(1.5,1.5,2)$, intensity values were normalized between 0 and 1,
and data augmentation was used. Random foreground patches of size 128x128x32
were extracted from each volume such that the center voxel of each patch
belonged to either the foreground or background class. Prior to training, the
models were initialized using LeCun’s method [30]. Both models were trained
for 500 epochs with batch size of 2, an initial learning rate of 1e-4, and
cosine annealing learning rate scheduler [31]. The best models were chosen by
the highest mean Dice score on the validation set. After the training phase,
both models were evaluated on the internal MSD Liver test set ($n=25$) and the
external BTCV test set ($n=30$) to measure the mean Dice score between the
ground truth segmentation and the model’s prediction segmentation. We
statistically compared the mean Dice scores from each test set across both
models using a two-tailed paired t-test. Statistical significance was also
defined as $p<0.05$. While the sample sizes for segmentation are small
compared to classification, power analysis shows us that a sample size of
$n=25$ is sufficient for a two-tailed paired t-test with significance level of
$p<0.05$, with an effect size of 0.8 (Cohen’s d for large effect sizes), to
ensure at least 80% statistical power.

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