# The preservability
of the curvature-adaptedness
along the mean curvature flow

Naoyuki Koike

###### Abstract

In this paper, we investigate the preservability of the curvature-adaptedness
along the mean curvature flow starting from a compact curvature-adapted
hypersurface in locally symmetric spaces, where the curvature-adaptedness
means that the shape operator and the normal Jacobi operator of the
hypersurface commute.

##  1 Introduction

In 1996, K. Smoczyk ([S]) proved that the Lagrangianity is preserved along the
mean curvature flow starting from a compact Lagrangian submanifold in a
Calabi-Yau manifold. He proved this fact by showing that the squared norms of
the $2$-forms on the submanifolds induced from the fundamental $2$-form of the
Calabi-Yau manifold remain to vanish. In this paper, we investigate the
preservability of the curvature-adaptedness along the mean curvature flow
starting from a compact curvature-adapted hypersurface in locally symmetric
spaces, where the curvature-adaptedness of a hypersurface means that the shape
operator and the normal Jacobi operator of the hypersurface commute. See the
following paragraph about the precise definition of the curvature-adaptedness.
The strategy of the proof of the main theorem (Theorem B) of this paper is to
find a sufficient condition for the squared norm of the commutator of the
shape operator and the normal Jacobi operator to remains to vanish by
investigating the evolution of the squared norm. Our strategy is similar to
that of [S].

Throughout this paper, we assume that all manifolds are oriented. We shall
state the definition of the curvature-adaptedness of the hypersurface. Let
$(\widetilde{M},\widetilde{g})$ be an $(n+1)$-dimensional Riemannian manifold,
$M$ be a $n$-dimensional compact manifold and $f$ be an immersion of $M$ into
$\widetilde{M}$. Also, let $\xi$ be the unit normal vector field of $f$
compatible with the orientations of $M$ and $\widetilde{M}$. Denote by
$\widetilde{R}$ the curvature tensor of $\widetilde{M}$. Also, denote by
$T^{\perp}_{x}M$ the normal space of $f$ at $x(\in M)$, by $A$ the shape
tensor of $f$ for $-\xi$ and by $\nabla^{\perp}$ the normal connection of $f$.
If the shape operator $A$ commutes with the normal Jacobi operator
$\widetilde{R}(\xi):=f_{\ast}^{-1}\circ\widetilde{R}(\cdot,\xi)\xi\circ
f_{\ast}$, then $f$ is said to be curvature-adapted. This notion was
introduced by J. Berndt and L. Vanhecke ([BV]). All hypersurfaces in real
space forms are curvature-adapted and the curvature-adapted hypersurfaces in
the a complex projective space and the complex hyperbolic space are called
Hopf hypersurfaces. Note that the condition of the curvature-adapetdness is
stricter as the curvature of the ambient space is more complicate.

Next we shall state the definition of an isoparametric hypersurface. A
($C^{\infty}$-)function $\psi$ over $\widetilde{M}$ is said to be
isoparametric if it satisfied the following conditions:

(i) $||d\psi||^{2}=a(\psi)$ holds for some $C^{\infty}$-function $a$ over
${\mathbb{R}}$,

(ii) $\widetilde{\triangle}\psi=b(\psi)$ holds for some continuous function
$b$ over ${\mathbb{R}}$,

where $\widetilde{\triangle}$ is the Laplace-Beltrami operator of
$\widetilde{M}$.

The regular level sets of an isoprarametric function are called isoparametric
hypersurfaces in $\widetilde{M}$. Since the regular level sets of an
isoprarametric function are parallel to one another and they are of constant
mean curvature, the mean curvature flow starting from an isoparametric
hypersurface consists of the parallel hypersurfaces of the hypersurface. In
this paper, if a hypersurface $M$ and the parallel hypersurfaces sufficiently
close to $M$ are of constant mean curvature, then we call $M$ a locally
isoparametric hypersurface.

In 2012, T. Murphy ([M]) studied curvature-adapted hypersurfaces in a compact
symmetric space. He proved that a curvature-adapted hypersurface in a compact
symmetric space is isoparametric if and only if both the shape operator and
the normal Jacobi operator of the hypersurface have constant eigenvalues. In
2014, the author ([K2]) proved that curvature-adapted submanifolds with
maximal flat section in a symmetric space are principal orbits of the isotropy
action of the symmetric space under certain conditions, where “submanifold
with maximal flat section” means that the normal umbrellas of the submanifold
are maximal dimensional flat totally geodesic submanifolds in the ambient
symmetric space. Since principal orbits of the isotropy action are curvature-
adapted isoparametric submanifolds and they are parallel to one another, the
mean curvature flow starting from a principal orbit of the isotropy action
consists of principal orbits of the isotropy action. Hence the curvature-
adaptedness is preserved along the mean curvature flow starting from the
principal orbit. From these facts, we can derive that the curvature-
adaptedness is preserved along the mean curvature flow starting from a
curvature-adapted submanifold with maximal flat section satisfying the
conditions.

Next we shall state the definitions of the mean curvature flow and the
backward mean curvature flow. Let $M,(\widetilde{M},\widetilde{g})$ and $f$ be
as above, and $\\{f_{t}\\}_{t\in[0,T)}$ be a $C^{\infty}$-family of immersions
of $M$ into $\widetilde{M}$ and $\xi_{t}$ be the unit normal vector field of
$f_{t}$ compatible with the orietations of $M$ and $\widetilde{M}$. Denote by
$H_{t}$ the mean curvature of $f_{t}$ for $-\xi_{t}$. Define a map
$F:M\times[0,T)\to V$ by $F(x,t):=f_{t}(x)$ ($(x,t)\in M\times[0,T)$). This
family $\\{f_{t}\\}_{t\in[0,T)}$ is called the mean curvature flow starting
from $f$ if $f_{0}=f$ and if the following evolution equation hold:

$\frac{\partial F}{\partial t}=-H_{t}\xi_{t}.$ $None$

Also, this family $\\{f_{t}\\}_{t\in[0,T)}$ is called the backward mean
curvature flow starting from $f$ if $f_{0}=f$ and if the following evolution
equation hold:

$\frac{\partial F}{\partial t}=H_{t}\xi_{t}.$ $None$

Note that G. Huisken ([Hu1, Hu2]) initiated the study of the mean curvature
flow starting from a hypersurface as the evolution of immersions. Also, B.
Andrews and C. Baker ([AB]) studied the mean curvature flow starting from a
submanifold (of general codimension) as the evolution of immersions in the
aspect of the theory of the vector bundle.

In this paper, we shall tackle the following question:

Question 1. In what case, is the curvature-adaptedness preserved along the
mean curvature flow starting from $f$ if $f$ is curvature-adapted?

First, we derive the following result for this question.

Theorem A. Assume that $(\widetilde{M},\widetilde{g})$ is an
$(n+1)$-dimensional locally symmetric space. Let $f$ be an immersion of an
$n$-dimensional compact manifold $M$ into $\widetilde{M}$ and
$\\{f_{t}\\}_{t\in[0,T)}$ the mean curvature flow starting from $f$. If $f$ is
curvature-adapted and if both the shape operator and the normal Jacobi
operator of $f$ have constant eigenvalues, then $f$ is locally isoprarametric
and $f_{t}$ remains to be curvature-adapted and locally isoparametric for all
$t\in[0,T)$.

We shall prepare some notations to state the main result in this paper. Let
$f$ be an immersion of an $n$-dimensional compact manifold $M$ into a
$(n+1)$-dimensional Riemannian manifold $\widetilde{M}$ and
$\\{f_{t}\\}_{t\in[0,T)}$ the mean curvature flow starting from $f$. Denote by
$g_{t}$ the induced metric on $M$ by $f_{t}$, and by $\nabla^{t}$ and $R_{t}$
the Levi-Civita connection and the curvature tensor of $g_{t}$, respectively.
Set $S_{t}:=[A_{t},\widetilde{R}(\xi_{t})]$ (the commutator of $A_{t}$ and
$\widetilde{R}(\xi_{t})$) and define non-negative functions $\rho_{t}$ ($0\leq
t<T$) over $M$ by

$\rho_{t}:=-{\rm Tr}(S_{t}^{2})(={\rm Tr}(S_{t}^{\ast_{t}}\circ
S_{t})=||S_{t}||_{t}^{2}(\geq 0)),$ $None$

where $A_{t}$ and $\widetilde{R}(\xi_{t})$ are the shape operator and the
normal Jacobi operator of $f_{t}$, respectively, and $S_{t}^{\ast_{t}}$ is the
adjoint operator of $S_{t}$ with respect to $g_{t}$. This function $\rho_{t}$
implies the gap from the curvature-adaptedness of $f_{t}$. In this paper, we
call this function a gap function. Define $(1,2)$-tensor fields
$\widetilde{R}_{i}(\xi_{t})$ ($i=1,3$) over $M$ by

$\widetilde{R}_{1}(\xi_{t})(X,Y):=f_{t\ast}^{-1}((\widetilde{R}(\xi_{t},f_{t\ast}X)f_{t\ast}Y)_{T})\qquad\,\,(X,Y\in
TM)$

and

$\widetilde{R}_{3}(\xi_{t})(X,Y):=f_{t\ast}^{-1}(\widetilde{R}(f_{t\ast}X,f_{t\ast}Y)\xi_{t})\qquad\,\,(X,Y\in
TM),$

where $(\bullet)_{T}$ is the $f_{t\ast}(TM)$-component of $(\bullet)$. Also,
for tangent vector fields $X$ and $Y$ on $M$, define a $(1,1)$-tensor field
$R_{t}(X,Y)A_{t}^{k}$ ($k=1,2$) on $M$ by

$R_{t}(X,Y)A_{t}^{k}:=\nabla^{t}_{X}(\nabla^{t}_{Y}(A_{t}^{k}))-\nabla^{t}_{Y}(\nabla^{t}_{X}(A_{t}^{k}))-\nabla^{t}_{[X,Y]}(A_{t}^{k}).$

Define skew-symmetric $(1,1)$-tensor fields $\widehat{S}_{t}$ ($0\leq t<T$)
over $M$ by

$\begin{array}[]{l}\displaystyle{\widehat{S}_{t}:=2\left[A_{t}^{2}+\widetilde{R}(\xi_{t}),{\rm
Tr}_{g_{t}}^{\bullet}(R_{t}(\cdot,\bullet)A_{t})(\bullet))\right]}\\\ \hskip
28.45274pt\displaystyle{+2\left[A_{t},\,(\widetilde{R}_{3}(\xi_{t})-\widetilde{R}_{1}(\xi_{t}))(\cdot,{\rm
Tr}_{g_{t}}^{\bullet}(\nabla^{t}_{\bullet}A_{t})(\bullet))\right]}\\\ \hskip
28.45274pt\displaystyle{+2{\rm
Tr}^{\bullet}_{g_{t}}\left[\nabla^{t}_{\bullet}A_{t},\,\nabla^{t}_{\bullet}\widetilde{R}(\xi_{t})\right].}\end{array}$
$None$

Note that, if ${\rm Ker}\,A_{t}=\\{0\\}$, then the tensor field
$\widetilde{R}_{3}(\xi_{t})-\widetilde{R}_{1}(\xi_{t})$ is described in terms
of $\nabla^{t}\widetilde{R}(\xi_{t})$ and $A_{t}^{-1}$ (see $(3.8)$) and hence
$\widehat{S}_{t}$ is described in terms of
$A_{t},A_{t}^{-1},\nabla^{t}A_{t},\nabla^{t}\nabla^{t}A_{t},\widetilde{R}(\xi_{t})$
and $\nabla^{t}\widetilde{R}(\xi_{t})$. Define a function
$\mu_{t}:M\to\mathbb{R}$ ($t\in[0,T)$) by

$\mu_{t}(x):=\left\\{\begin{array}[]{ll}\displaystyle{-\frac{\langle(\widehat{S}_{t})_{x},(S_{t})_{x}\rangle}{\|(S_{t})_{x}\|^{2}}}&(\|(S_{t})_{x}\|\not=0)\\\
\displaystyle{0}&(\|(S_{t})_{x}\|=0),\end{array}\right.$ $None$

where $\langle(\widehat{S}_{t})_{x},(S_{t})_{x}\rangle$ denotes ${\rm
Tr}((\widehat{S}_{t}^{\ast_{t}})_{x}\circ(S_{t})_{x})(=-{\rm
Tr}((\widehat{S}_{t})_{x}\circ(S_{t})_{x}))$.

We prove the following result for Question 1.

Theorem B. Let $(\widetilde{M},\widetilde{g})$ be an $(n+1)$-dimensional
locally symmetric space, $f$ a curvature-adapted immersion of an
$n$-dimensional compact manifold $M$ into $\widetilde{M}$, $\widehat{S}$ the
skew-symmetric $(1,1)$-tensor field on $M$ defined as in $(1.4)$ for $f$ and
$\\{f_{t}\\}_{t\in[0,T)}$ the mean curvature flow starting from $f$. If
$\widehat{S}\not=0$, then $f_{t}$ ($0<t<\varepsilon$) are not curvature-
adapted for some $\varepsilon>0$.

From Theorems A and B, we obtain the following result for this question.

Corollay C. Let $(\widetilde{M},\widetilde{g})$ be an $(n+1)$-dimensional
locally symmetric space, $f$ an immersion of an $n$-dimensional compact
manifold $M$ into $\widetilde{M}$ and $\widehat{S}$ the skew-symmetric
$(1,1)$-tensor filed on $M$ defined as in $(1.4)$ for $f$. Then, if $f$ is
curvature-adapted and if both the shape operator and the normal Jacobi
operator of $f$ have constant eigenvalues, then $\widehat{S}=0$ holds.

Naturally the following question arises.

Question 2. Let $f$ and $\\{f_{t}\\}_{t\in[0,T)}$ be as in Theorem B. Does
$f_{t}$ remain to be curvature-adapted for all $t\in[0,T)$ if $\widehat{S}=0$?

If this question were solved affirmatively, then we see that $\widehat{S}$ is
an obstruction for the curvature-adaptedness to be preserved along the mean
curvature flow starting from a curvature-adapted compact hypersurface in a
locally symmetric space.

We prove the following result for this question.

Theorem D. Let $(\widetilde{M},\widetilde{g})$ be an $(n+1)$-dimensional
locally symmetric space, $f$ a curvatue-adapted immersion of an
$n$-dimensional compact manifold $M$ into $\widetilde{M}$ and
$\\{f_{t}\\}_{t\in[0,T)}$ the mean curvature flow starting from $f$. If
$\widehat{S}_{0}=0$ and if

$\mathop{\sup}_{t\in[0,T)}\,\mathop{\sup}_{x\in M}\,\mu_{t}(x)<\infty,$

then $f_{t}$ remains to be curvature-adapted for all $t\in[0,T)$.

As a corollary of Theorem D, we obtain the following result.

Corollary E. Let $(\widetilde{M},\widetilde{g})$ be an $(n+1)$-dimensional
locally symmetric space, $f$ a curvatue-adapted immersion of an
$n$-dimensional compact manifold $M$ into $\widetilde{M}$ and
$\\{f_{t}\\}_{t\in[0,T)}$ the mean curvature flow starting from $f$. If
$\widehat{S}_{0}=0$ and if

$\mathop{\inf}_{t\in[0,T)}\,\mathop{\min}_{x\in
M}\,\langle(\widehat{S}_{t})_{x},(S_{t})_{x}\rangle\geq 0,$

then $f_{t}$ remains to be curvature-adapted for all $t\in[0,T)$.

Assume that $f$ is curvature-adapted and that $f_{t_{0}}$ is not curvature-
adapted for some $t_{0}\in[0,T)$. Set

$t_{\min}:=\inf\,\\{t\in[0,T)\,|\,f_{t}\,\,{\rm is}\,\,{\rm not}\,\,{\rm
curvature-adapted}\\}.$

Then, according to Theorem D, $\displaystyle{\mathop{\sup}_{x\in
M}\,\mu_{t}(x)}$ diverges to $+\infty$ as $t\downarrow t_{\min}$ (see Figure
1.1). This fact is restated in terms of “backward mean curvature flow” as
follows.

Theorem F. Let $(\widetilde{M},\widetilde{g})$ be an $(n+1)$-dimensional
locally symmetric space, $f$ an immersion of an $n$-dimensional compact
manifold $M$ into $\widetilde{M}$ and $\\{f^{b}_{t}\\}_{t\in[0,T)}$ the
backward mean curvature flow starting from $f$. Assume that $f$ is not
curvature-adapted and that $f^{b}_{t_{0}}$ is curvature-adapted for some
$t_{0}\in[0,T)$, where $t_{0}$ is the first time such that $f_{t}^{b}$ is
curvature-adapted. Then $\displaystyle{\lim_{t\uparrow
t_{0}}\,\mathop{\sup}_{x\in M}\,\mu_{t}(x)=\infty}$ holds.

$t$$t_{\min}$$T$$y$$\displaystyle{y=\mathop{\max}_{x\in
M}\,\rho_{t}(x)}$$\displaystyle{y=\mathop{\sup}_{x\in M}\,\mu_{t}(x)}$

Figure 1.1: The graph of $\displaystyle{\mathop{\sup}_{x\in M}\,\mu_{t}(x)}$

In the future, we plan to tackle the following question.

Question 3. Can we find a pinching condition of the norms
$\|A_{0}\|,\,\|\nabla^{0}A_{0}\|$ and $\|(\nabla^{0})^{2}A_{0}\|$ satisfying
$\displaystyle{\mathop{\sup}_{t\in[0,T)}\,\mathop{\sup}_{x\in
M}\,\mu_{t}(x)<\infty}$? Furthermore, if such a pinching condition were found,
does the mean curvature flow starting from a curvature-adapted immersion $f$
satisfying $\widehat{S}=0$ and the pinching condition asymptote to the mean
curvature flow starting from a curvature-adapted equifocal hypersurface?
Hence, does the flow collapse to a focal submanifold of the curvature-adapted
equifocal hypersurface? (see [TT] about the notion of an equifocal
hypersurface).

If the above questions are solved affirmatively, then we can derive that there
are only finitely many of diffeomorphism classes of compact hypersurfaces in a
simply connected compact symmetric space satisfying $\widehat{S}=0$ and the
pinching condition of the norms $\|A\|,\,\|\nabla A\|$ and $\|\nabla^{2}A\|$
by using the finiteness theorem for curvature-adapted equifocal hypersurfaces
in a simply connected compact symmetric space by J. Q. Ge and C. Qian ([GQ]).

In Section 2, we recall the evolution equations for the basic geometric
quantities along the mean curvature flow. In Section 3, we derive the
evolution equation for the normal Jacobi operator $\widetilde{R}(\xi_{t})$. In
Section 4, we prove Theorems A, B and D.

##  2 Evolution equations

Let $M$ and $(\widetilde{M},\widetilde{g})$ be as in Introduction. Assume that
$\widetilde{M}$ is a locally symmetric space. Denote by $\widetilde{\nabla}$
the Levi-Civita connection of $\widetilde{g}$. Also, denote by
$\widetilde{R},\widetilde{\rm Ric}$ and ${\widetilde{R}}^{S}$ the curvature
tensor, the Ricci tensor and the scalar curvature of $\widetilde{g}$. Let
$\\{f_{t}\\}_{t\in[0,T)}$ be the mean curvature flow starting from $f$. Define
a map $F:M\times[0,T)\to\widetilde{M}$ by $F(x,t):=f_{t}(x)$ ($(x,t)\in
M\times[0,T)$). Let $g_{t},\nabla^{t},R_{t},\xi_{t},A_{t}$ and $H_{t}$ be as
in Introduction, and $h_{t}$ be the second fundamental form of $f_{t}$ for
$-\xi_{t}$. Also, let $\pi_{M}$ be the projection of $M\times[0,T)$ onto $M$.
For a vector bundle $E$ over $M$, denote by $\pi_{M}^{\ast}E$ the induced
bundle of $E$ by $\pi_{M}$. Also denote by $E_{x}$ the fibre of $E$ over $x$
and by $\Gamma(E)$ the space of all sections of $E$. Define a section $g$ of
$\pi_{M}^{\ast}(T^{(0,2)}M)$ by $g(x,t)=(g_{t})_{x}$ ($(x,t)\in
M\times[0,T)$), where $T^{(0,2)}M$ is the $(0,2)$-tensor bundle of $M$.
Similarly, we define a section $R$ of $\pi_{M}^{\ast}(T^{(1,3)}M)$, $h$ of
$\pi_{M}^{\ast}(T^{(0,2)}M)$, a section $A$ of $\pi_{M}^{\ast}(T^{(1,1)}M)$, a
map $H:M\times[0,T)\to{\mathbb{R}}$ and a section $\xi$ of
$F^{\ast}T\widetilde{M}$ in terms of $h_{t},\ A_{t},\,\,H_{t}$ and $\xi_{t}$,
respectively. The bundle $\pi_{M}^{\ast}(TM)$ is regarded as a subbundle of
$T(M\times[0,T))$ under the identification of
$((x,t),v)\in(\pi_{M}^{\ast}TM)_{(x,t)}=\\{(x,t)\\}\times T_{x}M$ and
$v^{L}_{(x,t)}(\in T_{(x,t)}(M\times[0,T))$, where $v^{L}_{(x,t)}$ is the
horizontal lift of $v$ to $(x,t)$ with respect to $\pi_{M}$. Also, the fibre
$\pi_{M}^{\ast}(TM)_{(x,t)}$ is identified with $T_{x}M$ under the
identification of $((x,t),v)(\in\pi_{M}^{\ast}(TM)_{(x,t)})$ and $v$. For a
section $B$ of $\pi_{M}^{\ast}(T^{(r,s)}M)$, we define
$\displaystyle{\frac{\partial B}{\partial t}}$ by
$\displaystyle{\left(\frac{\partial B}{\partial
t}\right)_{(x,t)}:=\frac{dB_{(x,t)}}{dt}}$, where the right-hand side of this
relation is the derivative of the vector-valued function $t\mapsto
B_{(x,t)}\,(\in T^{(r,s)}_{x}M)$. For a tangent vector field $X$ on $M$ (or an
open set $U$ of $M$), we define a section $\overline{X}$ of $\pi_{M}^{\ast}TM$
(or $\pi_{M}^{\ast}TM|_{U}$) by $\overline{X}_{(x,t)}:=((x,t),X_{x})$
($(x,t)\in M\times[0,T)\,\,({\rm or}\,\,U\times[0,T))$). Define a connection
$\nabla$ of $\pi_{M}^{\ast}TM$ by

$(\nabla_{v}X)_{(\cdot,t)}:=\nabla^{t}_{v}X_{(\cdot,t)}\,\,\,{\rm
and}\,\,\,\nabla_{\frac{\partial}{\partial t}}X:=\frac{dX_{(x,\cdot)}}{dt}$

for $v\in(\pi_{M}^{\ast}TM)_{(x,t)}(=T_{x}M)$ and
$X\in\Gamma(\pi_{M}^{\ast}TM)$, where $\displaystyle{\frac{dX_{(x,t)}}{dt}}$
is the derivative of the vector-valued function $t\mapsto X_{(x,t)}\,(\in
T_{x}M)$. Let $\\{{\mathcal{S}}_{t}\\}_{t\in[0,T)}$ be a $C^{\infty}$-family
of a $(r,s)$-tensor fields on $M$ and ${\mathcal{S}}$ a section of
$\pi_{M}^{\ast}(T^{(r,s)}M)$ defined by
${\mathcal{S}}_{(x,t)}:=({\mathcal{S}}_{t})_{x}$. We define a section
$\triangle\,{\mathcal{S}}$ of $\pi_{M}^{\ast}(T^{(r,s)}M)$ by

$(\triangle\,{\mathcal{S}})_{(x,t)}:=\sum_{i=1}^{n}\nabla_{e_{i}}\nabla_{e_{i}}{\mathcal{S}},$

where $\nabla$ is the connection of $\pi_{M}^{\ast}(T^{(r,s)}M)$ (or
$\pi_{M}^{\ast}(T^{(r,s+1)}M)$) induced from $\nabla$ and
$\\{e_{1},\cdots,e_{n}\\}$ is an orthonormal base of $T_{x}M$ with respect to
$(g_{t})_{x}$. Also, we define a section $\triangle_{t}{\mathcal{S}}_{t}$ of
$T^{(r,s)}M$ by

$(\triangle_{t}{\mathcal{S}}_{t})_{x}:=(\triangle{\mathcal{S}})_{(x,t)}\quad\,\,(x\in
M).$

Let $E$ be a vector bundle over $M$. For a section ${\mathcal{S}}$ of
$\pi_{M}^{\ast}(T^{(0,r)}M\otimes E)$, we define $\displaystyle{{\rm
Tr}_{g}^{\bullet}\,{\mathcal{S}}(\cdots,\mathop{\bullet}^{j},\cdots,\mathop{\bullet}^{k},\cdots)}$
by

$({\rm
Tr}_{g}^{\bullet}\,{\mathcal{S}}(\cdots,\mathop{\bullet}^{j},\cdots,\mathop{\bullet}^{k},\cdots))_{(x,t)}=\sum_{i=1}^{n}{\mathcal{S}}_{(x,t)}(\cdots,\mathop{e_{i}}^{j},\cdots,\mathop{e_{i}}^{k},\cdots)$

$((x,t)\in M\times[0,T))$, where $\\{e_{1},\cdots,e_{n}\\}$ is an orthonormal
base of $T_{x}M$ with respect to $(g_{t})_{x}$,
$\displaystyle{{\mathcal{S}}(\cdots,\mathop{\bullet}^{j},\cdots,\mathop{\bullet}^{k},\cdots)}$
means that $\bullet$ is entried into the $j$-th component and the $k$-th
component of ${\mathcal{S}}$ and
$\displaystyle{{\mathcal{S}}_{(x,t)}(\cdots,\mathop{e_{i}}^{j},\cdots,\mathop{e_{i}}^{k},\cdots)}$
means that $e_{i}$ is entried into the $j$-th component and the $k$-th
component of ${\mathcal{S}}_{(x,t)}$. By using the normal Jacobi operator
$\widetilde{R}(\xi_{t})$ of $f_{t}$ we define a section $\widetilde{R}(\xi)$
of $\pi_{M}^{\ast}T^{(1,1)}M$ by
$\widetilde{R}(\xi)_{(x,t)}:=\widetilde{R}(\xi_{t})_{x}(=((x,t),\widetilde{R}(\xi_{t})_{x}))\,\,\,((x,t)\in
M\times[0,T))$. Also, by using the $(1,1)$-tensor field $\widetilde{\rm
Ric}^{\sharp}$ over $\widetilde{M}$ (which is define by
$\widetilde{g}(\widetilde{\rm Ric}^{\sharp}(X),Y)=\widetilde{\rm
Ric}(X,Y)\,\,\,(X,Y\in T\widetilde{M})$), we define a $(1,1)$-tensor field
$(\widetilde{\rm Ric}^{\sharp})^{T}_{t}$ over $M$ by $(\widetilde{\rm
Ric}^{\sharp})^{T}_{t}=f_{t\ast}^{-1}\circ{\rm pr}^{T}_{t}\circ\widetilde{\rm
Ric}^{\sharp}\circ f_{t\ast}$, where ${\rm pr}^{T}_{t}$ is the orthogonal
projection of $f_{t}^{\ast}T\widetilde{M}$ onto $f_{t\ast}(TM)$. Denote by
$(\widetilde{\rm Ric}^{\sharp})^{T}$ the section of $\pi_{M}^{\ast}T^{(1,1)}M$
defined by using $(\widetilde{\rm Ric}^{\sharp})^{T}_{t}$. Similarly, we
define a $(1,3)$-tensor field ${\widetilde{R}}^{T}_{t}$ over $M$ by
${\widetilde{R}}^{T}_{t}=f_{t\ast}^{-1}\circ{\rm
pr}^{T}_{t}\circ\widetilde{R}\circ(f_{t\ast}\times f_{t\ast}\times
f_{t\ast})$. Denote by ${\widetilde{R}}^{T}$ the section of
$\pi_{M}^{\ast}T^{(1,3)}M$ defined by using ${\widetilde{R}}^{T}_{t}$. Since
$\widetilde{M}$ is locally symmetric and irreducible,
$\widetilde{\nabla}\widetilde{R}=0$ holds and it is Einstein, that is,

$\widetilde{\rm Ric}^{\sharp}=\frac{{\widetilde{R}}^{S}}{n}{\rm id}.$ $None$

According to (i) of Lemma 3.3 in [Hu2], we have the following evolution
equation.

Lemma 2.1. The family $\\{g_{t}\\}_{t\in[0,T)}$ satisfies the following
evolution equation:

$\frac{\partial g}{\partial t}=-2H_{t}h_{t}.$

According to (ii) of Lemma 3.3 in [Hu2], we have the following evolution
equation.

Lemma 2.2. The family $\\{\xi_{t}\\}_{t\in[0,T)}$ satisfies the following
evolution equation:

$\frac{\partial\xi}{\partial t}=-F_{\ast}({\rm grad}_{g_{t}}H_{t}),$

where ${\rm grad}_{g_{t}}H_{t}$ is the element of $\pi_{M}^{\ast}(TM)$ such
that $dH_{t}(X)=g_{t}({\rm grad}_{g_{t}}H_{t},X)$ for any
$X\in\pi_{M}^{\ast}(TM)$.

According to (i) of Lemma 3.3 and Theorem 3.4 in [Hu2], we have the following
evolution equation.

Lemma 2.3. The family $\\{A_{t}\\}_{t\in[0,T)}$ satisfies the following
evolution equation:

$\begin{array}[]{l}\displaystyle{\frac{\partial A}{\partial
t}=\triangle_{t}A_{t}+\left({\rm Tr}(A_{t}^{2})+{\rm
Tr}\,\widetilde{R}(\xi_{t})\right)A_{t}+2A_{t}^{3}-2{\rm
Tr}(A_{t}^{2})A_{t}-\frac{2{\widetilde{R}}^{S}}{n}A_{t}}\\\ \hskip
28.45274pt\displaystyle{+A_{t}\circ\widetilde{R}(\xi_{t})+\widetilde{R}(\xi_{t})\circ
A_{t}+2{\rm
Tr}^{\bullet}_{g_{t}}R_{t}(\cdot,\bullet)(A_{t}(\bullet)).}\end{array}$

Proof. According to (i) of Lemma 3.3 and Theorem 3.4 in [Hu2] and $(2.1)$, we
have

$\begin{array}[]{l}\displaystyle{\frac{\partial A}{\partial
t}=\triangle_{t}A_{t}+\left({\rm Tr}(A_{t}^{2})+{\rm
Tr}\,\widetilde{R}(\xi_{t})\right)A_{t}-\frac{2{\widetilde{R}}^{S}}{n}A_{t}}\\\
\hskip
28.45274pt\displaystyle{+A_{t}\circ\widetilde{R}(\xi_{t})+\widetilde{R}(\xi_{t})\circ
A_{t}+2{\rm
Tr}^{\bullet}_{g_{t}}{\widetilde{R}}^{T}(\cdot,\bullet)(A_{t}(\bullet)).}\end{array}$

On the other hand, according to the Gauss equation, we have

${\widetilde{R}}^{T}(X,Y)Z=R_{t}(X,Y)Z+h_{t}(X,Z)A_{t}Y-h_{t}(Y,Z)A_{t}X$
$None$

for any tangent vector fields $X,Y$ and $Z$ on $M$. From these relations, we
can derive the desired evolution equation. ∎

According to Corollary 3.5 in [Hu2], we have the following evolution equation.

Lemma 2.4. The family $\\{H_{t}\\}_{t\in[0,T)}$ satisfies the following
evolution equation:

$\begin{array}[]{l}\displaystyle{\frac{\partial H}{\partial
t}=\triangle_{t}\,H_{t}+\left({\rm Tr}(A_{t}^{2})+{\rm
Tr}\,\widetilde{R}(\xi_{t})\right)H_{t}.}\end{array}$

##  3 Evolution of the normal Jacobi operator

We use the notations in Sections 1 and 2. Assume that $\widetilde{M}$ is a
locally symmetric space. In this section, we derive the evolution equation for
the family $\\{\widetilde{R}(\xi_{t})\\}_{t\in[0,T)}$ of the normal Jacobi
operators. Denote by $S$ the section of $\pi_{M}^{\ast}T^{(1,1)}M$ defined by
using $S_{t}$’s and by $\widetilde{R}_{i}(\xi)$ ($i=1,3$) the sections of
$\pi_{M}^{\ast}T^{(1,2)}M$ defined by using $\widetilde{R}_{i}(\xi_{t})$’s.
Denote by ${\widetilde{\nabla}}^{F}$ (resp. ${\widetilde{\nabla}}^{f_{t}}$)
the pull-back connection of $\widetilde{\nabla}$ by $F$ (resp. $f_{t}$). First
we prepare the following lemma.

Lemma 3.1. Let $\\{Z_{t}\\}_{t\in[0,T)}$ be a $C^{\infty}$-family of tangent
vector fields on $M$. Then we have

$\frac{\partial f_{t\ast}(Z_{t})}{\partial t}=f_{t\ast}\left(\frac{\partial
Z_{t}}{\partial t}\right)-(Z_{t}H_{t})\xi_{t}-H_{t}f_{t\ast}(A_{t}Z_{t}).$

Proof. Fix $x_{0}\in M$. Let $\\{\phi^{t}_{s}\\}_{s\in I}$ be the local one-
parameter transformation group of $Z_{t}$ and define a map
$\widetilde{\delta}:[0,T)^{2}\times I\to\widetilde{M}$ by
$\widetilde{\delta}(t,u,s):=f_{t}(\phi^{u}_{s}(x_{0}))$ and
$\delta:[0,T)\times I\to\widetilde{M}$ by
$\delta(t,s):=\widetilde{\delta}(t,t,s)$. Then we have

$\frac{\partial\delta}{\partial
t}=\left.\left(\frac{\partial\widetilde{\delta}}{\partial
t}+\frac{\partial\widetilde{\delta}}{\partial
u}\right)\right|_{u=t}=-(H_{t})_{\phi^{t}_{s}(x_{0})}(\xi_{t})_{\phi^{t}_{s}(x_{0})}+\left.f_{t\ast}\left(\frac{\partial\phi^{u}_{s}(x_{0})}{\partial
u}\right)\right|_{u=t}.$ $None$

Denote by ${\widetilde{\nabla}}^{\delta}$ the pull-back connection of
$\widetilde{\nabla}$ by $\delta$. Then we have

$\displaystyle\left(\frac{\partial f_{t\ast}(Z_{t})}{\partial
t}\right)_{(x_{0},t_{0})}$
$\displaystyle=\left({\widetilde{\nabla}}^{F}_{\frac{\partial}{\partial
t}}F_{\ast}Z\right)_{(x_{0},t_{0})}=\left.\left({\widetilde{\nabla}}^{\delta}_{\frac{\partial}{\partial
t}}\left.\frac{\partial\delta}{\partial
s}\right|_{s=0}\right)\right|_{t=t_{0}}$
$\displaystyle=\left.\left({\widetilde{\nabla}}^{\delta}_{\frac{\partial}{\partial
s}}\left.\frac{\partial\delta}{\partial
t}\right|_{t=t_{0}}\right)\right|_{s=0}.$

By substituting $(3.1)$ into this relation, we can derive

$\begin{array}[]{l}\displaystyle{\left(\frac{\partial
f_{t\ast}(Z_{t})}{\partial
t}\right)_{(x_{0},t_{0})}={\widetilde{\nabla}}^{f_{t_{0}}}_{(Z_{t_{0}})_{x_{0}}}(-H_{t_{0}}\xi_{t_{0}})+\left.\left({\widetilde{\nabla}}^{\widetilde{\delta}}_{\frac{\partial}{\partial
u}}\frac{\partial\widetilde{\delta}}{\partial
s}\right)\right|_{t=u=t_{0},s=0}}\\\
\displaystyle{=-((Z_{t_{0}})_{x_{0}}H_{t_{0}})(\xi_{t_{0}})_{x_{0}}-(H_{t_{0}})_{x_{0}}f_{t_{0}\ast}(A_{t_{0}}(Z_{t_{0}})_{x_{0}})+f_{t_{0}\ast}\left(\left.\frac{d(Z_{u})_{x_{0}}}{du}\right|_{u=t_{0}}\right).}\end{array}$

Therefore, the desired relation follows from the arbitrariness of
$(x_{0},t_{0})$. ∎

Since $(\widetilde{M},\widetilde{g})$ is an Einstein space, we have the
following relation.

Lemma 3.2. The following relation holds:

${\rm grad}_{g_{t}}H_{t}={\rm
Tr}_{g_{t}}^{\bullet}(\nabla^{t}_{\bullet}A_{t})(\bullet).$

Proof. According to the Codazzi equation, we have

$\widetilde{R}_{3}(\xi_{t})(X,Y)=-(\nabla^{t}_{X}A_{t})(Y)+(\nabla^{t}_{Y}A_{t})(X)\quad\,\,(X,Y\in
TM).$ $None$

From this relation and the Einsteinity of $(\widetilde{M},\widetilde{g})$, we
obtain

$\displaystyle{\rm grad}_{g_{t}}H_{t}$ $\displaystyle=(\bullet\,\mapsto\,{\rm
Tr}(\nabla^{t}_{\bullet}A_{t}))^{\sharp}={\rm
Tr}_{g_{t}}^{\bullet}(\nabla^{t}_{\bullet}A_{t})(\bullet)+\widetilde{\rm
Ric}^{\sharp}(\xi_{t})_{T}$ $\displaystyle={\rm
Tr}_{g_{t}}^{\bullet}(\nabla^{t}_{\bullet}A_{t})(\bullet)+\left(\frac{\widetilde{R}^{S}}{n}\,\xi_{t}\right)_{T}={\rm
Tr}_{g_{t}}^{\bullet}(\nabla^{t}_{\bullet}A_{t})(\bullet).$

∎

By using these lemmas, we can derive the following evolution.

Proposition 3.3. The family $\\{\widetilde{R}(\xi_{t})\\}_{t\in[0,T)}$
satisfies the following evolution equation:

$\begin{array}[]{l}\displaystyle{\frac{\partial\widetilde{R}(\xi)}{\partial
t}=\triangle_{t}\widetilde{R}(\xi_{t})+H_{t}S_{t}-\widetilde{R}(\xi_{t})\circ
A_{t}^{2}+2{\rm Tr}(A_{t}^{2})\widetilde{R}(\xi_{t})}\\\ \hskip
44.10185pt\displaystyle{-2\widetilde{R}_{3}(\xi_{t})(\cdot,{\rm
Tr}^{\bullet}_{g_{t}}(\nabla^{t}_{\bullet}A_{t})(\bullet))+2\widetilde{R}_{1}(\xi_{t})(\cdot,{\rm
Tr}^{\bullet}_{g_{t}}(\nabla^{t}_{\bullet}A_{t})(\bullet))}\\\ \hskip
44.10185pt\displaystyle{-2{\rm
Tr}^{\bullet}_{g_{t}}R_{t}(\cdot,A_{t}(\bullet))A_{t}(\bullet)-2{\rm
Tr}(A_{t}^{3})A_{t}+2A_{t}^{4}.}\end{array}$

Proof. Take $X$ be a tangent vector field on $M$ and $\overline{X}$ be the
section $\pi_{M}^{\ast}TM$ defined by $\overline{X}_{(x,t)}:=((x,t),X_{x})$.
By using Lemma 3.1 and $\frac{\partial\overline{X}}{\partial t}=0$, we can
show

$\begin{array}[]{l}\displaystyle{\frac{\partial(f_{t\ast}\circ\widetilde{R}(\xi_{t}))}{\partial
t}(X)=\frac{\partial f_{t\ast}(\widetilde{R}(\xi_{t})(\overline{X}))}{\partial
t}}\\\
\displaystyle{=f_{t\ast}\left(\frac{\partial\widetilde{R}(\xi_{t})}{\partial
t}(X)\right)-((\widetilde{R}(\xi_{t})(X))H_{t})\xi_{t}-H_{t}f_{t\ast}(A_{t}(\widetilde{R}(\xi_{t})(X)))}\\\
\displaystyle{\equiv
f_{t\ast}\left(\frac{\partial\widetilde{R}(\xi_{t})}{\partial
t}(X)\right)-H_{t}f_{t\ast}(A_{t}(\widetilde{R}(\xi_{t})(X)))\qquad\,\,({\rm
mod}\,\,{\rm Span}\\{\xi_{t}\\}).}\end{array}$ $None$

On the other hand, by using Lemmas 2.2, 3.1,
$\frac{\partial\overline{X}}{\partial t}=0$ and
$\widetilde{\nabla}\widetilde{R}=0$, we can show

$\begin{array}[]{l}\displaystyle{\frac{\partial(f_{t\ast}\circ\widetilde{R}(\xi_{t}))}{\partial
t}(X)=\frac{\partial F_{\ast}(\widetilde{R}(\xi)(\overline{X}))}{\partial
t}=\frac{\partial}{\partial t}\widetilde{R}(F_{\ast}(\overline{X}),\xi)\xi}\\\
\displaystyle{=\widetilde{R}\left(\frac{\partial f_{t\ast}(X)}{\partial
t},\xi_{t}\right)\xi_{t}+\widetilde{R}\left(f_{t\ast}(X),\frac{\partial\xi}{\partial
t}\right)\xi_{t}+\widetilde{R}\left(f_{t\ast}(X),\xi_{t}\right)\frac{\partial\xi}{\partial
t}}\\\
\displaystyle{=-\widetilde{R}\left((XH_{t})\xi_{t}+H_{t}f_{t\ast}(A_{t}X),\xi_{t}\right)\xi_{t}-\widetilde{R}\left(f_{t\ast}(X),f_{t\ast}({\rm
grad}_{g_{t}}H_{t})\right)\xi_{t}}\\\ \hskip
14.22636pt\displaystyle{-\widetilde{R}\left(f_{t\ast}(X),\xi_{t}\right)f_{t\ast}({\rm
grad}_{g_{t}}H_{t})}\\\
\displaystyle{=-f_{t\ast}\left(H_{t}\widetilde{R}(\xi_{t})(A_{t}X)+\widetilde{R}_{3}(\xi_{t})(X,{\rm
grad}_{g_{t}}H_{t})-\widetilde{R}_{1}(\xi_{t})(X,{\rm
grad}_{g_{t}}H_{t})\right).}\end{array}$ $None$

From $(3.3),\,(3.4)$ and Lemma 3.2, we derive

$\frac{\partial\widetilde{R}(\xi_{t})}{\partial
t}(X)=H_{t}S_{t}(X)-\widetilde{R}_{3}(\xi_{t})(X,{\rm
Tr}^{\bullet}_{g}(\nabla_{\bullet}A)(\bullet))+\widetilde{R}_{1}(\xi_{t})(X,{\rm
Tr}^{\bullet}_{g}(\nabla_{\bullet}A)(\bullet)).$ $None$

Fix $(x_{0},t_{0})\in M\times[0,T)$. Take any $v,w\in T_{x_{0}}M$. Let
$\widetilde{w}$ be a tangent vector field on a neighborhood of $x_{0}$ in $M$
with $\widetilde{w}_{x_{0}}=w$ and $(\nabla^{t_{0}}\widetilde{w})_{x_{0}}=0$.
Then we have

${\widetilde{\nabla}}^{f_{t_{0}}}_{v}(f_{t_{0}\ast}(\widetilde{R}(\xi_{t_{0}})(\widetilde{w})))=f_{t_{0}\ast}((\nabla^{t_{0}}_{v}\widetilde{R}(\xi_{t_{0}}))(w))-h_{t_{0}}(v,\widetilde{R}(\xi_{t_{0}})(w))(\xi_{t_{0}})_{x_{0}}.$
$None$

On the other hand, from $\widetilde{\nabla}\widetilde{R}=0$, we have

$\begin{array}[]{l}\hskip
14.22636pt\displaystyle{{\widetilde{\nabla}}^{f_{t_{0}}}_{v}(f_{t_{0}\ast}(\widetilde{R}(\xi_{t_{0}})(\widetilde{w})))={\widetilde{\nabla}}^{f_{t_{0}}}_{v}(\widetilde{R}(f_{t_{0}\ast}(\widetilde{w}),\xi_{t_{0}})\xi_{t_{0}})}\\\
\displaystyle{=\widetilde{R}({\widetilde{\nabla}}^{f_{t_{0}}}_{v}f_{t_{0}\ast}(\widetilde{w}),(\xi_{t_{0}})_{x_{0}})(\xi_{t_{0}})_{x_{0}}+\widetilde{R}(f_{t_{0}\ast}(w),{\widetilde{\nabla}}^{f_{t_{0}}}_{v}\xi_{t_{0}})\xi_{t_{0}}}\\\
\hskip
14.22636pt\displaystyle{+\widetilde{R}(f_{t_{0}\ast}(w),\xi_{t_{0}}){\widetilde{\nabla}}^{f_{t_{0}}}_{v}\xi_{t_{0}}}\\\
\displaystyle{=-h(v,w)\widetilde{R}((\xi_{t_{0}})_{x_{0}},(\xi_{t_{0}})_{x_{0}})(\xi_{t_{0}})_{x_{0}}+\widetilde{R}(f_{t_{0}\ast}(w),f_{t_{0}\ast}(A_{t_{0}}(v)))\xi_{t_{0}}}\\\
\hskip
14.22636pt\displaystyle{+\widetilde{R}(f_{t_{0}\ast}(w),\xi_{t_{0}})f_{t_{0}\ast}(A_{t_{0}}(v))}\\\
\displaystyle{=\widetilde{R}(f_{t_{0}\ast}(w),f_{t_{0}\ast}(A_{t_{0}}(v)))\xi_{t_{0}}+\widetilde{R}(f_{t_{0}\ast}(w),\xi_{t_{0}})f_{t_{0}\ast}(A_{t_{0}}(v)).}\end{array}$
$None$

From $(3.6)$ and $(3.7)$, we can derive

$(\nabla^{t_{0}}_{v}\widetilde{R}(\xi_{t_{0}}))(w)=\widetilde{R}_{3}(\xi_{t_{0}})(w,A_{t_{0}}(v))-\widetilde{R}_{1}(\xi_{t_{0}})(w,A_{t_{0}}(v)).$

From the arbitrarinesses of $v,w$ and $(x_{0},t_{0})$, we have

$(\nabla^{t}_{X}\widetilde{R}(\xi_{t}))(Y)=\widetilde{R}_{3}(\xi_{t})(Y,A_{t}X)-\widetilde{R}_{1}(\xi_{t})(Y,A_{t}X)$
$None$

for any $X,Y\in\Gamma(TM)$ and any $t\in[0,T)$. Let $v,w$ and $\widetilde{w}$
be as above. Also, let $\widetilde{v}$ be a tangent vector field on a
neighborhood of $x_{0}$ in $M$ with $\widetilde{v}_{x_{0}}=v$ and
$(\nabla^{t_{0}}\widetilde{v})_{x_{0}}=0$. Then we have

$\begin{array}[]{l}\displaystyle{f_{t_{0}\ast}\left((\nabla_{v}^{t_{0}}\nabla_{v}^{t_{0}}\widetilde{R}(\xi_{t_{0}}))(w)\right)=f_{t_{0}\ast}\left(\nabla_{v}^{t_{0}}((\nabla_{\widetilde{v}}^{t_{0}}\widetilde{R}(\xi_{t_{0}}))(\widetilde{w}))\right)}\\\
\displaystyle{\equiv\left({\widetilde{\nabla}}^{f_{t_{0}}}_{v}f_{t_{0}\ast}((\nabla_{\widetilde{v}}^{t_{0}}\widetilde{R}(\xi_{t_{0}}))(\widetilde{w}))\right)_{T}\qquad\,\,({\rm
mod}\,\,{\rm Span}\\{(\xi_{t_{0}})_{x_{0}}\\}).}\end{array}$ $None$

From $(3.8)$, $(3.9)$ and $\widetilde{\nabla}\widetilde{R}=0$, we can derive

$\begin{array}[]{l}\displaystyle{(\nabla_{v}^{t_{0}}\nabla_{v}^{t_{0}}\widetilde{R}(\xi_{t_{0}}))(w)=h_{t_{0}}(v,w)\widetilde{R}(\xi_{t_{0}})(A_{t_{0}}v)-2h_{t_{0}}(v,A_{t_{0}}v)\widetilde{R}(\xi_{t_{0}})(w)}\\\
\hskip
103.85237pt\displaystyle{+\widetilde{R}_{3}(\xi_{t_{0}})(w,(\nabla^{t_{0}}_{v}A_{t_{0}})(v))-\widetilde{R}_{1}(\xi_{t_{0}})(w,(\nabla^{t_{0}}_{v}A_{t_{0}})(v))}\\\
\hskip
103.85237pt\displaystyle{+2R_{t_{0}}(w,A_{t_{0}}v)(A_{t_{0}}v)+2h_{t_{0}}(A_{t_{0}}v,A_{t_{0}}v)A_{t_{0}}w}\\\
\hskip
103.85237pt\displaystyle{-2h_{t_{0}}(w,A_{t_{0}}v)A_{t_{0}}^{2}v.}\end{array}$

From the arbitrarinesses of $v,w$ and $(x_{0},t_{0})$, we have

$\begin{array}[]{l}\displaystyle{(\nabla_{X}^{t}\nabla_{X}^{t}\widetilde{R}(\xi_{t}))(Y)=h_{t}(X,Y)\widetilde{R}(\xi_{t})(A_{t}X)-2h_{t}(X,A_{t}X)\widetilde{R}(\xi_{t})(Y)}\\\
\hskip
99.58464pt\displaystyle{+\widetilde{R}_{3}(\xi_{t})(Y,(\nabla^{t}_{X}A_{t})(X))-\widetilde{R}_{1}(\xi_{t})(Y,(\nabla^{t}_{X}A_{t})(X))}\\\
\hskip
99.58464pt\displaystyle{+2R_{t}(Y,A_{t}X)(A_{t}X)+2h_{t}(A_{t}X,A_{t}X)A_{t}Y}\\\
\hskip 99.58464pt\displaystyle{-2h_{t}(Y,A_{t}X)A_{t}^{2}X}\end{array}$ $None$

for any $X,Y\in\Gamma(TM)$ and any $t\in[0,T)$. Hence we can derive

$\begin{array}[]{l}\displaystyle{(\triangle_{t}\widetilde{R}(\xi_{t}))(X)=\widetilde{R}(\xi_{t})(A_{t}^{2}X)-2{\rm
Tr}(A_{t}^{2})\widetilde{R}(\xi_{t})(X)}\\\ \hskip
81.09052pt\displaystyle{+\widetilde{R}_{3}(\xi_{t})(X,{\rm
Tr}_{g_{t}}^{\bullet}(\nabla^{t}_{\bullet}A_{t})(\bullet))-\widetilde{R}_{1}(\xi_{t})(X,{\rm
Tr}_{g_{t}}^{\bullet}(\nabla^{t}_{\bullet}A_{t})(\bullet))}\\\ \hskip
81.09052pt\displaystyle{+2{\rm
Tr}_{g_{t}}^{\bullet}R_{t}(X,A_{t}(\bullet))(A_{t}(\bullet))+2{\rm
Tr}(A_{t}^{3})A_{t}X-2A_{t}^{4}X}\end{array}$ $None$

for any $X\in\Gamma(TM)$ and any $t\in[0,T)$. From $(3.5)$ and $(3.11)$, we
can derive the desired evolution equation. ∎

##  4 Proofs of Theorems A, B and D

In this section, we use the notations in Sections 1-3. First we shall prove
Theorem A stated in Introduction. For its purpose, we shall show the following
lemma.

Lemma 4.1. Assume that $\widetilde{M}$ is a locally symmetric space and that
$f(:M\hookrightarrow\widetilde{M})$ is curvature-adapted. Also, let $f^{r}$ be
the end-point map of the hypersurface $f(M)$ for $r\xi$ (i.e.,
$f^{r}(x):=\exp^{\perp}(r\xi_{x})\,\,\,(x\in M)$), where $\exp^{\perp}$ is the
normal exponential map of $f$ and $r$ is a real number close to zero
sufficiently. Then the following statements (i) and (ii) hold.

(i) $f^{r}$ also is curvature-adapted.

(ii) Furthermore, if both the shape operator and the normal Jacobi operator of
$f$ have constant eigenvalues, then both the shape operator and the normal
Jacobi operator of $f^{r}$ also have constant eigenvalues.

Proof. Fix $x\in M$. Let $\gamma_{\xi_{x}}$ be the normal geodesic of $f(M)$
with $\gamma_{\xi_{x}}^{\prime}(0)=\xi_{x}$ and set
$\xi^{r}_{x}:=\gamma_{\xi_{x}}^{\prime}(r)$. Then, since $\widetilde{M}$ is
locally symmetric, it is shown that $\xi^{r}_{x}$ is a unit normal vector of
$f^{r}(M)$ at $x$. Denote by $A$ the shape operator of $f$ for $-\xi$ and by
$\widetilde{R}(\xi)$ the normal Jacobi operator of $f$. Also, denote by
$A^{r}$ the shape operator of $f^{r}$ for $-\xi^{r}(:x\mapsto\xi^{r}_{x})$ and
by $\widetilde{R}(\xi^{r})$ the normal Jacobi operator of $f^{r}$. Since $f$
is curvatrue-adapted, there exists a base $(e_{1},\cdots,e_{n})$ of $T_{x}M$
satisfying $A_{x}e_{i}=\lambda_{i}e_{i}$ and
$\widetilde{R}(\xi_{x})(e_{i})=\nu_{i}e_{i}$, where $\lambda_{i}$ and
$\nu_{i}$ are real numbers. Take a curve
$\alpha_{i}:(-\varepsilon,\varepsilon)\to M$ with
$\alpha_{i}^{\prime}(0)=e_{i}$ and define a map
$\delta_{i}:(-\varepsilon,\varepsilon)\times[0,r+\varepsilon)\to\widetilde{M}$
by $\delta_{i}(s,t):=\gamma_{\xi_{\alpha_{i}(s)}}(t)$
($(s,t)\in(-\varepsilon,\varepsilon)\times[0,r+\varepsilon)$). Define a vector
field $Y_{i}$ along $\gamma_{\xi_{x}}$ by
$Y_{i}:=\frac{\partial\delta_{i}}{\partial s}|_{s=0}$. Since $Y_{i}$ is the
Jacobi field along $\gamma_{\xi_{x}}$ with $Y_{i}(0)=f_{\ast}e_{i}$ and
$Y^{\prime}(0)=f_{\ast}A_{x}e_{i}$, it is described as

$Y_{i}(t)=\left(\cos(t\sqrt{\nu_{i}})-\frac{\lambda_{i}\sin(t\sqrt{\nu_{i}})}{\sqrt{\nu_{i}}}\right)P_{\gamma_{\xi_{x}}|_{[0,t]}}(f_{\ast}e_{i}),$
$None$

where $P_{\gamma_{\xi_{x}}|_{[0,t]}}$ is the parallel translation along
$\gamma_{\xi_{x}}|_{[0,t]}$. Note that, in case of $\nu_{i}\leq 0$,
$\cos(t\sqrt{\nu_{i}})=\cosh(t\sqrt{-\nu_{i}})$ and

$\frac{\sin(t\sqrt{\nu_{i}})}{\sqrt{\nu_{i}}}=\left\\{\begin{array}[]{cc}\displaystyle{\frac{\sinh(t\sqrt{-\nu_{i}})}{\sqrt{-\nu_{i}}}}&(\nu_{i}<0)\\\
\displaystyle{t}&(\nu_{i}=0).\end{array}\right.$

Here we used a general description of Jacobi fields in a symmetric space (see
Section 3 of [TT] or (1.2) of [K1]). In general, the description is valid in a
locally symmetric space. From $(4.1)$, we have

$f^{r}_{\ast}(e_{i})=Y_{i}(r)=\left(\cos(r\sqrt{\nu_{i}})-\frac{\lambda_{i}\sin(r\sqrt{\nu_{i}})}{\sqrt{\nu_{i}}}\right)P_{\gamma_{\xi_{x}}|_{[0,r]}}(f_{\ast}e_{i}).$
$None$

and

$f^{r}_{\ast}(A^{r}_{x}e_{i})=Y^{\prime}_{i}(r)=-\left(\sqrt{\nu_{i}}\sin(r\sqrt{\nu_{i}})+\lambda_{i}\cos(r\sqrt{\nu_{i}})\right)P_{\gamma_{\xi_{x}}|_{[0,r]}}(f_{\ast}e_{i}).$
$None$

Hence we obtain

$A^{r}_{x}e_{i}=-\frac{\sqrt{\nu_{i}}\tan(r\sqrt{\nu_{i}})+\lambda_{i}}{1-\lambda_{i}\tan(r\sqrt{\nu_{i}})/\sqrt{\nu_{i}}}\,e_{i}.$
$None$

Since $\widetilde{M}$ is locally symmetric, it is shown that
$P_{\gamma_{\xi_{x}}|_{[0,r]}}$ is equal to the differential $\phi_{\ast
f(x)}$ of a local isometry $\phi$ of a neighborhood of $f(x)$ onto a
neighborhood of $f^{r}(x)$ (see the discussion in Page 208 of [He] in the case
of a symmetric space). Since $\phi$ is a local isometry, it follows from
$(4.2)$ that

$\begin{array}[]{l}\displaystyle{\widetilde{R}(\xi^{r}_{x})(e_{i})=(f^{r}_{\ast})^{-1}(\widetilde{R}(f^{r}_{\ast}(e_{i}),\xi^{r}_{x})\xi^{r}_{x})}\\\
\displaystyle{=\left(\cos(r\sqrt{\nu_{i}})-\frac{\lambda_{i}\sin(r\sqrt{\nu_{i}})}{\sqrt{\nu_{i}}}\right)}\\\
\hskip
14.22636pt\displaystyle{\times(f^{r}_{\ast})^{-1}(\widetilde{R}(P_{\gamma_{\xi_{x}}|_{[0,r]}}(f_{\ast}e_{i}),P_{\gamma_{\xi_{x}}|_{[0,r]}}({\xi}_{x}))P_{\gamma_{\xi_{x}}|_{[0,r]}}({\xi}_{x}))}\\\
\displaystyle{=\left(\cos(r\sqrt{\nu_{i}})-\frac{\lambda_{i}\sin(r\sqrt{\nu_{i}})}{\sqrt{\nu_{i}}}\right)}\\\
\hskip
14.22636pt\displaystyle{\times(f^{r}_{\ast})^{-1}(\widetilde{R}(\phi_{\ast
x}(f_{\ast}e_{i}),\phi_{\ast x}({\xi}_{x}))\phi_{\ast x}({\xi}_{x}))}\\\
\displaystyle{=\left(\cos(r\sqrt{\nu_{i}})-\frac{\lambda_{i}\sin(r\sqrt{\nu_{i}})}{\sqrt{\nu_{i}}}\right)\times((f^{r}_{\ast})^{-1}\circ\phi_{\ast
x})(\widetilde{R}(f_{\ast}e_{i},\xi_{x})\xi_{x})}\\\
\displaystyle{=\left(\cos(r\sqrt{\nu_{i}})-\frac{\lambda_{i}\sin(r\sqrt{\nu_{i}})}{\sqrt{\nu_{i}}}\right)\times((f^{r}_{\ast})^{-1}\circ
P_{\gamma_{\xi_{x}}|_{[0,r]}}\circ f_{\ast})(\nu_{i}e_{i})}\\\
\displaystyle{=\nu_{i}e_{i}.}\end{array}$ $None$

Thus $A^{r}_{x}$ and $\widetilde{R}(\xi^{r}_{x})$ are simultaneously
diagonalized with respect to $(e_{1},\cdots,e_{n})$, that is, they commute to
each other. Hence $f^{r}$ is curvature-adapted. Thus the statement (i) has
been proved. The statement (ii) also follows from $(4.4)$ and $(4.5)$. ∎

By using this lemma, we shall prove Theorem A.

Proof of Theorem A. Let $\widetilde{M}$ and $f$ be as in the statement of
Theorem A. Since $A$ and $\widetilde{R}(\xi)$ have constant eigenvalues by the
assumption, it follows from (ii) of Lemma 4.1 that $A^{r}$ and
$\widetilde{R}(\xi^{r})$ have constant eigenvalues for any constant $r$
sufficiently close to $0$. Hence $f$ and $f^{r}$ are of constant mean
curvature. This implies that $f(M)$ and $f^{r}(M)$ are locally isoparametric.
Since $f$ and $f^{r}$ are of constant mean curvature, we see that the mean
curvature flow $\\{f_{t}\\}_{t\in[0,T)}$ is described as $f_{t}=f^{r_{t}}$ for
some $C^{\infty}$-correspndence $t\mapsto r_{t}$. Hence $f_{t}$ is curvature-
adapted and locally isoparametric by Lemma 4.1. ∎

Next we shall prove Theorem B stated in Introduction. For its purpose, we
first derive the following evolution equation for the family
$\\{S_{t}=[A_{t},\widetilde{R}(\xi_{t})]\\}_{t\in[0,T)}$ in terms of Lemma 2.3
and Proposition 3.3.

Lemma 4.2. The family $\\{S_{t}\\}_{t\in[0,T)}$ satisfies the following
evolution equation:

$\begin{array}[]{l}\displaystyle{\frac{\partial S}{\partial
t}-\triangle_{t}S_{t}=\left({\rm Tr}(A_{t}^{2})+{\rm
Tr}\,\widetilde{R}(\xi_{t})-\frac{2{\widetilde{R}}^{S}}{n}\right)S_{t}}\\\
\hskip 69.70915pt\displaystyle{+H_{t}[A_{t},S_{t}]-S_{t}\circ
A_{t}^{2}+[A_{t},\widetilde{R}(\xi_{t})^{2}]+2[A_{t}^{3},\widetilde{R}(\xi_{t})]-\widehat{S}_{t},}\end{array}$
$None$

where $\widehat{S}_{t}$ is as in $(1.4)$.

Proof. Clearly we have

$\begin{array}[]{l}\displaystyle{\frac{\partial S}{\partial
t}-\triangle_{t}S_{t}=\left[\frac{\partial A}{\partial
t}-\triangle_{t}A_{t},\widetilde{R}(\xi_{t})\right]+\left[A_{t},\frac{\partial\widetilde{R}(\xi_{t})}{\partial
t}-\triangle_{t}\widetilde{R}(\xi_{t})\right]}\\\ \hskip
70.27834pt\displaystyle{-2{\rm
Tr}_{g_{t}}^{\bullet}\left[\nabla^{t}_{\bullet}A_{t},\nabla^{t}_{\bullet}\widetilde{R}(\xi_{t})\right].}\end{array}$

By substituting the evolution equations in Lemma 2.3 and Proposition 3.3 into
this relation, we can derive the desired evolution equation for
$\\{S_{t}\\}_{t\in[0,T)}$. ∎

By using this lemma, we prove Theorem B.

Proof of Thoerem B Since $f(=f_{0})$ is curvature-adpated, we have $S_{0}=0$.
Hence, from the evolution equation $(4.6)$, we obtain
$\displaystyle{\left.\frac{\partial S}{\partial
t}\right|_{t=0}=-\widehat{S}_{0}\not=0}$. Therefore we can derive the
statement of Theorem B. ∎

Denote by $P(S_{t})$ the $(-1)$-multiple of the right-hand side of $(4.6)$.
Let $\rho$ be a function over $M\times[0,T)$ defined by using $\rho_{t}$’s.
From $(4.6)$, we can derive the following evolution equation for
$\\{\rho_{t}\\}_{t\in[0,T)}$ directly.

Lemma 4.3. The family $\\{\rho_{t}\\}_{t\in[0,T)}$ satisfies the following
evolution equation:

$\frac{\partial\rho}{\partial t}-\triangle_{t}\rho_{t}=2{\rm Tr}(P(S_{t})\circ
S_{t})+2{\rm Tr}{\rm
Tr}_{g_{t}}^{\bullet}(\nabla^{t}_{\bullet}S_{t}\circ\nabla^{t}_{\bullet}S_{t}).$

For $(1,1)$-tensor fields $\Phi$ and $\Psi$ over $M$, we denote ${\rm
Tr}(\Phi^{\ast_{t}}\circ\Psi)$ by $\langle\Phi,\Psi\rangle_{t}$ and ${\rm
Tr}(\Phi^{\ast_{t}}\circ\Phi)$ by $||\Phi||_{t}^{2}$, where $\Phi^{\ast_{t}}$
is the adjoint operator of $\Phi$ with respect to $g_{t}$. Define
$||\widetilde{R}||(:M\to{\mathbb{R}})$ by

$||\widetilde{R}||(x):=\max\\{||\widetilde{R}(v_{1},v_{2})v_{3}||\,\,|\,v_{i}\in
T_{x}\widetilde{M}\,\,{\rm
s.t.}\,\,||v_{i}||=1\,\,\,(i=1,2,3)\\}\quad\,\,(x\in M),$

where $||\bullet||:=\sqrt{\widetilde{g}(\bullet,\bullet)}$. Note that
$||\widetilde{R}||$ is constant in the case where $\widetilde{M}$ is a
Riemannian homogeneous space. Clearly we have the following inequalities.

Lemma 4.4. (i) ${\rm Tr}(\widetilde{R}(\xi_{t})^{k})\leq
n||\widetilde{R}||^{k}$ ($k\in{\mathbb{N}}$),

(ii) ${\rm Tr}_{g_{t}}^{\bullet}{\rm
Tr}(\widetilde{R}_{i}(\xi_{t})(\bullet)\circ\widetilde{R}_{i}(\xi_{t})(\bullet))\leq
n^{2}||\widetilde{R}||^{2}\quad(i=1,3)$.

By using Lemmas 4.3 and 4.4, we can derive the following estimate of the
functions $\rho_{t}$.

Proposition 4.5. Assume that $\widetilde{M}$ is a locally symmetric space and
that

$\mathop{\sup}_{t\in[0,T)}\,\mathop{\sup}_{x\in M}\,\mu_{t}(x)<\infty,$

where $\mu_{t}$ is as in $(1.5)$. Fix any $T_{0}\in[0,T)$. Then
$\rho_{t}\,\,(0\leq t\leq T_{0})$ are estimated from above as follows:

$\rho_{t}\leq\left(\mathop{\max}_{x\in M}\,\rho_{0}(x)\right)\cdot
e^{C_{1}(T_{0})t}\qquad\,\,(0\leq t<T_{0}),$

where $C_{1}(T_{0})$ is defined by

$C_{1}(T_{0}):=4(2n+1)\left(\mathop{\max}_{(x,t)\in
M\times[0,T_{0}]}||A_{t}||_{t}(x)\right)^{2}+10n||\widetilde{R}||+2\mathop{\sup}_{t\in[0,T_{0}]}\,\mathop{\sup}_{x\in
M}\,\mu_{t}(x).$

Proof. Since $S_{t}$ is skew-symmetric, so is also $\nabla^{t}_{X}S_{t}$ for
any $X\in\Gamma(TM)$. Hence $S_{t}^{2}$ and $(\nabla^{t}_{X}S_{t})^{2}$ are
non-positive opearators. Therefore we obtain

$\rho_{t}=-{\rm Tr}\,S_{t}^{2}\geq 0\quad\,\,$ and $\quad\,\,{\rm Tr}{\rm
Tr}_{g}^{\bullet}(\nabla^{t}_{\bullet}S_{t}\circ\nabla^{t}_{\bullet}S_{t})\leq
0.$
---
$None$

Hence, from Lemma 4.3, we have

$\frac{\partial\rho}{\partial t}-\triangle_{t}\rho_{t}\leq 2{\rm
Tr}(P(S_{t})\circ S_{t}).$ $None$

For simplicity, we set $C_{A}(T_{0}):=\max_{(x,t)\in
M\times[0,T_{0}]}||A_{t}||_{t}(x)$.

Now we shall calculate $P(S_{t})$. Clearly we have

$(R_{t}(X,Y)A^{k}_{t})(Z)=R_{t}(X,Y)(A_{t}^{k}Z)-A_{t}^{k}(R_{t}(X,Y)Z)$
$None$

for tangent vector fields $X,Y$ and $Z$ on $M$. From $(2.2)$, we have

${\rm Tr}_{g_{t}}^{\bullet}\widetilde{R}^{T}(X,\bullet)\bullet={\rm
Tr}_{g_{t}}^{\bullet}R_{t}(X,\bullet)\bullet+A_{t}^{2}X-H_{t}A_{t}X.$

On the other hand, from $(2.1)$, we have

${\rm
Tr}_{g_{t}}^{\bullet}\widetilde{R}^{T}(X,\bullet)\bullet=\frac{\widetilde{R}^{S}}{n}X-\widetilde{R}(\xi_{t})(X).$

Hence we have

${\rm
Tr}_{g_{t}}^{\bullet}R_{t}(X,\bullet)\bullet=\frac{\widetilde{R}^{S}}{n}X-\widetilde{R}(\xi_{t})(X)-A_{t}^{2}X+H_{t}A_{t}X.$
$None$

By using $(4.9)$ and $(4.10)$, we can show

$\begin{array}[]{l}\displaystyle{{\rm
Tr}_{g_{t}}^{\bullet}R_{t}(X,A_{t}(\bullet))(A_{t}(\bullet))={\rm
Tr}_{g_{t}}^{\bullet}R_{t}(X,\bullet)(A_{t}^{2}(\bullet))}\\\
\displaystyle{={\rm
Tr}_{g_{t}}^{\bullet}(R_{t}(X,\bullet)A_{t}^{2})(\bullet)+{\rm
Tr}_{g}^{\bullet}A_{t}^{2}(R_{t}(X,\bullet)\bullet)}\\\ \displaystyle{={\rm
Tr}_{g_{t}}^{\bullet}(R_{t}(X,\bullet)A_{t}^{2})(\bullet)+\frac{\widetilde{R}^{S}}{n}A_{t}^{2}X}\\\
\hskip
14.22636pt\displaystyle{-(A_{t}^{2}\circ\widetilde{R}(\xi_{t}))(X)-A_{t}^{4}X+H_{t}A_{t}^{3}X.}\end{array}$

Also, we have

$R_{t}(X,\cdot)A_{t}^{2}=R_{t}(X,\cdot)A_{t}\,\circ A_{t}+A_{t}\circ
R_{t}(X,\cdot)A_{t}.$

Hence we have

$[A_{t},{\rm
Tr}_{g_{t}}^{\bullet}R_{t}(X,A_{t}(\bullet))(A_{t}(\bullet))]=[A_{t}^{2},{\rm
Tr}_{g_{t}}^{\bullet}(R_{t}(X,\bullet)A_{t})(\bullet)]-A_{t}^{2}\circ S_{t}.$
$None$

Also, by using $(4.9)$ and $(4.10)$, we show

$\begin{array}[]{l}\displaystyle{{\rm
Tr}_{g_{t}}^{\bullet}R_{t}(X,\bullet)(A_{t}(\bullet))={\rm
Tr}_{g_{t}}^{\bullet}(R_{t}(X,\bullet)A_{t})(\bullet)+\frac{\widetilde{R}^{S}}{n}A_{t}X}\\\
\hskip
106.69783pt\displaystyle{-(A_{t}\circ\widetilde{R}(\xi_{t}))(X)-A_{t}^{3}X+H_{t}A_{t}^{2}X.}\end{array}$

Hence we obtain

$\begin{array}[]{l}\displaystyle{[\widetilde{R}(\xi_{t}),{\rm
Tr}_{g_{t}}^{\bullet}R_{t}(X,\bullet)(A_{t}(\bullet))]=[\widetilde{R}(\xi_{t}),{\rm
Tr}_{g_{t}}^{\bullet}(R_{t}(X,\bullet)A_{t})(\bullet)]-\frac{\widetilde{R}^{S}}{n}S_{t}}\\\
\hskip
142.26378pt\displaystyle{+S_{t}\circ\widetilde{R}(\xi_{t})+[A_{t}^{3},\widetilde{R}(\xi_{t})]-H_{t}[A_{t}^{2},\widetilde{R}(\xi_{t})].}\end{array}$
$None$

In the sequel, we omit the subscript “$t$”. By using $(4.11)$ and $(4.12)$, we
can derive

$\begin{array}[]{l}\displaystyle{P(S)=\widehat{S}-\left({\rm Tr}(A^{2})+{\rm
Tr}\,\widetilde{R}(\xi)\right)S-H[A,S]-2H[A^{2},\widetilde{R}(\xi)]}\\\ \hskip
37.27303pt\displaystyle{+S\circ A^{2}-2A^{2}\circ
S+2S\circ\widetilde{R}(\xi)-[A,\widetilde{R}(\xi)^{2}]}\end{array}$ $None$

and hence

$\begin{array}[]{l}\displaystyle{{\rm Tr}(P(S)\circ
S)=-\langle\widehat{S},S\rangle+\left({\rm Tr}(A^{2})+{\rm
Tr}\,\widetilde{R}(\xi)\right)\rho-H{\rm Tr}([A,S]\circ S)}\\\ \hskip
74.54607pt\displaystyle{-2H{\rm Tr}([A^{2},\widetilde{R}(\xi)]\circ S)-{\rm
Tr}(A^{2}\circ S^{2})}\\\ \hskip 74.54607pt\displaystyle{+2{\rm
Tr}(\widetilde{R}(\xi)\circ S^{2})-{\rm Tr}([A,\widetilde{R}(\xi)^{2}]\circ
S).}\end{array}$ $None$

Now we shall estimate ${\rm Tr}(P(S)\circ S)$ from above. By using Lemma 4.4,
we have

$\left({\rm Tr}(A^{2})+{\rm
Tr}\,\widetilde{R}(\xi)\right)\rho\leq(||A||^{2}+n||\widetilde{R}||)\rho\leq(C_{A}(T_{0})^{2}+n||\widetilde{R}||)\rho,$
$None$ ${\rm Tr}(([A,S]\circ S))={\rm Tr}(A\circ S^{2}-S\circ A\circ S)=0,$
$None$ $\begin{array}[]{l}\displaystyle{-2H{\rm
Tr}([A^{2},\widetilde{R}(\xi)]\circ S)=-2H{\rm Tr}((A\circ S+S\circ A)\circ
S)}\\\ \displaystyle{=-4H{\rm Tr}(A\circ S^{2})\leq 4n||A||^{2}\rho\leq
4nC_{A}(T_{0})^{2}\rho,}\end{array}$ $None$ $-{\rm Tr}(A^{2}\circ
S^{2})\leq||A||^{2}\rho\leq C_{A}(T_{0})^{2}\rho,$ $None$ ${\rm
Tr}(\widetilde{R}(\xi)\circ S^{2})\leq n||\widetilde{R}||\rho$ $None$

and

$\begin{array}[]{l}\displaystyle{-{\rm Tr}([A,\widetilde{R}(\xi)^{2}]\circ
S)=-{\rm Tr}(A\circ\widetilde{R}(\xi)^{2}\circ S)+{\rm
Tr}(\widetilde{R}(\xi)^{2}\circ A\circ S)}\\\ \displaystyle{=-{\rm
Tr}(S\circ(A\circ\widetilde{R}(\xi))\circ\widetilde{R}(\xi)))+{\rm
Tr}(S\circ(\widetilde{R}(\xi)\circ A)\circ\widetilde{R}(\xi))}\\\ \hskip
14.22636pt\displaystyle{-{\rm
Tr}((S\circ\widetilde{R}(\xi))\circ(A\circ\widetilde{R}(\xi)))+{\rm
Tr}((S\circ\widetilde{R}(\xi))\circ(\widetilde{R}(\xi)\circ A))}\\\
\displaystyle{=-{\rm Tr}(S^{2}\circ\widetilde{R}(\xi))+{\rm
Tr}(S\circ\widetilde{R}(\xi)\circ S)}\\\ \displaystyle{=-2{\rm
Tr}(S^{2}\circ\widetilde{R}(\xi))\leq 2n||\widetilde{R}||\rho}\end{array}$
$None$

on $M\times[0,T_{0}]$. Also, we have

$-\langle\widehat{S},S\rangle\leq
2\left(\mathop{\sup}_{t\in[0,T_{0}]}\,\mathop{\sup}_{x\in
M}\,\mu_{t}(x)\right)\cdot\rho$ $None$

on $M\times[0,T_{0}]$. From $(4.14)-(4.21)$, we obtain

$2{\rm Tr}(P(S)\circ S)\leq C_{1}(T_{0})\rho\,\,\,\,{\rm
on}\,\,M\times[0,T_{0}],$

where $C_{1}(T_{0})$ is the positive constant as in the statement of
Proposition 4.5. Hence, from $(4.8)$, we obtain

$\frac{\partial\rho}{\partial t}-\triangle\rho\leq
C_{1}(T_{0})\rho\,\,\,\,{\rm on}\,\,M\times[0,T_{0}].$

Furthermore, set $\widehat{\rho}_{t}:=e^{-C_{1}(T_{0})t}\rho_{t}$. Then we
have

$\frac{\partial\widehat{\rho}}{\partial t}-\triangle\widehat{\rho}\leq 0.$

Hence, by the maximum principle, we obtain
$\widehat{\rho}_{t}\leq\max\,\widehat{\rho}_{0}$, which is equivalent to the
inequality in the statement. ∎

Proof of Theorem D. Take any $T_{0}\in[0,T)$. Since $f$ is curvature-adapted,
we have $\rho_{0}=0$. Hence, it follows from Proposition 4.5 that $\rho_{t}=0$
holds for all $t\in[0,T_{0}]$. Therefore, from the arbitrariness of $T_{0}$,
we can conclude $\rho_{t}=0$ holds for all $t\in[0,T)$. ∎

Proof of Corollary E. Since

$\mathop{\inf}_{t\in[0,T)}\,\mathop{\min}_{x\in
M}\,\langle(\widehat{S}_{t})_{x},(S_{t})_{x}\rangle\geq 0$

by the assumption, we have

$\mathop{\sup}_{t\in[0,T)}\,\mathop{\sup}_{x\in M}\,\mu_{t}(x)\leq 0.$

Hence we can derive the statement of Corollary E from Theorem D. ∎

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