数理解析研究所講究録
第 1929 巻 2014 年 75-90

75The regularized mean curvature flow forinvariant hypersurfaces in a Hilbert spacewith an almost free Lie group action
Naoyuki Koike
Department of Mathematics, Faculty of Science

Tokyo University of Science

1

Introduction

From 1984, Huisken and many geometers studied the mean curvature flow for a hypersurface
or a submanifold (of higher codimension) in a Euclidean space as an evolution of the immersion.
The existenceness (and the uniqueness) of the mean curvature flow for an initial hypersurface (or:submanifold)in short time is assured under the assumption of the compactness of$M$ or under the assumptions of the invariantness of $f(M)$ by some Lie group action consisting ofisometries ofand the compactness of the $f(M)/G.$The study of the mean curvature flow for a submanifold $M$ in (general) Riemannian manifold
also has been done by many geometers. The evolutions of various geometric quantities (tensor
fields) along the mean curvature flow are obtained by calculating the evolutions of their components
with respect to local coordinates of $M$ and. In particular, in the case whereis an Euclidean
space, it is simpler to treat because we can use the fact thatis a linear space.
In order to define and study the mean curvature flow for an infinite dimensional submanifold: $M\mapsto V$ in a Hilbert space , we must first define the mean curvature vector of . The
mean curvature vector of should be defined by using the traces of the shape operators offor unit normal vectors but how the trace should be defined is important problem. The geodesic
closed ball with respect to the induced metric on the submanifold is not compact. Hence in order toassure the existenceness (and the uniqueness) of the mean curvature flow for an initial submanifold: $M\mapsto V$ in short time, we must impose the conditions of the invariantness of $f(M)$ by some
infinite dimensional Lie group action consisting of isometries ofand the compactness of the$f(M)/G$ . Also, since $M$ is a Hilbert manifold, we cannot use a local coordinate. Hence we shallcalculate the evolutions of various geometric quantities by using technique in the theory of thevector bundle.
Under the above background, we studied the regularized mean curvature flow for a -invariant
regularizable hypersurface in a Hilbert spaceequipped with an almost free Hilbert Lie groupisometric action $Gc\sim V$ whose orbits are minimal. See Sections 2 and 3 about the definitions of
the regularizable hypersurface and the regularized mean curvature flow. This study can be applied
to the study of the mean curvature flow in the orbit space $V/G$ (which is a Riemannian orbifold).
$f$$M\mapsto \mathbb{R}^{m}$$\mathbb{R}^{m}$$a$$\overline{M}$$\overline{M}$$\overline{M}$$\overline{M}$

$V$

$f$

$f$

$f$

$f$

$f$

$V$

$G$

$V$

76
2

Regularizable submanifolds
In this section, we shall state the definition of a regularizable submanifolds in(separable)$M$
Hilbert space. Letbe (separable) Hilbert space,
be a Hilbert manifold andbe animmersion of $M$ into . Denote by
andthe normal bundle, the shape tensor
and the normal exponential map of , respectively. If the following three conditions hold, then
: $M\mapsto V$ is called a proper Fredholm submanifold:
$V$

$a$

$a$

$f$

$V$

$T^{\perp}M,$$A$$\exp^{\perp}$

$f$

$f$

(i) $co\dim f(M)<\infty,$(ii) the restriction of$\exp^{\perp}$(iii) the diﬀerential ofto the unit normal ball bundle of is proper,
at each point of
is a Redholm operator.
$f$$T^{\perp}M$$\exp^{\perp}$

Note that the shape operators $A_{v}(v\in T^{\perp}M)$ of a proper Fredholm submanifold are compact
operators. This notion was introduced by C. L. Terng ([Te]) in 1989. In 2006, E. Heintze, X. Liu
and C. Olmos ([HLO]) defined the regularized traceof the shape operator
as follows:$Tn_{r}A_{v}$$A_{v}$$Tr_{r}A_{v}:=\sum_{i=1}^{\infty}(\mu_{i}^{+}+\mu_{i}^{-})$(

Assume that$\mu_{1}^{-}\leq\mu_{2}^{-}\leq\cdots\leq 0\leq\cdots\leq\mu_{2}^{+}\leq\mu_{1}^{+}$: the spectrum of$A_{v}$): $M\mapsto V$ is proper Fredholm. Furthermore, if there exist the regularized traceofand the (usual) trace offor any unit normal vector of , then : $M\mapsto V$ is calleda regularizable submanifold. This notion was introduced by E. Heintze, X. Liu and C. Olmos([HLO]). Let : $M\mapsto V$ be a regularizable submanifold. The regularized mean curvature vectorof is defined as the normal vector field $H$ of satisfying$f$$A_{v}$$A_{v}^{2}$

$v$

$f$

$f$

$f$

$f$

$f$

$\langle H, v\rangle=Tr_{r}A_{v}(\forall v\in T^{\perp}M)$

,

where
the inner product of . The norm of $H$ is called the regularized mean curvature of $f.$
In particular, if $H=0$ , then : $M\mapsto V$ is said to be minimal. On the other hand, the regularized
Laplacianof the vector-valued function is defined by$\langle,$$V$$\rangle is$$f$$\triangle_{r}f$$f$$\langle\triangle_{r}f, v\rangle=T\tau_{r}\langle(\nabla df)(\cdot, v\rangle^{\#}(\forall v\in T^{\perp}M)$,

where is the Riemannian connection of the induced metric on $M$ by andthe $(1, 1)$ -tensor field on $M$ defined by $g_{t}(\langle(\nabla df)(\cdot, v\rangle^{\#}(X), Y)=\langle(\nabla df)(X, Y),$
It is easy to show that $\triangle_{r}f=H$ holds.
$\nabla$$g$$f$$\langle(\nabla df)(\cdot,$$v\rangle$$v\rangle^{\#}$

is$(X, Y\in TM)$

Example 2.1. Let be a compact semi-simple Lie group equipped with a bi-invariant metric andbe an embedded submanifold in . The parallel transport map :foris definde by$G$$M(\subset G)$$G$$\phi$$H^{0}([0,1], \mathfrak{g})arrow G$$G$$\phi(u):=g_{u}(1)(u\in H^{0}([0,1], \mathfrak{g}))$$(g_{u}\in H^{1}([0,1], G)s.t.

whereg_{u}(0)=e, (R_{g_{u}(t)})_{*}^{-1}(g_{u}’(t))=u(t)(\forall t\in[O, 1)$ ,is the (separable) Hilbert space of all-paths in the Lie algebra of $G$ and$G$$H^{1}([0,1], G)$ is the Hilbert Lie group of all-paths in . Then it is shown that $\overline{M}:=\phi^{-1}(M)$ isa regularizable submanifold in(see [HLO]). The relation between the focal structures$H^{0}$$H^{0}([0,1], \mathfrak{g})$$\mathfrak{g}$$H^{1}$$H^{0}([0,1], \mathfrak{g})$77
is as in Figure 1. In the case where $M$ is curvature-adapted $(i.e., R(v)(T_{x}M)\subset$$T_{x}M,$ $[A_{v}, R(v)]=0(\forall x\in M,$
we shall state the relation between the spectrums of
the shape operators of $M$ and, whereis the curvature tensor of and $R(v)$ is the normal
Jacobi operator for $(i.e., R(v) :=R v)v$ ). Take a unit normal vector of $M$ at $x\in M$ . Let
be the horizontal lift of to $u\in\phi^{-1}(x)$ . Denote by
and the shape operators of $M$ and$:=Ker(A_{v}-\lambda id)$
respectively. Set$:=Ker(R(v)-\mu id)$Spec A andSpec R $(v))$ .
Then Specis described asof $M$ and$\overline{M}$$\forall v\in T_{x}M$$\overline{M}$

$R$

$G$

$v$

$v_{u}^{L}$$v$$\tilde{A}$

$A$

$v$

$D_{\lambda}^{A}$$(\lambda\in$$)$$\overline{M},$$(\mu\in$$D_{\mu}^{R}$$\overline{A}_{v_{u}^{L}}\backslash \{O\}$Spec$\tilde{A}_{v_{u}^{L}}\backslash \{O\}$$=\{\lambda|\lambda\in$Spec A$s.t. D_{\lambda}^{A}\cap D_{0}^{R}\neq\{O\}\}$$\cup\{\frac{\mu}{\arctan(\mu/\lambda)+j\pi}|(\lambda, \mu)\in$Spec R $(v)$ s.t.

$\cup\{\frac{\mu}{j\pi}|\mu\in$Spec A$\cross$

Spec R $(v)s.t.$$D_{\mu}^{R}\cap T_{x}^{\perp}M\neq\{0\},$$D_{\lambda}^{A}\cap D_{\mu}^{R}\neq\{0\},$$j\in \mathbb{Z}\}$$j\in \mathbb{Z}\backslash \{0\}\}.$

From this description, it follows that the regularized trace ofexists. Also, it follows that theregularized mean curvature vector ofis the horizontal lift of the mean curvature vector of $M.$$\tilde{A}_{v_{u}^{L}}$$\overline{M}$$X\in$(the nullity space of focal points $p_{i}(i\in \mathbb{N})$“the nullity space of a focal point
are linearly independent.)$\tilde{X}_{i}(i\in \mathbb{N})\in$($\tilde{X}_{i}(i\in \mathbb{N})$$\tilde{p}_{i}$Figure 1.

3

Regularized mean curvature flow

In this section, we shall state the definition of a regularized mean curvature flow in (separable)
Hilbert space. Let
and $M$ be as in the previous section. Let $f_{t}(0\leq t<T)$ be a-family of$M$
regularizable immersions of
into . Denote by
the regularized mean curvature vector of$arrowV$$map\cdot
F:M\cross[0,T$$F(x,t)=f_{t}(x)((x,t)\in M\cross[0, T))$ . We call $f_{t}s(0\leq t<T)$
Define a)by
the regularized mean curvature flow if the following evolution equation holds:
$a$

$V$

$C^{\infty}$

$V$

(2.1)

$H_{t}$$\frac{\partial F}{\partial t}=\triangle_{t}^{r}f_{t},$$f_{t}.$

78
where

$\triangle_{t}^{r}f_{t}$is the regularized Laplacian of$f_{t}$. In general, the existenceness and$(i.e., \triangle_{t}^{r}f_{t}=H_{t})$the uniqueness (in short time) of solutions of this evolution equation satisfying any initial condition
has not been shown yet. For we cannot apply the Hamilton‘s result ([Ha]) to this evolution equation
because it is regarded as the evolution equation for sections of the infinite dimensional vector bundle$M\cross V$ over $M$ .
However we can show the existenceness and the uniqueness (in short time) of
solutions of this evolution equation in the following special case. We consider the case where
equips an almost free and isometric Hilbert Lie group action $G\sim V$ with minimal regulariazable
fibres and where : $M\mapsto V$ is a -invariant embedded hypersurface in such that $f(M)/G$ iscompact. Then it is shown that the rgularized mean curvature flow for uniquely exists in short
time.
$V$

$V$

$G$

$f$

$f$

$/f(M)$$C\underline{f}$

$V$

$M$
$\downarrow\phi$$V/G$$\backslash$$(\phi\circ f)(M)$

Figure 2.
Example 3.1. Let $G$ be a compact semi-simple Lie group equipped with a bi-invariant metric
and $K$ be a closed subgroup of . Also, let and be the Lie algebras of and $K$ , respectively.
. Also, let be a discrete subgroup
Assume that
admits a reductive decomposition
$G$ .
$P(G,
\Gamma\cross
K)$
by
of
We define a Hilbert Lie group
$G$

$G$

$e$

$\mathfrak{g}$$(\mathfrak{g}, f)$$\Gamma$$\mathfrak{g}=e+\mathfrak{p}$$P(G, \Gamma\cross K) :=\{g\in H^{1}([0,1], G)|(g(0), g(1))\in\Gamma\cross K\}.$

This group $P(G, \Gamma\cross K)$ acts onas the action of a Gauge action on the space of theconnections, where $H^{1}([0,1], G)$ is the Hilbert Lie group of all-paths in $G$ . This action is$H^{0}([0,1], \mathfrak{g})$$H^{1}$an almost free and isometric action whose orbits are minimal regularizable submanifolds and theorbits space
is equal to$H^{0}([0,1], \mathfrak{g})/P(G, \Gamma\cross K)$$\Gamma\backslash G/K.$

79
The mean curvature flow for suborbifolds

4

In this section, we shall define the notion of the mean curvaure flow for a suborbifold in a
Riemannian orbifold. First we recall the notions of a Riemannian orbifold and a suborbifold
following to [AK,GKP,Sh,Th]. Let $M$ be a paracompact Hausdorﬀ space anda triplesatisfying the following conditions:$(U, \phi,\tilde{U}/\Gamma)$(i) $U$ is an open set of $M,$

is an open set of
and is a finite subgroup of thegroup $Dﬃ^{k}(U)$ of(iii) is a homeomorphism of onto(ii)$\hat{U}$$\mathbb{R}^{n}$$\Gamma$$C^{k}$-diﬀeomorphism$\hat{U},$$U$$\phi$$\hat{U}/\Gamma.$

Such a triple
is called an -dimensional orbifold chart. Let
be a family of -dimensional orbifold charts of $M$ satisfying the following conditions:$(U, \phi,\hat{U}/\Gamma)$$\Lambda\}$$n$$\mathcal{O}$$:=\{(U_{\lambda}, \phi_{\lambda},\hat{U}/\Gamma_{\lambda})|\lambda\in$

$n$

(O1)
(02) For any

is an open covering of $M,$
withand any
an -dimensional orbifold chartsuch that$\{U_{\lambda}|\lambda\in\Lambda\}$$\lambda,$$\mu\in\Lambda$$U_{\lambda}\cap U_{\mu}\neq\emptyset$$n$$\rho_{\lambda}$:$x\in U_{\lambda}\cap U_{\mu}$and$\hat{W}\mapsto\hat{U}_{\lambda}$:

satisfying, whereand, respectively.
$\rho_{\mu}$$\hat{W}\mapsto\hat{U}_{\mu}$$\pi_{\Gamma_{\lambda}},$$\Gamma_{\lambda},$$\Gamma_{\mu}$and$C^{k}$$\phi_{\lambda}^{-1}\circ\pi_{\Gamma_{\lambda}}\circ\rho_{\lambda}=\psi^{-1}0\pi_{\Gamma’}$$\phi_{\mu}^{-1}\circ\pi r_{\mu}\circ\rho_{\mu}=\psi^{-1}\circ\pi_{\Gamma’}$$\Gamma’$, there exists-embeddings
andare the orbit maps of$(W, \psi,\hat{W}/\Gamma’)$$\pi_{\Gamma’}$$\pi_{\Gamma_{\mu}}$

Such a family
is called an -dimensional
is called-orbifold atlas of $M$ and the pairan -dimensional -orbifold. Let
be an n-dimensional orbifold chart around $x\in M$ . Then the groupis unique forup to the conjugation, where is a point ofwith. Denote bytheconjugate class of this group, This conjugate class is called the local group at . If the localgroup at is not trivial, then is called a singular point of. Denote by Sing(orSing (M) ) the set of all singular points of. This set Singis called the singular set of$C^{k}$$\mathcal{O}$$n$$C^{k}$$n$$(M, \mathcal{O})$$(U_{\lambda}, \phi_{\lambda},\hat{U}_{\lambda}/\Gamma_{\lambda})$$(\Gamma_{\lambda})_{\hat{x}}$$x$$\hat{U}_{\lambda}$$\hat{x}$$:=\{b\in\Gamma_{\lambda}|b(\hat{x})=\hat{x}\}$$(\phi_{\lambda}^{-1}0\pi_{\Gamma_{\lambda}})(\hat{x})=x$$(\Gamma_{\lambda})_{x}$$(\Gamma_{\lambda})_{\hat{x}}$

$x$

$x$

$(M, \mathcal{O})$$x$$(M, \mathcal{O})$$(M, \mathcal{O})$$(M, \mathcal{O})$$(M, \mathcal{O})$

.

Let
and
be orbifolds, anda map from $M$ to $N$ . If, for each $x\in M$
and each pair of an orbifold chartofaround and an orbifold chartofaround $f(x)(f(U_{\lambda})\subset V_{\mu})$ , there exists a
-map
:
with
$(M, \mathcal{O}_{M})$$(N, \mathcal{O}_{N})$$f$$(U_{\lambda}, \phi_{\lambda},\hat{U}_{\lambda}/\Gamma_{\lambda})$$(V_{\mu}, \psi_{\mu},\hat{V}_{\mu}/\Gamma_{\mu}’)$$(M, \mathcal{O}_{M})$$x$$C^{k}$$(N, \mathcal{O}_{N})$, then $f$ is called a$\hat{f_{\lambda,\mu}}$$\hat{U}_{\lambda}arrow\hat{V}_{\mu}$-orbimap (or simply a-map). Alsois called a local lift of with respect to. Furthermore, if each
andlocal lift
is an immersion, then is called a-orbiimmersion (or simply a-immersion)$C^{k}-($
andis called a. Similarly, if each local lift\’immersed) suborbifold in
is a submersion, then is called a-orbisubmersion.
Now we shall define the notion of the mean curvature flow for a-suborbifold in a
Riemannian orbifold. Let $f_{t}(0\leq t<T)$ be a-family of-orbiimmersions of a-orbifold$(x_{0},t_{0})\inM\cross[O,T$ )into a-Riemannian orbifold. Assume that, for eachand each pair of an orbifold chartofaroundand an orbifold chartofaroundsuch thatfor any $t\in[t_{0}, t_{0}+\epsilon$ ):suﬃciently small positive number), there exists local lifts:
of $f_{t}(t\in[t_{0}, t_{0}+\epsilon))$$f\circ\phi_{\lambda}^{-1}\circ\pi r_{\lambda}=\psi_{\mu}^{-1}\circ\pi_{\Gamma_{\mu}’}\circ\hat{f_{\lambda,\mu}}$$f$$C^{k}$$(U_{\lambda}, \phi_{\lambda},\hat{U}_{\lambda}/\Gamma_{\lambda})$$C^{k}$$(V_{\mu}, \psi_{\mu},\hat{V}_{\mu}/\Gamma_{\mu}’)$$C^{k}$$\hat{f_{\lambda,\mu}}$$f$$(M, \mathcal{O}_{M})$$\hat{f_{\lambda,\mu}}$$C^{k}$$(N, \mathcal{O}_{N}, g)$$f$$\hat{f_{\lambda,\mu}}$$C^{k}$$C^{\infty}$$C^{\infty}$$(M, \mathcal{O}_{M})$$C^{\infty}$$(N, \mathcal{O}_{N})$$C^{\infty}-$$C^{\infty}$$(N, \mathcal{O}_{N}, g)$$(U_{\lambda}, \phi_{\lambda},\hat{U}_{\lambda}/\Gamma_{\lambda})$$(V_{\mu}, \phi_{\mu},\hat{V}_{\mu}/\Gamma_{\mu}’)$$C^{\infty}$$f_{t_{0}}(x_{0})$$(M, \mathcal{O}_{M})$$x_{0}$$f_{t}(U_{\lambda})\subset V_{\mu}$$(\hat{f_{t}})_{\lambda,\mu}$$\hat{U}_{\lambda}arrow\hat{V}_{\mu}$$(\epsilon$

$a$

80
such that they give the mean curvature flow in
we call $f_{t}(0\leq t<T)$ the mean curvature flow in$(\hat{V}_{\mu},\hat{g}_{\mu})$, where.

$\hat{g}_{\mu}$is the local lift of

$g$

to

$\hat{V}_{\mu}$. Then$(N, \mathcal{O}_{N}, g)$

Theorem 4.1([K3]). For any-orbiimmersion
of a compact-orbifold into a
Riemannian orbifold, the mean curvature flow starting from exists uniquely in short time.
$C^{\infty}$$C^{\infty}$$f$$C^{\infty}-$

$f$

be a
-orbiimmersion of an -dimensional compact-orbifoldinto an$(n+r)$ -dimensional. Fix $x_{0}\in M$ . Take an orbifold chart-Riemmannian orbifoldofaroundand an orbifold chartofaround$f(x_{0})$ such thatand thatis relative compact. Also, let:
be alocal lift of anda local lift of (to ). Since
is relative compact, there exists the mean:
curvature flow
starting from:
. Since-invariant,
is projetable toandisalso are projectable to maps ofinto. Denote by
maps
these
of
. It is clear that $(f_{\lambda,\mu})_{t}(0\leq t<T)$
into
is the mean curvature flow starting from. Hence, it follows from the arbitrariness ofandthe compactness of $M$ that the mean curvature flow starting from exists uniquely in short time.
q.e.

Proof. Let$C^{\infty}$$f$$C^{\infty}$$(U_{\lambda}, \phi_{\lambda},\hat{U}_{\lambda}/\Gamma_{\lambda})$$C^{\infty}$$n$$(N, \mathcal{O}_{N}, g)$$(M, \mathcal{O}_{M})$$(V_{\mu}, \psi_{\mu},\hat{V}_{\mu}/\Gamma_{\mu}’)$$x_{0}$$\hat{U}_{\lambda}$$\hat{f_{\lambda,\mu}}$$f(U_{\lambda})\subset V_{\mu}$$f$$f|_{U_{\lambda}}$$U_{\lambda}$$V_{\mu}$$(N, \mathcal{O}_{N})$$\hat{U}_{\lambda}\mapsto\hat{V}_{\mu}$$\hat{U}_{\lambda}$$\hat{V}_{\mu}$$\hat{g}_{\mu}$$(\hat{f_{\lambda,\mu}})_{t}$$(M, \mathcal{O}_{M})$$g$$\hat{U}_{\lambda}\mapsto(\hat{V}_{\mu\rangle}\hat{9}_{\mu})(0\leq t<T)$$\hat{g}_{\mu}$$\hat{f_{\lambda,\mu}}$$\hat{U}_{\lambda}\mapsto(\hat{V}_{\mu},\hat{g}_{\mu})$$\hat{f_{\lambda,\mu}}$$(\hat{f_{\lambda,\mu}})_{t}(0\leq t<T)$$\Gamma_{\mu}’$$(f_{\lambda,\mu})_{t}s$$U_{\lambda}$$V_{\mu}$$f|_{U_{\lambda}}$$x_{0}$$f$$d.$$ﬄ\rfloor 4.1.$$f(M)$$\sim$
$f$

$M$

$N$

Figure 3.

$\downarrow id \downarrow \pi_{\Gamma’}(\Gamma’\cong \mathbb{Z}_{4})$$\mapsto f$$\backslash s:H$

$M(=\mathbb{R})$$N(=\mathbb{R}^{2}/\Gamma’)$Figure 4.

81$\hat{f_{t}}(\mathbb{R})(t>0)$$\hat{f}(\mathbb{R})$time goes bytime goes byII

II

Figure 5.

$ﬃ^{1\rfloor 4.2}.$$f(M)$$M$$N$

Figure 6.

$\mathbb{R}^{2}$$\backslash s:\hat{H}$(a local lift of $H$ )$\downarrow\pi_{\Gamma’}(\Gamma’\cong \mathbb{Z}_{2}\oplus \mathbb{Z}_{2})$$\downarrow\pi_{\Gamma}(\Gamma=\mathbb{Z}_{2})$$-$$\mapsto f$$\backslash s:H$

$M(=\mathbb{R}/\Gamma)$$N(=\mathbb{R}^{2}/\Gamma’)$Figure 7.

82$\hat{f_{t}}(\mathbb{R})(t>0)$$\hat{f}(\mathbb{R})$time goes byII$\prime I$$h_{\wedge}\sim\sim-arrow A_{-}\sim$$f(M) f_{t}\overline{(}M) (t>0)$
Figure 8.

Evolution equations

5

Let $G\sim V$ be an isometric almost free action with minimal regularizable orbit of a Hilbert Lie. The orbit space $V/G$ is agroupequipped with an inner producton a Hilbert space(finite dimensional)-orbifold. Let : $Varrow V/G$ be the orbit map and set $N$ $:=V/G$ . Give
$N$ the Riemannian orbimetric such thatis a Riemannian orbisubmersion. Let : $M\mapsto V$ bea -invariant submanifold such that $(\phi\circ f)(M)$ is compact. For this immersion , we can takean orbiimmesion of a compact orbifold
into $N$ and an orbisubmersion:
with. Let $\overline{f}_{t}(0\leq t<T)$ be the mean curvature flow for . The existenceness and theuniqueness of this flow in short time is assured by Theorem 4.1. Define a map) $arrow N$
byDenote by $H$ the regularized mean curvature vectorof andthe mean curvature vector of . Since has minimal regularizable fibres, $H$ is the: $[0, T$ ) $arrow N$ byhorizontal lift of . Take
. Define a curveand: $[0, T$ ) $arrow V$ be the horizontal lift of for $f(u)$ . Define an immersion $f_{t}:M\mapsto V$ by
and let$f_{t}(u)=(c_{x})_{u}^{L}(t)(u\in\overline{M})$and a map $F:M\cross[0, T$ ) $arrow V$ by $F(u, t)=f_{t}(u)((u, t)\in M\cross[0, T))$ .
$V$

$G$

$\langle,$$C^{\infty}$$\rangle$$\phi$$f$$\phi$$G$$f$$\overline{M}$$\overline{f}$$\phi_{M}$$\phi\circ f=\overline{f}\circ\phi_{M}$$Marrow\overline{M}$$\overline{f}$$\overline{F}:\overline{M}\cross[0, T$$\overline{F}(x, t)$$:=\overline{f}_{t}(x)((x, t)\in\overline{M}\cross[0,$ $T$$\overline{H}$$f$$\overline{f}$$x\in\overline{M}$$\overline{H}$$\phi$$u\in\phi_{M}^{-1}(x)$$(c_{x})_{u}^{L}$$c_{x}(t)$$c_{x}$$:=\overline{f}_{t}(x)$$c_{x}$

Proposition 5.1 ([K3]). The flow $f_{t}(0\leq t<T)$ is the regularized mean curvature flow for $f.$the mean curvature vector ofandthe regularized mean curvature
$(u,
t)\in
M\cross[0,T$$x:=\phi_{M}(u)$). Set. It is clear that
vector of . Take any. Hence,since each fibre of is regularizable and minimal,
coincides with one of the horizontal lifts

Proof. Denote by$\overline{H}_{t}$$\overline{f}_{t}$$H_{t}$$\phi\circ f_{t}=\overline{f}_{t}\circ\phi_{M}$$f_{t}$$(H_{t})_{u}$$\phi$$\partial F$of$(\overline{H}_{t})_{x}$to $f_{t}(u)$ . On the other hand, from the definition of $F$ , we have

which is one of the horizontal lifts of$(\overline{H}_{t})_{x}$to$f_{t}(u)$$\overline{\partial t}^{(u,t)}=((c_{x})_{u}^{L})’(t)$. These facts together with,$\frac{\partial F}{\partial t}(u, 0)=H_{u}$. Thus it follows from the arbitrariness of $(u, t)$ that $f_{t}(0\leq t<T)$
q.e.
is the regularized mean curvature flow for . This completes the proof.
implies that$\frac{\partial F}{\partial t}(u, t)=(H_{t})_{u}$

$d.$

$f$

Assume that the codimension of$f$is equal to one. Denote by$\tilde{\mathcal{H}}$(resp.

$\tilde{\mathcal{V}}$) the horizontal (resp.

83
vertical) distribution of . Denote by(resp.
) the orthogonal projection of $TV$ onto
(resp.
). For simplicity, for $X\in TV$ , we denote(resp.
) by(resp.
).
$M$
Define a distribution
byonand a distribution
on $M$ by. Note thatindependentofisthe choice of $t\in[0, T$ ).
Denote by
and the induced metric, the second fundamental form, the shape tensor
and the regularized mean curvature vector and the unit normal vector field of , respectively. The
group
acts on $M$ through . Since : $Varrow V/G$ is a -orbibundle andis a connection$G\capM$of the orbibundle, it follows from Proposition 5.1 that this action
is independent of the$t\in[0,T$). It is clear that quantities
choice ofare -invariant. Also, let
be theandRiemannian connection of . Let
be the projection of $M\cross[0, T$ ) onto $M$ . For a vector bundle$E$
over $M$ , denote by
the induced bundle of $E$ by. Also denote by
the space of all$E$$\pi_{M}^{*}(T^{(0,2)}M)$
sections of . Define a section of
where
by $g(u, t)=(g_{t})_{u}((u, t)\in M\cross[O,$$T^{(0,2)}M$$\pi_{M}^{*}(T^{(0,2)}M)$$M$, a sectionis the $(0,2)$ -tensor bundle of . Similarly, we define a section of$\pi_{M}^{*}(T^{(1,1)}M)$$F$$H$$F^{*}TV$$TV$, sections
ofby . We regard $H$ andand of the induced bundleofas -valued functions over $M\cross[0, T$ ) under the identification of $T_{F(u,t)}Vs((u, t)\in M\cross[0, T))$and . Define a subbundle).
(resp. V) of(resp.
by(resp.
) the orthogonal projection of $\pi_{M}^{*}(TM)$ onto
Denote by(resp. ). For simplicity,
for $X\in\pi_{M}^{*}(TM)$ , we denote). The bundle $\pi_{M}^{*}(TM)$(resp.
(resp.
) by$\tilde{\mathcal{H}}$$\phi$$pr_{\tilde{\mathcal{V}}}$$pr_{\overline{\mathcal{H}}}$$\tilde{\mathcal{V}}$$pr_{\overline{\mathcal{V}}}(X)$$pr_{\overline{\mathcal{H}}}(X)$$f_{t*}((\mathcal{V}_{t})_{u})=\tilde{\mathcal{V}}_{f_{t}(u)}(u\in M)$$g_{t},$$h_{t},$$X_{\tilde{\mathcal{V}}}$$X_{\overline{\mathcal{H}}}$$f_{t*}((\mathcal{H}_{t})_{u})=f_{t*}(T_{u}M)\cap\tilde{\mathcal{H}}_{f_{t}(u)}(u\in M)$$\mathcal{H}_{t}$$A_{t},$ $H_{t}$$\mathcal{V}_{t}$$\mathcal{V}_{t}$$\xi_{t}$$f_{t}$$G$$f_{t}$$\tilde{\mathcal{H}}$$G$$\phi$$g_{t},$$G$$H_{t}$$h_{t}A_{t}\rangle$$\nabla^{t}$$\pi_{M}$$g_{t}$$\pi_{M}^{*}E$$\Gamma(E)$$\pi_{M}$

$T$

$g$

$h$

$A$

$\xi$

$\xi$

$V$

$V$

$\pi_{M}^{*}TM$$\mathcal{H}$$:=(\mathcal{H}_{t})_{u}$$\mathcal{H}_{(u,t)}$$\mathcal{H}$$pr_{\mathcal{H}}$$v_{(u,t)}$$:=(\mathcal{V}_{t})_{u}$$\mathcal{V}$$pr_{\mathcal{V}}$$pr_{\mathcal{H}}(X)$$pr_{\mathcal{V}}(X)$$X_{\mathcal{V}}$$X_{\mathcal{H}}$$\partial B$

is regarded as a subbundle of $T(M\cross[O, T For a$ section $B of \pi_{M}^{*}(T^{(r,s)}M)$ , we define$( \frac{\partial B}{\partial t})_{(u,t)}$$:= \frac{dB_{(u,t)}}{dt}$$\overline{\partial t}$by, where the right-hand side of this relation is the derivative of the vector-

valued function $t\mapsto B_{(u,t)}(\in T_{u}^{(r,s)}M)$ . Also, we define a section$B_{\mathcal{H}}$of $\pi_{M}^{*}(T^{(r,s)}M)$ by$B_{\mathcal{H}}=(pr_{\mathcal{H}}\otimes\cdots\otimes pr_{\mathcal{H}})\circ B\circ(pr_{\mathcal{H}}\otimes\cdots\otimes pr_{\mathcal{H}})$.

$(s-$ times)$(r-$ times)

The restriction ofto( -times) is regarded as a section of the $(r, s)$ -tensor bundleof . This restriction also is denoted by the same symbol. For a tangent vector field $X$$M$$M$$(or \pi_{M}^{*}TM|_{U})$ by$:=X_{u}$(or an open set
onof ), we define a sectionof$((u, t)\in M\cross[0, T))$ . Denote byof
the Riemannian connection of . Define a connectionby$\mathcal{H}\cross\cdots\cross \mathcal{H}$$B_{\mathcal{H}}$$\mathcal{H}^{(r,s)}$$s$$\mathcal{H}$$B_{\mathcal{H}}$$U$$\overline{X}$$\pi_{M}^{*}TM$$\tilde{\nabla}$$\overline{X}_{(u,t)}$$V$$\nabla$$\pi_{M}^{*}TM$$(\nabla xY)_{(\cdot,t)}:=\nabla_{X}^{t}Y_{(\cdot,t)}$andfor $X\in T_{(u,t)}(M\cross\{t\})$ and $Y\in\Gamma(\pi_{M}^{*}TM)$ , where$\nabla_{す_{}\overline{t}^{\partial}}Y:=\frac{dY_{(u,)}}{dt}$$\underline{dY_{(u,t)}}$$dt$

is the derivative of the vector-valued

function $t\mapsto Y_{(u,t)}(\in T_{u}M)$ . Define a connectionbyof
for $X\in$$T(M\cross[0_{\}}T))$ and. Similarly, define a connection
forof V by$X\in T(M\cross[0, T))$ and. Now we shall derive the evolution equations for some geometricquantities. First we derive the following evolution equation for$\nabla^{\mathcal{H}}$$\mathcal{H}$$\nabla^{\mathcal{V}}$$Y\in\Gamma(\mathcal{H})$$\nabla_{X}^{\mathcal{H}}Y$$:=(\nabla_{X}Y)_{\mathcal{H}}$$\nabla_{X}^{\mathcal{V}}Y$$:=(\nabla_{X}Y)_{\mathcal{V}}$$Y\in\Gamma(\mathcal{V})$$g_{\mathcal{H}}.$

Lemma 5.2([K3]). The sections$(g_{\mathcal{H}})_{t}s$of $\pi_{M}^{*}(T^{(0,2)}M)$ satisfy the following evolution equation:$\frac{\partial g_{\mathcal{H}}}{\partial t}=-2||H||h_{\mathcal{H}},$

where $||H||$$:=\sqrt{\langle H,H\rangle}.$84
Proof Take

$X,$ $Y\in\Gamma(TM)$. We have$==\{\begin{array}{l}\frac{\partial g_{\mathcal{H}}}{\frac{p}{\partial t}(}(X^{-},\overline{Y})=\frac{\partial}{F\partial\}}g_{\mathcal{H}}(X^{-},\overline{Y})=\frac{\partial}{\partial t,(}g(\overline{X}_{\mathcal{H}}’\overline{Y}_{\mathcal{H}})X_{\mathcal{H}}^{-}F),\overline{Y}_{\mathcal{H}}+\langle X_{\mathcal{H}}^{-}F,\frac{\partial}{\partial t}\overline{Y}_{\mathcal{H}}F)\rangle=\frac{\partial}{\partial t}\langle F_{*}\overline{X}_{\mathcal{H}}, F_{*}\overline{Y}_{\mathcal{H}}\rangle\overline{X}_{\mathcal{H}}(\frac{\partial F}{\partial t})+[\frac{\partial}{\partial t}, X_{\mathcal{H}}^{-}]F, \overline{Y}_{\mathcal{H}}F\rangle+\langle X_{\mathcal{H}}^{-}F, \overline{Y}_{\mathcal{H}}(\frac{\partial F}{\partial t})+[\frac{\partial}{\partial t},-Y_{\mathcal{H}}]F\rangle\end{array}$$=\langle\overline{X}_{\mathcal{H}}(||H||\xi) , \overline{Y}_{\mathcal{H}}F\rangle+\langle\overline{X}_{\mathcal{H}}F, \overline{Y}_{\mathcal{H}}(||H||\xi)\rangle$$=-||H||g(A\overline{X}_{\mathcal{H}},\overline{Y}_{\mathcal{H}})-||H||g(\overline{X}_{\mathcal{H}}, A\overline{Y}_{\mathcal{H}})=-2||H||h_{\mathcal{H}}(X^{-}, Y)-,$

where we use$[ \frac{\partial}{\partial t},\overline{X}_{\mathcal{H}}]\in \mathcal{V}$and$\overline{Y}_{\mathcal{H}}]\in \mathcal{V}$$[ \frac{\partial}{\partial t}$). Thus we obtain the desired evolution equation.
q.e.

Next we derive the following evolution equation forLemma 5.3([K3]). The unit normal vector fields$\xi_{t}$$d.$$\xi.$’s satisfy the following evolution equation:$\frac{\partial\xi}{\partial t}=-F_{*}(grad_{g}||H||)$,

where $grad_{9}(||H||)$ is the element of $\pi_{M}^{*}(TM)$ such that $d||H||(X)=g(grad_{g}||H||, X)$ for any$X\in\pi_{M}^{*}(TM)$ .

Proof Since$M\cross[O, T)$

expanding$\langle\xi,$. Let, we have$\langle g^{\partial},$$\xi\rangle=0$$1\partial\partial t$

is tangent to $f_{t}(M)$ . Take any. By the Fourier
with respect to$(u_{0}, t_{0})\in$$T_{u_{0}}M$$\{e_{i}\}_{i=1}^{\infty}$$\doteqdot^{\partial_{t}}|_{t=t_{0}}$$( \frac{\partial\xi}{\partial t}$. Hencebe an orthonormal base of, we have$\xi\rangle=1$$g_{(u_{O:}t_{O})}$$)_{(u_{0:}t_{0})}= \sum\langle(\frac{\partial\xi}{\partial t})_{(u_{0:}t_{0})},f_{t_{O}*}(\overline{e}_{i}|_{t=t_{0}})\rangle f_{t_{0}*}(\overline{e}_{i}|_{t=t_{0}})$$=- \sum^{=}-\sum\{\begin{array}{l}\xi_{(u_{0_{i}}t_{0})}, \frac{\partial f_{t*}(\overline{e}_{i})}{\partial t}|_{t=t_{0}}\rangle f_{t_{O}*}(\overline{e}_{\’{i}}|_{t=t_{0}})=-\sum\langle\xi_{(u_{O:}t_{0})}, \frac{\partial}{\partial t}(\overline{e}_{i}F)|_{t=t_{0}}\rangle f_{t_{0}*}(\overline{e}_{i}|_{t=t_{0}})\xi_{(u_{0},t_{0})}, \overline{e}_{i}(\frac{\partial F}{\partial t}|_{t=t_{0}})\rangle f_{t_{0}*}(\overline{e}_{i}|_{t=t_{0}})=-\sum\langle\xi_{(u_{0},t_{0})},(\overline{e}_{i}H)|_{t=t_{0}}\rangle f_{t_{0}*}(\overline{e}_{i}|_{t=t_{0}})\end{array}$$=- \sum(\overline{e}_{i}||H||)|_{t=t_{0}}f_{t_{0}*}(\overline{e}_{i}|_{t=t_{0}})=-\sum g_{(u_{0:}t_{0})}(grad_{g_{(ut)}00}||H_{(u_{O}t_{O}} \overline{e}_{i}|_{t=t_{O}})f_{t_{O}*}(\overline{e}_{i}|_{t=t_{0}})$$=-f_{t_{O}*}(grad_{g(u0\ell_{0})}||H_{(u_{0:}t_{0})}||)=-(F_{*}(grad_{g}||H||))_{(u0_{:}t_{0})},$

where we use$(k\in \mathbb{N})$$[ \frac{\partial}{\partial t},\overline{e}_{i}]=0$. Here we note that$\sum(\cdot)_{i}$means$\lim_{karrow\infty}\sum_{i\in S_{k}}(\cdot)_{i}$as$S_{k}$$:= \{i||(\cdot)_{i}|>\frac{1}{k}\}$. This completes the proof.

q.e.

-family of $a(r, s)$ -tensor fields on $M$ and a section of
Let $S_{t}(0\leq t<T)$ be a$:=(S_{t})_{u}$ . We define a section
defined byof $\pi_{M}^{*}(T^{(r,s)}M)$ by$c\infty$$S$$S_{(u,t)}$$d.$$\pi_{M}^{*}(T^{(r,s)}M)$$\triangle_{\mathcal{H}}S$$( \triangle_{\mathcal{H}}S)_{(u,t)}:=\sum_{i=1}^{n}\nabla_{e_{i}}\nabla_{e_{1}}S,$$(or \pi_{M}^{*}(T^{(r,s+1)}M))$ induced from V andwhere V is the connection ofan orthonormal base of. Also, we define a section
with respect to$\pi_{M}^{*}(T^{(r,s)}M)$$\mathcal{H}_{(u,t)}$$(g_{\mathcal{H}})_{(u,t)}$$\{e_{1}, \cdots, e_{n}\}$$\triangle_{\mathcal{H}}S_{\mathcal{H}}-$ofis$\mathcal{H}^{(r,s)}$85by$( \triangle_{\mathcal{H}}-S_{\mathcal{H}})_{(u,t)}:=\sum_{\iota’=1}^{n}\nabla_{e_{i}}^{\mathcal{H}}\nabla_{e_{i}}^{\mathcal{H}}S_{\mathcal{H}},$

whereis the connection of(or) induced fromLet
be the section of $T^{*}V\otimes T^{*}V\otimes TV$ defined by$\nabla^{\mathcal{H}}$$\mathcal{H}^{(r,s)}$$\mathcal{H}^{(r,s+1)}$$\nabla^{\mathcal{H}}$and$\{e_{1}, \cdots, e_{n}\}$is as above.

$\mathcal{A}^{\phi}$$\mathcal{A}_{X}^{\phi}Y:=(\tilde{\nabla}_{X_{\overline{\mathcal{H}}}}Y_{\overline{\mathcal{H}}})_{\tilde{\nu}}+(\tilde{\nabla}_{X_{\overline{\mathcal{H}}}}Y_{\overline{\mathcal{V}}})_{\tilde{\mathcal{H}}} (X, Y\in TV)$

Also, letbe the section of $T^{*}V\otimes T^{*}V\otimes TV$ defined by$\mathcal{T}^{\phi}$$\mathcal{T}_{X}^{\phi}Y:=(\tilde{\nabla}_{X_{\overline{\mathcal{V}}}}Y_{\tilde{\mathcal{H}}})_{\tilde{\nu}}+(\tilde{\nabla}_{X_{\overline{\mathcal{V}}}}Y_{\overline{\mathcal{V}}})_{\overline{\mathcal{H}}} (X, Y\in TV)$

Also, let$\mathcal{A}_{t}$.

.

be the section of $T^{*}M\otimes T^{*}M\otimes TM$ defined by$(\mathcal{A}_{t})_{X}Y:=(\nabla_{X_{7t_{t}}}^{t}Y_{\mathcal{H}_{t}})_{\mathcal{V}_{t}}+(\nabla_{X_{\mathcal{H}_{t}}}^{t}Y_{\mathcal{V}_{t}})_{\mathcal{H}_{t}} (X, Y\in TM)$.

Also let be the section of $\pi_{M}^{*}(T^{*}M\otimes T^{*}M\otimes TM)$ defined in terms of $\mathcal{A}_{t}s(t\in[0, T))$ . Also,
letbe the section of $T^{*}M\otimes T^{*}M\otimes TM$ defined by$\mathcal{A}$$\mathcal{T}_{t}$$(\mathcal{T}_{t})x^{Y}:=(\nabla_{x_{v_{t}}}^{t}Y_{\mathcal{V}_{f}})_{\mathcal{H}_{t}}+(\nabla_{x_{v_{t}}}^{t}Y_{\mathcal{H}_{t}})\nu_{t} (X, Y\in TM)$

Also let
we have$\mathcal{T}$be the section of $\pi_{M}^{*}(T^{*}M\otimes T^{*}M\otimes TM)$ defined in terms of.

$\mathcal{T}_{t}s(t\in[0,$ $T$

Clearly$F_{*}(\mathcal{A}_{X}Y)=\mathcal{A}_{F_{*}X}^{\phi}F_{*}Y$for $X,$$Y\in \mathcal{H}$and$F_{*}(\mathcal{T}_{W}X)=\mathcal{T}_{F_{*}}^{\phi}{}_{W}F_{*}X$for$X\in \mathcal{H}$

we defineand$W\in \mathcal{V}$

. Let

$E$

be a vector bundle over $M$ . For a section$Tr_{9\mathcal{H}}S(\cdots, oj, \cdots, k, \cdots)$$S$of $\pi_{M}^{*}(T(0,r)M\otimes E)$ ,by$( Tr_{9H}S(\cdots, j, \cdots, k, \cdots))_{(u,t)}=\sum_{i=1}^{n}S_{(u,t)}(\cdots, e_{i}^{j}, \cdots, e_{i}^{k}, \cdots)$

where$((u, t)\in M\cross[0,$ $T$$S(\cdots, j, \cdots, k, \cdots)$$\{e_{1}, \cdots, e_{n}\}$means that$j$$S_{(u,t)}(\cdots, e_{i}, \cdots, e_{i}^{k}, \cdots)$$\mathcal{H}_{(u,t)}$with respect to$(g_{\mathcal{H}})_{(u,t)},$is entried into the j-th component and the k-th component ofmeans thatis entried into the j-th component and the k-th$\bullet$andcomponent ofThen we have the following relation.
$S$

is an orthonormal base of$e_{i}$$S_{(u,t)}.$Lemma 5.4([K3]). Let$S$

be a section of $\pi_{M}^{*}(T(0,2)M)$ which is symmetric with respect toThen we have$(\triangle_{\mathcal{H}}S)_{\mathcal{H}}(X, Y)=(\triangle_{\mathcal{H}}^{\mathcal{H}}S_{\mathcal{H}})(X, Y)$$-2b_{g_{\mathcal{H}}}((\nabla.S)(\mathcal{A}.X, Y))-2^{r}R_{9\mathcal{H}}((\nabla.S)(\mathcal{A}.Y, X))$$-Tr_{9\mathcal{H}}S(\mathcal{A}.(\mathcal{A}.X), Y)-Tr_{g_{\mathcal{H}}}S(\mathcal{A}.(\mathcal{A}.Y), X)$$-Tr_{g_{\mathcal{H}}}S((\nabla.\mathcal{A}).X, Y)-Tr_{9\mathcal{H}}S((\nabla.\mathcal{A}).Y, X)$$-2h_{g_{\mathcal{H}}}\cdot S(\mathcal{A}.X, \mathcal{A}_{\bullet}Y)$

$g.$

86
for $X,$$Y\in \mathcal{H}$, whereProof. Take any$\nabla$is the connection of$(u_{0}, t_{0})\in M\cross[0, T$. Take any $X,$
andwith
. Then, it follows fromofthat
respect to$(g_{\mathcal{H}})_{(u_{0:}t_{0})}$$\tilde{X}_{(u_{0}.t_{0})}=X$$(u_{0}, t_{0})$). Let$\pi_{M}^{*}(T^{(1,2)}M)$

induced from$\nabla.$withbe an orthonormal base of. Let
be a section of on a neighborhood of
Similarly.
we define and . Let $W=X,$ $Y$ or$W$ and the skew-symmetricness$\{e_{1}, \cdots, e_{n}\}$$\mathcal{H}_{(u_{0:}t_{0})}$$\overline{X}$$\mathcal{H}$$Y\in \mathcal{H}_{(u_{0:}t_{0})}$$\tilde{Y}$$(\nabla^{\mathcal{H}}X)_{(u_{0:}t_{0})}=0$$(\nabla_{e_{i}}^{\mathcal{H}}\overline{W})_{(u_{0:}t_{0})}=0,$$\tilde{e}_{i}$$(\nabla_{e_{1}}\overline{W})_{(u_{O,}.t_{0})}=\mathcal{A}_{e},$$e_{i}$$\mathcal{A}|_{\mathcal{H}x\mathcal{H}}$$( \triangle_{\mathcal{H}}S)_{\mathcal{H}}(X, Y)=\sum_{i=1}^{n}(\nabla_{e}.\nabla_{e_{i}}S)(X, Y)$$= \sum_{i=1}^{n}(\nabla_{e_{1}}^{\mathcal{H}}\nabla_{e_{1}}^{\mathcal{H}}S_{\mathcal{H}})(X, Y)-2\sum_{i=1}^{n}((\nabla_{e_{t}}S)(\mathcal{A}_{e_{i}}X, Y)+(\nabla_{e_{i}}S)(\mathcal{A}_{e_{t}}Y, X))$$- \sum_{i=1}^{n}(S(\mathcal{A}_{e_{i}}(\mathcal{A}_{e_{\mathfrak{i}}}X), Y)+S(\mathcal{A}_{e_{i}}(\mathcal{A}_{e_{i}}Y), X))-2\sum_{i=1}^{n}S(\mathcal{A}_{e_{i}}X, \mathcal{A}_{e_{i}}Y)$$- \sum_{i=1}^{n}(S((\nabla_{e_{1}}\mathcal{A})_{e_{i}}X, Y)+S((\nabla_{e_{\iota}}\mathcal{A})_{e}:Y, X))$.

The right-hand side of this relation is equal to the right-hand side of the relation in the statement.
q.e.
This completes the proof.
$d.$Also we have the following Simons-type identity.
Lemma 5.5 ([K3]). We have$\triangle_{\mathcal{H}}h=\nabla d||H||+||H||(A^{2})_{\#}-(Tr(A^{2})_{\mathcal{H}})h,$where$(A^{2})_{\#}$$\pi_{M}^{*}TM)$is the element of$\Gamma(\pi_{M}^{*}T^{(0,2)}M)$

defined by $(A^{2})_{\#}(X, Y)$$:=g(A^{2}X, Y)$(X, $Y\in$.

Proof Take

$X,$ $Y,$ $Z,$ $W\in\pi_{M}^{*}(TM)$. Since the ambient space $V$ is flat, it follows from the Ricci’s

identity, the Gauss equation and the Codazzi equation that$(\nabla_{X}\nabla_{Y}h)(Z, W)-(\nabla_{Z}\nabla_{W}h)(X, Y)=(\nabla_{X}\nabla_{Z}h)(Y, W)-(\nabla_{Z}\nabla_{X}h)(Y, W)$$=h(X, Y)h(AZ, W)-h(Z, Y)h(AX, W)+h(X, W)h(AZ, Y)-h(Z, W)h(AX, Y)$ .
By using this relation, we obtain the desired relation.
Note. In the sequel, we omit the notation

Define a section$\mathcal{R}$of$\pi_{M}^{*}(\mathcal{H}^{(0,2)})$$F_{*}$

q.e.

$d.$

for simplicity.

by$\mathcal{R}(X, Y) :=\ulcorner rr_{g_{\mathcal{H}}}^{\bullet}/h(\mathcal{A}.(\mathcal{A}.X), Y)+R_{g_{\mathcal{H}}}^{\cdot}h(\mathcal{A}.(\mathcal{A}.Y), X)$$+Tr_{g^{r}\succ t}h((\nabla.\mathcal{A}).X, Y)+h_{g_{\mathcal{H}}}h((\nabla.\mathcal{A}).Y, X)$$+2R_{g_{7\{}}(\nabla.h)(\mathcal{A}.X, Y)+2H_{9\sim}(\nabla.h)(\mathcal{A}.Y, X)$$+2R_{g_{\mathcal{H}}}h(\mathcal{A}.X, \mathcal{A}.Y) (X, Y\in \mathcal{H})$.

From Lemmas 5.3, 5.4 and 5.5, we derive the following evolution equation for$(h_{\mathcal{H}})_{t}$

)

$s.$

87
Theorem 5.6([K3]). The sections$(h_{\mathcal{H}})_{t}$’s of $\pi_{M}^{*}(T^{(0,2)}M)$ satisfies the following evolution equa-tion:$\frac{\partial h_{\mathcal{H}}}{\partial t}(X, Y)=(\triangle_{\mathcal{H}}^{\mathcal{H}}h_{\mathcal{H}})(X, Y)-2||H||((A_{\mathcal{H}})^{2})_{\#}(X, Y)-2||H||((\mathcal{A}_{\xi}^{\phi})^{2})_{\#}(X, Y)$$+Tr((A_{\mathcal{H}})^{2}-((\mathcal{A}_{\xi}^{\phi})^{2})_{\mathcal{H}})h_{\mathcal{H}}(X, Y)-\mathcal{R}(X, Y)$

for $X,$$Y\in \mathcal{H}.$Proof. Take

$X,$ $Y\in \mathcal{H}_{(u,t)}$(5.1). Easily we have

$AX=A_{\mathcal{H}}X+\mathcal{A}_{\xi}^{\phi}X$

and$(A^{2})_{\mathcal{H}}X=(A_{\mathcal{H}})^{2}X-(\mathcal{A}_{\xi}^{\phi})^{2}X,$

where we use$(\tilde{\nabla}_{W}\xi)_{\overline{\mathcal{H}}}=(\tilde{\nabla}_{\xi}W+[W, \xi])_{\overline{\mathcal{H}}}=(\overline{\nabla}_{\xi}W)_{\tilde{\mathcal{H}}}=\mathcal{A}_{\xi}W$

for$W\in\Gamma(\overline{\mathcal{V}})$because of$[W, \xi]\in\Gamma(\tilde{\mathcal{V}})$(5.2). Also, since$[ \frac{\partial}{\partial t},\overline{X}_{\mathcal{H}}]\in \mathcal{V}$, we have$[ \frac{\partial}{\partial t}, X_{\mathcal{H}}^{-}]=2||H||\mathcal{A}_{\xi}^{\phi}X_{\mathcal{H}}^{-}.$

From Lemma 5.3, (5.1) and (5.2), we have$= \langle,\mathcal{H}(\overline{Y}_{\mathcal{H}}F)\rangle\langle\xi,\frac{\partial(}{\partial t}(X_{\mathcal{H}}^{-}(\overline{Y}_{\mathcal{H}})\rangle\frac{\partial h_{\mathcal{H}}}{\frac{}{}\partial\xi_{x^{-}}^{t} ,\partial t}(X,Y)=\frac{\partial}{\partial t,+}(h_{\mathcal{H}}X^{-},\overline{Y}))=\frac{\partial}{F)\partial t}\langle\xi,\overline{X}_{\mathcal{H}}(\overline{Y}_{\mathcal{H}}F)\rangle$$=- \langle F_{*}(grad_{9}||H||) , \tilde{\nabla}_{X}F_{*}\overline{Y}_{\mathcal{H}}\rangle+\langle\xi, X(\overline{Y}_{\mathcal{H}}(\frac{\partial F}{\partial t}))\rangle$$+ \langle\xi, X([\frac{\partial}{\partial t},\overline{Y}_{\mathcal{H}}]F)\rangle+\langle\xi, [\frac{\partial}{\partial t},\overline{X}_{\mathcal{H}}](\overline{Y}_{\mathcal{H}}F)\rangle$$=-g(grad_{g}||H||, \nabla_{X}\overline{Y}_{\mathcal{H}})+X(\overline{Y}_{\mathcal{H}}||H||)-||H||\langle\xi, \overline{\nabla}_{X}F_{*}(A(\overline{Y}_{\mathcal{H}})\rangle$$+ \langle\xi, \overline{\nabla}_{X}F_{*}([\frac{\partial}{\partial t},\overline{Y}_{\mathcal{H}}])\rangle+\langle\xi, \tilde{\nabla}_{[_{\delta^{\partial}:}\overline{t}X_{\mathcal{H}}^{-}]}F_{*}\overline{Y}_{\mathcal{H}}\rangle$$=(\nabla d||H||)(X, Y)-||H||h_{\mathcal{H}}(X, A_{\mathcal{H}}Y)+||H||h(X, \mathcal{A}_{\xi}^{\phi}Y)+2||H||h(\mathcal{A}_{\xi}^{\phi}X, Y)$$=(\nabla d||H||)(X, Y)-||H||g_{\mathcal{H}}((A_{\mathcal{H}})^{2}X, Y)-3||H||g((\mathcal{A}_{\xi}^{\phi})^{2}X, Y)$

From this relation and the Simons-type identity in Lemma 5.5, we have$\frac{\partial h_{\mathcal{H}}}{\partial t}=\triangle_{\mathcal{H}}h-2||H||((A_{\mathcal{H}})^{2})_{\#}-2||H||((\mathcal{A}_{\xi}^{\phi})^{2})_{\#}$(5.3)$+Tr((A_{\mathcal{H}})^{2}-((\mathcal{A}_{\xi}^{\phi})^{2})_{\mathcal{H}})h_{\mathcal{H}}.$

Substituting the relation in Lemma 5.4 into (5.3), we obtain the desired relation.
For$\mathcal{R}$, we can show the following fact.

Lemma 5.7([K3]). For$X\in \mathcal{H}$, we have$\mathcal{R}(X, X)=4h_{g_{\mathcal{H}}}\langle \mathcal{A}^{\phi}X, \mathcal{A}^{\phi}(A_{\mathcal{H}}X)\rangle+4^{r}R_{g_{\mathcal{H}}}\langle \mathcal{A}^{\phi}X, \mathcal{A}_{X}^{\phi}(A_{\mathcal{H}}\bullet)\rangle$$+3’R_{9\mathcal{H}}\langle(\tilde{\nabla}.\mathcal{A}^{\phi})_{\xi}X, \mathcal{A}^{\phi}X\rangle+2R_{g_{\mathcal{H}}}^{\cdot}\langle(\tilde{\nabla}.\mathcal{A}^{\phi}).X, \mathcal{A}_{\xi}^{\phi}X\rangle$$+Tr_{g_{\mathcal{H}}}\langle \mathcal{A}^{\phi}X, (\tilde{\nabla}_{X}\mathcal{A}^{\phi})_{\xi}\bullet\rangle$

q.e.

$d.$

88
and hence$h_{g_{\mathcal{H}}}^{\bullet}\mathcal{R}(\bullet, \bullet)=0.$

By using Theorem 5.6 and Lemma 5.7, we can show the following evolution equation for

Corollary 5.8([K3]). The norms $||H_{t}||s$ of$H_{t}$$||H_{t}||’ s.$satisfy the following evolution equation:$\frac{\partial||H||}{\partial t}=\triangle_{\mathcal{H}}||H||+||H||Tr(A_{\mathcal{H}})^{2}-3||H||R((\mathcal{A}_{\xi}^{\phi})^{2})_{\mathcal{H}}.$

Remark 5.1. $Rom$ the evolution equations obtained in this section, the evolution equations for thecorresponding geometric quantities ofare derived, respectively. In the case where$V/G$
is (complete) Riemannian manifold, the above derived evothe -action is free and hence
lution equations coincide with the evolution equations for the corresponding geometric quantities
along the mean curvature flow in a complete Riemannian manifold which were given by Huisken[Hu2]. That is, the discussion in this section give a new proof of the evolution equations in [Hu2] inthe case where the ambient complete Riemannian manifold occurs as $V/G$ . In the proof of [Hu2],
one need to take local coordinates of the ambient space to derive the evolution equations. On
the other hand, in our proof, one need not take local coordinates of the ambient spaceand canidentify the tangent space of the ambient space at each point with . These are an advantage
of our proof.
$\overline{f}_{t}(:\overline{M}\mapsto V/G)$

$G$

$a$

$V$

$V$

$V$

Horizontally strongly convexity preservability theorem

6

Let $G\cap V$ be an isometric almost free action with minimal regularizable orbit of a Hilbert Liegroup on a Hilbert space equipped with an inner productand : $Varrow V/G$ the orbit
map. Denote by the Riemannian connection of . Set $n:=\dim V/G-1$ . Let $M(\subset V)$ be a-invariant hypersurface in such that $\phi(M)$ is compact. Let be an inclusion map of $M$ into
and $f_{t}(0\leq t<T)$ the regularized mean curvature flow for . We use the notations in Section
5. In the sequel, we omit the notation
for simplicity. Set
$G$

$V$

$\langle,$$\tilde{\nabla}$$G$$\phi$$\rangle$

$V$

$V$

$f$

$V$

$f$

$f_{t*}$$L:= \max|\langle \mathcal{A}_{X_{1}}^{\phi}((\tilde{\nabla}_{X_{2}}\mathcal{A}^{\phi})_{X_{3}}X_{4})(X_{1},\cdots,X_{5})\in\tilde{\mathcal{H}}_{1}^{5}, X_{5}$$:=\{X\in\tilde{\mathcal{H}}|||X||=1\}$ . Assume that $L<\infty$ . Note that $L<\infty$ in
where
the case where $V/G$
is compact. Then we obtain the following horizontally strongly convexity preservability theorem by$\tilde{\mathcal{H}}_{1}$using the evolution equations in Section 5 and the maximum principle forsymmetric $(0,2)$ -tensor fields on $M$ (see [K3]).

Theorem 6.1([K3]). If $M$ satisfies
holds for all $t\in[0, T$ ).

$||H_{0}||^{2}(h_{\mathcal{H}})_{(\cdot,0)}>2n^{2}L(g_{\mathcal{H}})_{(\cdot,0)}$$c\infty$-family of $G$-invariant, then $T<\infty$ holds and$||H_{t}||^{2}(h_{\mathcal{H}})_{(\cdot,t)}>2n^{2}L(g_{\mathcal{H}})_{(\cdot,t)}$Strongly convex preservability theorem in the orbit space

7
Let
and

$V,$

$R_{N}$

and be as in the previous section. Set $N:=V/G$ and $n:=\dim V/G-1$ . Denote by
the Riemannian orbimetric and the curvature orbitensor of $N$ . Also,
the Riemannian$G$$\phi$$g_{N}$$\nabla^{N}$

89. Since the Riemannian manifold $(N\backslash Sing(N), g_{N}|_{N\backslash Sing(N)})$ is locallyconnection ofhomogeneous, the norm
(with respect to) is constant over $N\backslash Sing(N)$ .
of$:=||\nabla^{N}R_{N}||$. Assume that $L_{N}<\infty$ . Let
Set
be a compact suborbifold of codimension one$N$in immersed by andandthe mean curvature flow for . Denote by
be the induced orbimetric, the second fundamental orbiform, the shape orbitensor and the mean
curvature orbifunction of , respectively, and the unit normal vector field of$Rom$ Theorem 6.1, we obtain the following strongly convexity preservability theorem for compact
suborbifolds in $N.$
$g_{N}|_{N\backslash Sing(N)}$$\nabla^{N}R_{N}$$||\nabla^{N}R_{N}||$$g_{N}$$\overline{M}$$L_{N}$$\overline{f}$$\overline{f}_{t}(t\in[0, T))$$\overline{f}_{t}$$\overline{f}$$\overline{h}_{t},$$\overline{\xi}_{t}$Theorem 7.1 ([K3]). If satisfies
holds for all $t\in[0, T$ ).
$\overline{f}$$\overline{H}_{t}$$\overline{A}_{t}$$\overline{g}_{t},$$\overline{f}_{t}|_{\overline{M}\backslash Sing(\overline{M})}.$$||\overline{H}_{0}||^{2}\overline{h}_{0}>n^{2}L_{N}\overline{g}_{0}$, then $T<\infty$ holds and$||\overline{H}_{t}||^{2}\overline{h}_{t}>$$n^{2}L_{N}g_{t}$Proof Set

and define : $Marrow V$ by $f(x, u)=u((x, u)\in M)$ .
It is clear that is an immersion. Denote by
the regularized mean curvature vector of . Define$:=\overline{f}_{t}(x)(t\in[0, T))$ and let: $[0, T$ ) $arrow N$ by
be the horizontal lift ofa curvefor$M\mapstoV$$:=(c_{x})_{u}^{L}(t)((x,
u)\in
M)$
$u\in\phi^{-1}(f(x))$, where. Define an immersion.
:
by $f_{t}(x, u)$Then $f_{t}(t\in[0, T))$ is the regularized mean curvature flow for (see the proof of Proposition 5.1).
Denote byandthe induced metric, the second fundamental form, the shape tensor andthe mean curvature vector of , respectively. By the assumption,satisfies
Also, we can show $L_{N}=2L$ by long calculation, where is as in the prevoius section. From thesefacts, we can show that
. Hence, it follows from Theorem 6.1
satisfies$f_{t}(t\in[0,
T))$
that
. Furthermore, it follows from this fact thatsatisfies
satisfies.
q.e.d.
$M$ $:=\{(x, u)\in\overline{M}\cross V|\overline{f}(x)=\phi(u)\}$
$f$

$f$

$f$

$H_{0}$$(c_{x})_{u}^{L}$$c_{x}(t)$$c_{x}$$u$$c_{x}$$f_{t}$$f$$g_{t},$$h_{t},$$A^{t}$$H_{t}$$||\overline{H}_{0}||^{2}\overline{h}_{0}>n^{2}L_{N\overline{9}0}.$$\overline{f}_{0}$$f_{t}$$L$$f_{0}$$||H_{0}||^{2}(h_{\mathcal{H}})_{0}>2n^{2}L(g_{\mathcal{H}})_{0}$$||H_{t}||^{2}(h_{\mathcal{H}})_{t}>2n^{2}L(g_{\mathcal{H}})_{t}$$\overline{f}_{t}(t\in[0, T))$$||\overline{H}_{t}||^{2}\overline{h}_{t}>n^{2}L_{N}\overline{g}_{t}$Remark 7.1. In the case where the -action is free and hence $N$ is (complete) Riemannian
manifold, Theorem 6.1 implies the strongly convexity preservability theorem by G. Huisken (see
[Hu2, Theorem 4.2]).
$G$

$a$

Acknowledgement
The author is grateful to JSPS for support (Grant-in-Aid for Science Research (C), no.25400076).

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Department of Mathematics, Faculty of Science
Tokyo University of Science
Tokyo 162-8601
Japan
E–mail address: koike@ma.kagu.tus.ac.jp