数理解析研究所講究録
第 1929 巻 2014 年 54-63

54

EXAMPLES OF GENERALIZED LAGRANGIAN MEAN CURVATUREFLOWS IN TORIC ALMOST CALABI-YAU MANIFOLDSHIKARU YAMAMOTO
ABSTRACT. In the former half of this paper, Section 1 and Section 2, we summarize
some basic notions and facts of calibrated geometries and special Lagrangian geometries
alld Lagrangiall mean curvature flows. In the later half, Section 3, we review the resultsof [10] which give some examples of generalized Lagrangian mean curvature flows in toricalmost Calabi-Yau manifolds.

1. CALIBRATED GEOMETRIESIn this section, we briefly sketch some basic notions of calibrated geometries introduced
by Harvey and Lawson [3]. Let $(M, g)$ be a real -dimensional Riemannian manifold.
$n$Definition 1.1. A closed -form$k$$\varphi$

on $M$ is called a calibration if it satisfies(1)$\varphi|_{\xi}\leq dV_{\xi},$

in $M$ and all -dimensional oriented subvector spaces
Let , . . . , be an oriented orthogonal basis of with respect to the metric . Then the. Note that the notion of calibration depends
inequality (1) means thaton ambient Riemannian metrics. Let be a calibration on $(M, g)$ and -from.

for all points

$k$

$p$

$\xi\subset T_{p}M.$$\xi$$e_{k}$$e_{1}$$g$$\varphi(e_{1}, \ldots, e_{k})\leq 1$$k$$\varphi$

Definition 1.2. A real -dimensional oriented submanifold $L\subset M$ is called a calibrated
submanifold or -submanifold if all its tangent spaces attain the equality of (1), that is,
we have$k$$\varphi$

(2)

where

$\varphi|_{L}=dV_{9}|_{L},$$dV_{g1_{L}}$is the volume form on$L$with the induced Riemannian metric$g|_{L}.$:The notion of calibrated submanifolds can be easily extended for an immersion
from a real -dimensional oriented manifold . Actually we call an immersion: $Larrow M$ also a calibrated submanifold or -submanifold if $F^{*}\varphi=dV_{F^{*}g}$ . In [3],Harvey and Lawson studied geometries of calibrated submanifolds and showed that calibrated submanifolds are homologically volume minimizing in $(M, g)$ . Hence it is clear thatcalibrated submanifolds are automatically minimal submanifolds. This is the most important property of calibrated submanifolds. The precise statement and its very fundamental
proof is the following.
be a Riemannian manifold with a
Theorem 1.3 (Harvey and Lawson [3]). Let$L\subset
M$
compactabecalibration and-submanifold. Then for any submanifold $L’$ in$Vo1_{g}(L)\leq
Vo1_{9}(L’)$ , where$[L]$ , the homolog classis the volume ofof , we havethe submanifold measured by the metric 9.
Proof. We have $Vo1_{g}(L)=\int_{L}\varphi=\int_{L},$ $\varphi\leq Vo1_{g}(L’)$ . The first equality follows from (2).
The middle equality follows from that the pairing of a closed form and a homology class,
it is given by the integration, does not depend on the choice of representations. The final
inequality follows from (1).
$F$

$Larrow\Lambda I$

$L$

$k$

$F$

$\varphi$$(\Lambda I, g)$$\varphi$$\varphi$$L$$Vo1_{g}(\cdot)$$\square$55
As explained in Section 4.2 in Joyce [5], interesting calibrations can be constructednaturally if the ambient Riemannian manifold $(M, g)$ has a special holonomy. One ofsuch examples is a K\"ahler manifold. Actually, let $(M, \omega, 9, J)$ be a complex -dimensional
K\"ahler manifold with a symplectic form , Riemannian metric and complex structure
Then one can see thatfor some $1\leq p\leq m$ becomes a calibration on $(M, g)$ by
Wirtinger’s inequality. It is well-known that the -submanifolds are just the canonicallyoriented complex submanifolds of complex dimension in $M$ . Hence it follows thatthe complex submanifolds are homologically volume minimizing in the K\"ahler manifold.
Another such example is an (almost) Calabi-Yau manifold, a main subject in this paper,
which is explained in the next section and calibrated submanifolds in it are called special
Lagrangian submanifolds.
$m$$\omega$$\varphi$

$J.$

$g$

$:=\omega^{p}/p!$$\varphi$$p$

2. ALMOST CALABI-YAU MANIFOLDS AND LAGRANGIAN SUBMANIFOLDSFirst of all, we introduce the notion of almost Calabi-Yau manifolds following Definition
8.4.3 of Joyce [5]. Almost Calabi-Yau manifolds are ambient spaces for special Lagrangiansubmanifolds, weighted hamiltonian stationary Lagrangian submanifolds and generalized
Lagrangian mean curvature flows defined also in this section. Let $(M, \omega, g, J)$ be a complex-dimensional K\"ahler manifold.
$m$

Definition 2.1. If there exists a non-vanishing holomorphic $(m, 0)$ -form over $M$ , wecall this quintuplet
an almost Calabi-Yau manifold and a holomorphic
volume form over M.
$\Omega$$(M, \omega, g, J, \Omega)$$\Omega$

It is clear that the canonical line bundle of an almost Calabi-Yau manifold is trivial
and its 1st Chern class, denoted by $c_{1}(M)$ , is zero, since its holomorphic volume form
gives a global trivialization of it. On an almost Calabi-Yau manifold, wedefine a real valued function :by$(M, \omega, 9, J, \Omega)$$\psi$(3)$Marrow \mathbb{R}$$e^{2m\psi} \frac{\omega^{m}}{m!}=(-1)^{\frac{m(m-1)}{2}}(\frac{i}{2})^{m}\Omega\wedge\overline{\Omega}.$

Then one can easily see that the Ricci form
given by$\rho(\omega)$of this almost Calabi-Yau manifold is$\rho(\omega)=2mi\partial\overline{\partial}\psi.$

Thus it follows that is a Ricci flat K\"ahler metric if and only if is a constant. Especially
if $\psi=0$ , we calla Calabi-Yau manifold.
Note that if an almost Calabi-Yau manifold $M$ is compact then we can give a new
K\"ahler structure on it so that $M$ becomes a Calabi-Yau manifold by Calabi Ansatz sinceO. However this given Calabi-Yau metric is not explicit in general. On theother hand, there are many examples of almost Calabi-Yau metrics which have explicit
forms. Hence, we prefer to work on almost Calabi-Yau manifolds rather than Calabi-Yau
manifolds to observe some concrete examples of Lagrangian submanifolds in these.
The most typical example of almost Calabi-Yau manifolds is a complex spacewiththe standard structure described precisely below, and actually it is a Calabi-Yau manifold.
$\omega$$\psi$$(M, \omega, 9, J, \Omega)$$c_{1}(\Lambda I)=$$\mathbb{C}^{m}$

Example 2.2. Let
standard complex structure
form bybe the standard complex coordinates onwith the. If we define a K\"ahler form
and a holomorphic volume$(z_{1}, \ldots, z_{m})$$J$$\mathbb{C}^{m}$$\omega$$\Omega$$\omega$$:= \frac{i}{2}\sum_{j=1}^{m}dz_{j}\wedge d\overline{z}_{j}$and$\Omega$$:=dz_{1}\wedge\cdots\wedge dz_{m},$

56
then
clidean metric on$(\mathbb{C}^{m}, \omega, g, J, \Omega)$

is an (almost) Calabi-Yau manifold, where$g$is the standard Eu-$\mathbb{R}^{2m}\cong \mathbb{C}^{n\iota}.$

is equal to
In the equality (3), the term
whereis equal toclear that
. Hence the equality (3) is reformulateted as$\omega^{m}/m!$$e^{2m\psi}\omega^{m}/m!$, the volume form of , and it isis the conformal rescaling of with$dV_{g}$$\tilde{g}:=e^{2\psi}g$$dV_{\overline{g}}$

$g$

$g$

$\psi$$m(m-1)( \frac{i}{2})^{m}\Omega\wedge\overline{\Omega}.$$dV_{\overline{g}}=(-1)\overline{2}$, note that the reft hand side is the norm of
By this equality, we haverespect to the metric , not . Then we can see that$|\Omega|_{\overline{g}}=\backslash \Gamma 2^{n\iota}$$\tilde{g}$$\Omega$

with

$g$

$\varphi_{\beta}:={\rm Re}(e^{-i\beta}\Omega)$. This method is also
, for allis a calibration on Riemannian manifold[3].
LawsonHarvey
paper
of
and
in
the
mentioned in Section V.3$(M,\tilde{g})$$\beta\in \mathbb{R}$Definition 2.3. We call a -submanifold in Riemannian manifold
grangian submanifold with phase$(M_{\tilde{9}})$$\varphi_{\beta}$

a special La-$e^{i\beta}.$

The name of special Lagrangian submanifold comes from the fact that a special La,
grangian submanifold is a Lagrangian submanifold in the symplectic manifold
Take
proof
following.
the
the
of
is
The
outlinebythedefinition.
though it is non-trivialbea point in $M$ and a real $m$-dimensional subvector space in $T_{p}M$ . Let , . . . ,by. Then we define
an orthogonal basis of with respect to the metric, whereis the dual. Note that here we do not assume that is oriented,(up tohence
is defined up to sign. We define a complex number)bymultiplication by$(M, \omega)$

$\xi$

$p$

$\tilde{9}(=e^{2\psi}g)$$\xi$$e_{1}^{*}\wedge\cdots\wedge e_{n}^{*},$$e_{m}$$e_{1}$$dV_{\xi}$$\xi$$e_{j}^{*}$$\alpha_{\xi}\in \mathbb{C}/\{\pm 1\}$$dV_{\xi}$$\pm 1$$\Omega|_{\xi}=\alpha_{\xi}dV_{\xi}.$

if and only if is a Lagrangian subvector, andThen one can show that-submanifold, then it is necessary that is. Hence if is aspace ina Lagrangian submanifold.
(it is defined modulo
For a Lagrangian subvector space , we take the argument ofand denote it by$|\alpha_{\xi}|=1$$|\alpha_{\xi}|\leq 1$$L$$(T_{p}M_{:}\omega_{p})$

$\xi$

$L$

${\rm Re}(e^{-i\beta}\Omega)$$\xi$$\alpha_{\xi}$$\pi \mathbb{Z})$$\theta_{\xi}$$:=\arg\alpha_{\xi}\in \mathbb{R}/\pi \mathbb{Z}.$Definition 2.4. For a Lagrangian submanifoldby$L$, we define a function$\theta_{L}$:$Larrow \mathbb{R}/\pi \mathbb{Z}$$\theta_{L}(p):=\theta_{T_{p}L},$and call it the Lagrangian angle of$L.$:
has a lift
is oriented then the Lagrangian angle
Note that if. One can show that a special Lagrangian submanifold with phase:
is a constant . We call theis a Lagrangian submanifold whose Lagrangian angle
the Maslov class of . It is equivalent to that thecohomology class::
has a lift
Maslov class is zero and the Lagrangian angle
and if these two hold then is called graded. Hence it is clear that a special Lagrangiansubmanifold is graded, since its Lagrangian angle is constant.
As explained in Section 1, a special Lagrangian submanifold is volume-minimizing in its
homology class (note that the volume is measured by the metric g) since it is a calibrated. Furthermore the converse
submanifold. Especially it is a minimal submanifold inis true under the Lagrangian assumption, that is, if is a minimal Lagrangian submanifold
inthen it is a special Lagrangian submanifold. We explain this below. Let be$L$$\theta_{L}$$\theta_{L}$$Larrow \mathbb{R}/\pi \mathbb{Z}$$Larrow \mathbb{R}/2\pi \mathbb{Z}$$e^{i\beta}$$\theta_{L}$$\beta$$L$$[d\theta_{L}]\in H^{1}(L, \mathbb{R})$$Larrow \mathbb{R}/\pi \mathbb{Z}$$\theta_{L}$$\theta_{L}$$Larrow \mathbb{R},$$L$$(M,\tilde{g})$$L$$(M,\tilde{g})$

$L$

57
-dimensional submanifold in $M$ . We denote the mean curvature vector field of. Thenbyand the one defined bydefined by the ambient metic by
there is a relation asa real

$L$

$m$

$g$

$\tilde{9}(=e^{2\psi_{9)}}$$H_{L}$$\tilde{H}_{L}=e^{-2\psi}(H_{L}-m\nabla\psi^{\perp})$$\tilde{H}_{L}$,) of the gradient of defined by thewhereis the normal part (with respect toequation (3). Here we introduce the notion of generalized mean curvature vector field
following Behrndt [1].
Definition 2.5. We call a vector field defined by$TL$$\nabla\psi^{\perp}$$\psi$$K_{L}:=e^{2\psi}\tilde{H}_{L}=H_{L}-m\nabla\psi^{\perp}$

the generalized
curvature vector field on. If
coincides withNote that if $M$ is a Calabi-Yau manifold, that is, $\psi=0$ thenis a Lagrangian submanifold then the (generalized) mean curvature vector field and the
Lagrangian angle has a relation stated precisely below, it is proved in Proposition 2.17 inand for Calabi-Yau case seeisHarvey and Lawson $[3, III.2.D.]$ for the case thatthe paper of Thomas and Yau [8], and Behrndt proved in almost Calabi-Yau cases.
Proposition 2.6 (Behrndt [1]). On a Lagrangian submanifold , we have$L.$

$7r\iota ean$

$H_{L}$$K_{L}$$\Gamma_{\lrcorner}$$\mathbb{C}^{m}$

$L$

(4)

$K_{L}=J\nabla\theta_{L}.$, that is,Hence we see that if a Lagrangian submanifold is minimal inis constant by the identity
it is equivalent to that $K_{L}=0$ , then the Lagrangian angle(4). Here we summarize some equivalent conditions for special Lagrangian submanifolds.
in $M$ , the following four
Proposition 2.7. For an real $m$ -dimen ional submanifold
conditions are equivalent.
(1)
is a special Lagrangian submanifold with phasefor some(2)
is a Lagrangian submanifold whose Lagrangian angle is a constant(3)
is a minimal Lagrangian submanifold in(4)
is a Lagrangian submanifold andNote that in the conditions (2) $-(4)$ it follows that is orientable, and we have to admit
an orientation on so that
becomes the volume form on with the induced
metricLocally, a Lagrangian submanifold is expressed as a graph of 1-form $df$ of some functionin (4) in:
under a Darboux chart. Then the condition
Proposition 2.7 is expressed as a non-linear second order elliptic equation of , MongeAmp\‘ere type, as explained in [5]. Hence, to construct examples of special Lagrangiansubmanifolds in a given almost Calabi-Yau manifold is a diﬃcult problem in general. Asa method to avoid this diﬃculty, in this paper, we focus on the condition (2) in Proposition
2.7, the minimality of special Lagrangian submanifolds. To get minimal submanifold, we
can consider mean curvature flows. This is a parabolic equation like a heat equation. The
precise definition is the following. Let $(N, h)$ be an -dimensional Riemannian manifold,: $Larrow N$ be an immersion.
be an -dimensional manifold andDefinition 2.8. We say that a smooth 1-parameter family of immersions $F:L\cross[O, T$ )$N$ , which is continuous up to $t=0$ , is evolving by mean curvature flow with the initial: $Larrow N$ if it satisfies
condition$\tilde{H}_{L}=0,$$(M,\tilde{g})$$\theta_{L}$$L$$e^{i\beta}$$L$$\beta\in \mathbb{R}.$$L$$\theta$$L$$\beta.$$(\Lambda I_{\tilde{9})}.$${\rm Im}(e^{-i\beta}\Omega)|_{L}=0.$

$L$

$L$

$L$

$L$

${\rm Re}(e^{-i\beta}\Omega)|_{L}$$\tilde{g}|_{L}.$$f$${\rm Im}(e^{-i\beta}\Omega)|_{L}=0$$(\mathbb{R}^{m}\supset)\Omegaarrow \mathbb{R}$

$f$

$n$

$L$

$k$

$F_{0}$$arrow$$F_{0}$(5)$\frac{\partial F}{\partial t}|_{t=s}=H(F_{6})fors\in(O, T)$

and

$F$

$0$

) $=F_{0}.$58$:=F$ s) by
Here we denote the mean curvature vector field of the immersion$H(F_{s})$ .
It is known that there is the short-time existence and uniqueness result for themean curvature flow in the case that $M$ is compact. It is proved by Hamilton’s theorem [2]used for the short-time existence and uniqueness result for the Ricci flow. Mean curvature
gradient flow of the volume functional. We
flows appear naturally as the backward
explain this below. Let $Imm(L, N)$ be the set of all immersion maps from to $N$ . Then
we define the volume functional$F_{s}$$L^{2}$$L$$Vo1_{h}:Imm(L, N)arrow \mathbb{R}$by$Vo1_{h}(F):=\int_{L}1dV_{F^{*}h},$it is just the volume of $F(L)$ measured by the metric . Then it is well-known that thefirst variation ofis given as follows.
$h$$Vo1_{h}$Proposition 2.9. Let : $Larrow(N, h)$ be an immersion andsmooth 1-parameter family of immersions $(that is a curve in Imm(L, N)$ ) with
$F$$\{F_{s} :$F_{0}=F$andLarrow N\}_{s\in(-\epsilon,\epsilon)}$be a$V:= \frac{\partial F}{\partial s}|_{s=0}$

Then we have$\frac{d}{ds}Vo1_{h}(F_{S})=-s=0\int_{L}h(V, H(F))dV_{F^{n}h}.$By this proposition, it is clear that minimal submanifolds are critical points of thevolume functional, and the mean curvature flow is the backward
gradient flow of thevolume functional and the volume is monotone decreasing along a mean curvature flow.
This is one of characterizations of mean curvature flows.
Let us come back to the case that $(N, h)=(AI^{2m},\tilde{g})$ . In this case, as an analog ofmean curvature flows, Behrndt introduces generalized mean curvature flows in [1]. Let
be a real $m$-dimensional manifold.
$L^{2}$$L$

Definition 2.10. We say that a smooth 1-parameter family of immersions : $L\cross[O, T$ )$M$ , which is continuous up to $t=0$ , is evolving by generalized mean curvature
flow with
the initial condition: $Larrow M$ if it satisfies$F$$arrow$$F_{0}$(6)$\frac{\partial F}{\partial t}t=s=K(F_{s})$for $s\in(O, T)$

and

$F$

$0$

) $=F_{0}.$Note that in the original definition of [1] Behrndt considers the normal part of
The advantage of considering a generalized mean curvature flow in an almost Calabi-Yau
manifold is that the Lagrangian condition is preserved along the flow. It was first proved
by Smoczyk in [7] for Calabi-Yau cases by the parabolic maximum principle for the normofon , and Behrndt generalized this result for almost Calabi-Yau cases.
$\partial F/\partial t.$$F_{t}^{*}\omega$$L$

Proposition 2.11. Let $F:L\cross[O, T$ ) $arrow M$ be a solution of generalized mean curvature
flows. If the initial condition : $Larrow M$ is a Lagrangian immersion, then is also a
Lagrangian immersion for every $t\in[O, T$ ).
$F_{0}$$F_{t}$

If : $L\cross[O, T)arrow M$ is a solution of generalized mean curvature flows and each is a
Lagrangian immersion for every $t\in[O, T$ ), then we call it the generalized Lagrangian mean
curvature flow. Hence we hope that if there exists a long time solution $F:L\cross[O, \infty$ ) $arrow M$
of generalized mean curvature flows for a given Lagrangian immersion: $Larrow M$$tarrow\infty$
converges to some immersion as
andthen we can get a special Lagrangian: $Larrow M$ , since it is also a Lagrangian submanifold by Proposition 2.11
immersion$F$$F_{t}$$F_{0}$$F_{t}$$F_{\infty}$

59
and $K(F_{\infty})=0$ (see also Proposition2.7). Actually in some cases this hope is confirmed
to be true however in generic cases generalized mean curvature flows develop singularities
in finite time. In this paper, we construct examples of generalized Lagrangian mean
curvature flows in toric almost Calabi-Yau manifolds which have finite time singularities
and can be continued over singular times in some sense.
Before we step into the next section, we define weighted hamiltonian stationary Lagrangian submanifolds. It can be considered as a weak notion of special Lagrangian submanifold. Remember that a special Lagrangian submanifold (or immersion $F:Larrow\Lambda l$ )is a minimal Lagrangian submanifold (or immersion) by Proposition2.7. Hence it is a critical point of the weighted volume functional;$L$$Vo1_{\overline{g}}:Imm(L, N)arrow \mathbb{R}$by$Vo1_{\overline{g}}(F):=\int_{L}1dV_{F\tilde{g}},$along all infinitesimal deformations as submanifolds. In some sense, a weighted hamiltonian stationary Lagrangian submanifolds is also a critical point of the weighted volume
functional, however its variations are restricted to only Hamiltonian deformations. The
precise meaning is the following. First, Let: $Larrow f|I$ be a Lagrangian immersion to. Next take an infinitesimal Hamiltonianan almost Calabi-Yau manifold
deformationof , that is, we assume that it satisfies $F_{0}=F$ andthere exists a function
such that$F$$(M, \omega, g, J, \Omega)$$\{F_{s} :Larrow M\}_{s\in(-\epsilon,\epsilon)}$$F$$f\in C^{\infty}(L)$(7)$\frac{\partial F}{\partial t}=J\nabla ft=0^{\cdot}$

Under these assumptions, taking the first variation of the weighted volume functional byusing Proposition 2.9, we have$\frac{d}{ds}|_{s=0}Vo1_{\tilde{9}}(F_{s})=-\int_{L}\tilde{9}(J\nabla f,\tilde{H}(F))dV_{F^{*}\overline{g}}=-\int_{L}g(J\nabla f, K(F))dV_{F^{*}\overline{g}}.$In the second equality, we used the relations
andcan use the relation $K(F)=J\nabla\theta_{F}$ by (4). Hence we have$\tilde{9}=e^{2\psi_{9}}$$\tilde{H}=e^{-2\psi}K$. Furthermore we$\frac{d}{ds}|_{s=0}Vo1_{\overline{g}}(F_{S})=-\int_{L}9(J\nabla f, J\nabla\theta_{F})dV_{F^{*}\tilde{g}}$$=- \int_{L}\langle df, d\theta_{F}\rangle_{g}dV_{F^{*}\overline{g}}$$=- \int_{L}f\triangle_{\psi}\theta_{F}dV_{F\tilde{g}},$$:=\triangle u-m9(\nabla\psi, \nabla u)$ , hereisis the weighted Laplacian on defined by
where
$(L,
F_{9}^{*})$. Thus we can see that the first variations
the Laplacian on Riemannian manifoldof the weighted volume functional along all Hamiltonian deformations are zero if and only
if the Lagrangian angle:
is weighted harmonic, that is,$L$$\triangle\psi$$\theta_{F}$$\triangle$$\triangle\psi u$$Larrow \mathbb{R}/\pi \mathbb{Z}$$\triangle_{\psi}\theta_{F}=0.$

Definition 2.12. We call a Lagrangian submanifold with weighted harmonic Lagrangian
angle a weighted hamiltonian stationary Lagrangian submanifold.
It is clear that a special Lagrangian submanifold is a weighted hamiltonian stationaryis aLagrangian submanifold, since its Lagrangian angle is constant. Note that sinceconstant
wheneverimply
is-valued function the condition
does notis compact. For example,
is (weighted) hamiltonian stationary, this is locally
Lagrangian submanifold since its Lagrangian angle is given by$\theta_{F}$$\mathbb{R}/\pi \mathbb{Z}$$:=\{e^{i\theta}|\theta\in \mathbb{R}\}\subset \mathbb{C}$$L$$\theta_{F}$$\triangle_{\psi}\theta_{F}=0$$S^{1}$$a$$\theta(e^{i\theta})$$:=\theta$

60
linear and the second derivative is clearly zero, however this is not a special Lagrangiansubmanifold since this is not minimal.

3. EXAMPLES OF GENERALIZED LAGRANGIAN MEAN CURVATURE FLOWS IN TORICALMOST CALABI-YAU MANIFOLDSIn this section, we review examples of generalized Lagrangian mean curvature flows intoric almost Calabi-Yau manifolds constructed in [10]. First of all, we introduce somebasic notions of toric K\"ahler geometries. Let $(M^{2m}, \omega, g, J)\wedge T^{m}$ be a complex maction. Then we have a moment
dimensional toric K\"ahler manifold with a Hamiltonialmap :and its moment polytope $:=\mu(M)$ . Here we assume that is givenby$T^{m}$$\mu$$Marrow \mathbb{R}^{m}$$\triangle$$\Delta$$\triangle=\{y\in \mathbb{R}^{m}|\langle y, \lambda_{i}\rangle\geqq\kappa_{i}, i=1, .

.

.

, d\}$and constants . Note that on afor some primitive integral inward normal vectors
toric K\"ahler manifold $(M^{2m}, \omega, 9, J)$ , there exists an anti-holomorphic and anti-symplectic
involution : $Marrow M(\sigma^{2}=id)$ . We call the set of fixed points of a real form, anddenote it by$\lambda_{i}$$\kappa_{i}$$\sigma$$\sigma$$M^{\sigma}:=\{p\in M|\sigma(p)=p\}.$

We restrict the moment map

$\mu$

:

to

$Marrow\triangle$$M^{\sigma}$, and denote it by$\mu^{\sigma}:=\mu|_{M^{\sigma}}:M^{\sigma}arrow\triangle.$, the most typical
Note that this is a-fold ramified covering map over . In
K\"ahlermanifolds, the involution is just the complex conjugation given
example of toric, and the real form
is just the real planebyWe can construct a Lagrangian submanifold in $M$ by an aﬃne plane in . We explain
this construction below. Fix $0\leq k\leq m$ arbitrary. Let $A(V, c)$ $:=V+c$ be a -dimensional
is a -dimensional subspace andis a vector.
, whereaﬃne plane in
$A(V,
c)$
.
put
Then
the
interiorofwe
We assume thatintersects$2^{m}$$\mathbb{C}^{m}$$\triangle$$\sigma$$\sigma(z_{1}, \ldots, z_{n/})$$:=(\overline{z}_{1}, .

.

.

$\mathbb{R}^{nl}\subset \mathbb{C}^{n\iota}.$$(\mathbb{C}^{nt})^{\sigma}$, \overline{z}_{n\iota})$$\triangle$

$k$

$k$

$V\subset \mathbb{R}^{m}$$\mathbb{R}^{m}$$c\in \mathbb{R}^{m}$$\triangle$$M^{\sigma}(V, c):=(\mu^{\sigma})^{-1}(\Delta\cap A(V, c$$T(V^{\perp})$$:=V^{\perp}/(V^{\perp}\cap \mathbb{Z}^{n\iota})\cong T^{m-k}\subset T^{m}$$L(V, c) :=T(V^{\perp})\cdot M^{\sigma}(V, c)$and.

is the orthogonal complement of . Note that we assume thatHereinand we also assume that
becomes a
is isomorphic to a subtorus,-orbit ofsmooth real -dimensional submanifold in M. Then $L(V, c)$ , the
$M$
$A(V,
c)$
automatically. Especially, if we takebecomes a Lagrangian submanifold inas-dimensional aﬃne plane, that is a point in , then $L(V, c)$ becomes just a torus fiber of$\mu^{-1}(c)\cong T^{m}$ , and if we take $A(V, c)$ as a $m$ -dimensional aﬃne plane, that is just, then$L(V, c)$ becomes just the real form
. These two are typical Lagrangian submanifolds
in $M$ . Hence, roughly speaking, $L(V, c)$ is a hybrid (or interpolation) of a torus fiber, and $m-k$ is the dimension of torus factors in $L(V, c)$and the real form
$V^{\perp}$$V$$T^{nt-k}$$V^{\perp}/(V^{\perp}\cap \mathbb{Z}^{m})$$M^{\sigma}(V, c)$$T^{n\iota}$$T(V^{\perp})$$k$$M^{\sigma}(V, c)$

$a$

$c$

$\triangle$$\mathbb{R}^{n\iota}$$M^{\sigma}$$T^{m}$$M^{\sigma}$

is the standard toric K\"ahler manifold with a
Example 3.1. The complex space. For example, letbe a primitive
moment map
integral vector and define the $(m-1)$ -dimensional vector space by$\mathbb{C}^{n\iota}$$\mu(z_{1}, \ldots, z_{m})$$:= \frac{1}{2}(|z_{1}|^{2}, \ldots, |z_{\pi\iota}|^{2})$$\xi\in \mathbb{Z}^{m}$$V$$V:=\{y\in \mathbb{R}^{m}|\langle y, \xi\rangle=y_{1}\xi_{1}+\ldots y_{m}\xi_{m}=0\}.$61
Fix a vector
Putby. Then we have an $(m-1)$ -dimensional aﬃne plane $A(V, c)$ $:=V+c.$. Then $L(V, c)$ becomes a-invariant Lagrangian submanifold defined$c\in \mathbb{R}^{m}$$:=2\langle c,$$\kappa$$\xi\rangle$$T^{1}$$L(V, c)=\{(x_{1}e^{2\pi i\xi_{1}\theta}, \ldots, x_{m}e^{2\pi i\xi_{m}\theta})\in \mathbb{C}^{m}|\theta\in \mathbb{R}, \xi_{1}x_{1}^{2}+\cdots+\xi_{m}x_{m}^{2}=\kappa\},$where $x=(x_{1}, \ldots, x_{m})\in \mathbb{R}^{m}$ . Lagrangian submanifolds defined as the form above areconstructed and studied by Joyce [4, Example 9.4].
To make sense of notions of special Lagrangian submanifolds, weighted Hamiltonian
stationary Lagrangian submanifolds and generalized Lagrangian mean curvature flows, we.
have to admit an almost Calabi-Yau structure on the toric K\"ahler manifold$M$is trivial if and only if there exists a
It is known that the canonical line bundle offor all $i=1$ , . . . , . From now on, we assume thatsuch that
vector in
the existence of such . Actually, using , a nonvanishing holomorphic volume form
on $M$ is given by$(\lrcorner \mathfrak{h}I^{2m}, \omega, 9, J)$$\mathbb{Z}^{m}$$\langle\gamma,$$\gamma$$d$$\lambda_{i}\rangle=1$$\Omega_{\gamma}$$\gamma$$\gamma$$\Omega_{\gamma}=e^{\gamma_{1}w_{1}+\cdots+\gamma_{m}w_{m}}dw_{1}\wedge\cdots\wedge dw_{m},$-orbit overare logarithmic holomorphic coordinates on an open dense
where$M$ , that is,$:=\log z_{i}$ for the standard holomorphic coordinatesof-orbit over $M$ , however it can be exNote thatis only defined on an open dense
tended globally as a non-vanishing holomorphic $(m, 0)$ -form on $M$ . We call this quintupleta toric almost Calabi-Yau manifold.
$(w_{i})_{i=1}^{n\prime}$$T_{\mathbb{C}}^{m}$$(\mathbb{C}^{*})^{m}\cong T_{\mathbb{C}}^{m}.$$z_{i}$$w_{i}$$T_{C}^{m}$$\Omega_{\gamma}$$(M, \omega, g, J, \Omega_{\gamma})\cap T^{m}$

Example 3.2. We check the$\mathbb{C}^{m}$case. The moment polytope$\triangle$of$\mathbb{C}^{m}$is given by$\triangle:=\{y\in \mathbb{R}^{m}|\langle y, e_{i}\rangle\geq 0(i=1, \ldots, m)\},$

where$\langle\gamma,$onby. If we take $\gamma=(1, \ldots, 1)$ , then it satisfies thatis the standard basis ofbe the standard holomorphic coordinates
for all $i=1$ , . . . , . Let, we can define the logarithmic holomorphic coordinates. On$:=\log z_{i}$ . Thenbecomes$\mathbb{R}^{m}$$e_{i}$$d$$e_{i}\rangle=1$$\mathbb{C}^{\gamma n}$$w_{i}$$(z_{1}, \ldots, z_{m})$$(w_{1}, \ldots, w_{m})$$(\mathbb{C}^{*})^{m}$$\Omega_{\gamma}$$e^{w_{1}+\cdots+w_{m}}dw_{1}\wedge\cdots\wedge dw_{n\iota},$and this coincides with the standard holomorphic volume form

$dz_{1}\wedge\cdots\wedge dz_{m}.$For a given -dimensional aﬃne plane $A(V, c)$ $:=V+c$ , let $L(V, c)$ be the Lagrangiansubmanifold constructed above and we denote the normal part of with respect to by. Then main results in the paper [10] are stated as follows.
$k$$\gamma$$V$$\gamma^{\perp_{V}}$, the Lagrangian
Theorem 3.3. In the toric almost Calabi-Yau manifold
a weighted Hamiltonian stationary Lagrangian submanifold and its
submanifold
Lagrangian angle of $L(V, c)$ is given by
for $b\in V^{\perp}and$$p\in M^{\sigma}(V, c)$ .
Hence it is clear that $L(V, c)$ is a special Lagrangian submanifold if andonly if$(M, \omega, g, J, \Omega_{\gamma})$$L(V, c)i_{\mathcal{S}}$$\theta([b]\cdot p)=2\pi\langle\gamma,$ $b \rangle+\frac{\pi}{2}(m-k)$$\theta$$\gamma^{\perp_{V}}=0.$, then one parameter family of Lagrangianisasolutionof generalized Lagrangian mean curvaturesubmanifolds
flows with singularities and topological changes with the initial condition $L(V, c)$ . Hereis the first time such that $L(V, c(t))$ becomes the empty set.

Theorem 3.4. If we put $c(t)$$:=c-t\gamma^{\perp_{V}}$$a$$\{L(V.c(t))\}_{t\in[0,T)}$$T$

Here the precise definition of a generalized Lagrangian mean curvature flow with singularities and topological changes is the following.

62
be a rea12 -dimensional almost Calabi-Yau manifold
Definition 3.5. Leta solution ofandbe a one parameter family of subsets in $M$ . Then we calla generalized Lagrangian mean curvature flow with singularities and topological changes
if there exists a real -dimensional manifold and a solution of generalized Lagrangianmean curvature flows $F:L\cross Iarrow M$ such that : $Larrow M$ is an embedding into andis zero, that is,-dimensional Hausdorﬀ measure ofand$(M, \omega, g, J, \Omega)$$m$$\{L_{t}\}_{t\in T}$$\{L,\}_{\iota\in I}$

$L$

$m$

$L_{t}$$F_{t}$$L_{t}\backslash F_{t}(L)$$m$$\mathcal{H}^{m}(L_{t}\backslash F_{t}(L))=0.$$F_{t}(T,)\subset L_{t}$is almost parametrized by a smooth solution of generalized LaIt means that
grangian mean curvature flows.
$\{L,\}_{\in J}$

Example 3.6. In Example 3.1, we consider a$T^{1}$-invariant Lagrangian submanifold$\Gamma_{\lrcorner}(V, =\{(x_{1}e^{2\pi i\xi_{1}\theta}, \ldots, x_{m}e^{2\pi i\xi_{m}\theta})\in \mathbb{C}^{m}|\theta\in \mathbb{R}, \xi_{1}x_{1}^{2}+\cdots+\xi_{m}x_{m}^{2}=\kappa\}.$inconstructed bysubmanifold if and only if$\mathbb{C}$’$\xi\in \mathbb{Z}^{m}$. By applying Theorem 3.3, $L(V, c)$ is a special Lagrangian$\xi_{1}+\cdots+\xi_{m}=0.$

Actually, this is proved by Joyce in [4]. Next, we apply Theorem 3.4. Put $c=0$ in
Example 3.1. Remember that $\gamma=(1, \ldots, 1)$ incase. Then we have $c(t)=c-t\gamma^{1_{V}}=$$-t((\xi_{1}+\cdots+\xi_{r,z})/|\xi|^{2})\xi$ and$\mathbb{C}^{m}$$L(V, c(t))=\{(x_{1}e^{2\pi i\xi_{1}s}, \ldots, x_{rn}e^{2\pi i\xi_{m}s})\in \mathbb{C}^{m}|0\leq s\leq 1,$$\sum_{j=1}^{m}\xi_{j}x_{j}^{2}=-2t\sum_{j=1}^{m}\xi_{j}, x=(x_{1}, \ldots, x_{m})\in \mathbb{R}^{m}\}.$

By Theorem 3.4, we can see that $\{L(V, c(t))\}_{t\in[0,T)}$ is a solution of (generalized) Lagrangian mean curvature flows with singularities and topological changes with the initialin Theorem 1.1 in [6], and Lee
condition $L(V, c)$ . Actually, $L(V, c(t))$ coincides withand Wang proved thatform an eternal solution
is Hamiltonian stationary andfor Brakke flows. Hence our two theorems above can be considered as some kind of generalization of results of Joyce [4] and Lee and Wang [6] into the one in toric almost
Calabi-Yau manifolds.
$V_{t}$$V_{t}$$\{V_{t}\}_{t\in \mathbb{R}}$$\mathbb{C}^{nz}$Finally, we give an example of generalized Lagrangian mean curvature flows with singularities and topological changes.

Example 3.7. Let
be the total space of the canonical line bundle ofa moment polytope is given bywhere$\lrcorner \mathfrak{h}I:=K_{\mathbb{P}^{2}}$$\triangle=\{y\in \mathbb{R}^{3}|\langle y, \lambda_{i}\rangle\geq\kappa_{i}, i=1, .

.

.

$\mathbb{P}^{2}$. Then, 4\}$$\lambda_{1}=(0,0,1) , \lambda_{2}=(1,0,1) , \lambda_{3}=(0,1,1) , \lambda_{4}=(-1, -1,1)$

and $\kappa_{1}=\kappa_{2}=\kappa_{3}=0,$ $\kappa_{4}=-1$ . Then $M$ is a toric almost Calabi-Yau manifold sincefor all $i=1$ , 2, 3, 4. For example, takewe can take $\gamma=(0,0,1)$ so that$:=\{y\in \mathbb{R}^{m}|\langle y, \xi\rangle=0\}.$$\xi=(3,1,5)$ and consider a 2-dimensional subvector space
Take $c=(O, 0,1)$ . Then we have$\langle\gamma,$$\lambda_{i}\rangle=1$$V$$A(V, c(t))=\{y\in \mathbb{R}^{m}|\langle y, \xi\rangle=5-5t\}$byand denote the intersection of $A(V, c(t))$ andby
for $i=1$ , 2, 3, 4.
By simple calculation, one can easily see that whenwithis a triangle, whenand hence$\triangle$$F_{i}$$\triangle(V, c(t))$. We write each facet of$\triangle$$:=\{y\in\triangle|\langle y, \lambda_{i}\rangle=\kappa_{i}\}$$F_{2},$$F_{3}$$F_{4}$$\triangle(V, c(t))$

then $A(V, c(t))$ intersects
then $A(V, c(t))$ across $(1, 0,0)$$0 \leq t<\frac{2}{5}$$t= \frac{2}{5}$63acrossathen A $(V,c(t))$ intersects
vertex o $f\triangle andatopo1ogica1$ change h $appens’$ when
$F_{3}andF_{4}$ hence $\triangle(V,c(t))isa$ square,when tthen A $(V,c(t))$
with F$\frac{4}{5}<t<1$$(0,1,0)$ a vertex ofthen $A(V, c(t))$
and a topological change happens, whenisintersects withso
is a triangle, and when $t=1$ thenandone point $\{(0,0, O)\}$ this means that $L(V, c(t))$ vanishes. Hence a solution $\{L(V, c(t))\}_{t\in I}$
of generalized Lagrangian mean curvature flows with singularities and topological changes
andexists for $t\in I=[0$ , 1). It forms singularities and topological changes when
, and vanishes when $t=1.$, by the same arguOne can see the topology of $L(V, c(t))$ , the -orbit ofment as explained in the proof of Proposition A.3 in [9]. In fact the topology of, is, and is
is
when
when
when $\frac{4}{5}<t<1.$$\frac{2}{5}<t<\frac{4}{=5}$$,F_{2},$$\frac{4}{5}$$\triangle$$F_{1},$$F_{2}$$F_{3}$$\triangle(V, c(t))$$\triangle(V, c(t))$$t= \frac{2}{5}$$t= \frac{4}{5}$$S^{1}$$M^{\sigma}(V, c(t))$$M^{\sigma}(V, c(t))$$S^{2}$$0 \leq t<\frac{2}{5}$$T^{2}$$\frac{2}{5}<t<\frac{4}{5}$$S^{2}$REFERENCES[1] T. Behrndt. Generalized Lagrangian mean curvature flow in K\"ahler manifolds that are almost Einstein.
In Complex and Diﬀerential Geometry, volume 8 of Springer Proceedings in Mathematics, pages 65-79.
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[2] R. Hamilton. Three-manifolds with positive Ricci curvature. J. Diﬀerential Geom., $17(2):255-306,$

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[3] R. Harvey and H. B. Lawson, Jr. Calibrated geometries. Acta Math., 148:47-157, 1982.
[4] D. Joyce. Special Lagrangian $m$-folds inwith symmetries. Duke Math. J., $115(1):1-51$ , 2002.
$\mathbb{C}^{m}$[5] D. Joyce. Riemannian Holonomy Groups and Calibrated Geometry. Oxford Graduate Texts in Mathematics, 12. Oxford University Press, Oxford, 2007.
[6] Y. I. Lee and M.-T. Wang. Hamiltonian stationary cones and self-similar solutions in higher dimensions.
. Amer. Math. Soc., $362(3):1491-1503$ , 2010.
[7] K. Smoczyk. A canonical way to deform a Lagrangian submanifold. arXive:9605005v2, 1996.
[S] R. P. Thomas and S.-T. Yau. Special Lagrangians, stable bundles and mean curvature flow. Comm.
Anal. Geom., $10(5):1075-1113$ , 2002.
[9] H. Yamamoto. Special Lagrangians and Lagrangian self-similar solutions in cones over toric Ssaki
manifolds. arXive:1203.3934v2, 2013.
[10] H. Yamamoto. Weighted Hamiltonian stationary Lagrangian submanifolds and generalized Lagrangian mean cu1vature flows in toric almost Calabi-Yau manifolds. arXive: 1311.7541vl, 2013.
$\mathcal{I}Vans$GRADUATE SCHOOL OF MATHEMATICAL SCIENCES, THE UNIVERSITY OF ToKyo, 3-8-1 KOMABAMEGURO-KU TOKYO 153-8914, JAPAN-mail address: yamamoto@ms.u-tokyo.ac.jp$E$