# Calabi-Yau structures
on the complexifications of
rank two symmeric spaces

Naoyuki Koike

###### Abstract

In this paper, we prove that there exists a $C^{\infty}$-Calabi-Yau structure
on the complexification $G^{\mathbb{C}}/K^{\mathbb{C}}$ of a rank two
symmetric space $G/K$ of compact type. The proof is performed by deriving a
relation (which differs from the known relation somewhat) between the complex
Hessian of a real-valued $C^{\infty}$-function $\psi$ on
$G^{\mathbb{C}}/K^{\mathbb{C}}$ and the Hessian of the function
$\rho:=\psi|_{{\rm Exp}_{o}(\mathfrak{a})}\circ{\rm Exp}_{o}|_{\mathfrak{a}}$
on a maximal abelian subspace $\mathfrak{a}$ of the normal space
$T_{o}^{\perp}(G\cdot o)(=T_{o}(G^{d}\cdot o)\subset\mathfrak{g}^{d})$ of the
orbit $G\cdot o(=G/K)$ in $G^{\mathbb{C}}/K^{\mathbb{C}}$ at the base point
$o$, where ${\rm Exp}_{o}$ is the exponential map of
$G^{\mathbb{C}}/K^{\mathbb{C}}$ at $o$ and $\mathfrak{g}^{d}$ is the Lie
algebra of the dual $G^{d}$ of $G$. This relation is derived in a new
calculation method by using the explicit descriptions of the shape operators
of the orbits of the isotropy action $K\curvearrowright G^{d}/K$ and the
Hermann type action $G\curvearrowright G^{\mathbb{C}}/K^{\mathbb{C}}$.

##  1 Introduction

In this paper, a $C^{\infty}$-Calabi-Yau structure ($r\geq 0,\,\alpha>0$) of a
$2n$-dimensional manifold $M$ means a quadruple $(J,g,\omega,\Omega)$
satisfying the following three conditions:

(i) $(J,g)$ is a $C^{\infty}$-K$\ddot{\rm a}$hler structure of $M$;

(ii) $\omega$ is the $C^{\infty}$-K$\ddot{\rm a}$hler form of $(J,g)$;

(iii) $\Omega$ is a nonvanishing holomorphic $(n,0)$-form on $M$ satisfying

$\omega^{n}=(-1)^{n(n-1)/2}n!\left(\frac{\sqrt{-1}}{2}\right)^{n}\Omega\wedge\overline{\Omega}\quad({\rm
on}\,\,M).$ $None$

It follows from $(1.1)$ that $(M,J,g)$ is a $C^{\infty}$-Ricci-flat
K$\ddot{\rm a}$hler manifold and that, for any real constant $\theta$, a
$n$-form ${\rm Re}(e^{\sqrt{-1}\theta}\Omega)$ is a calibration on $(M,g)$.
Hence, special Lagrangian submanifolds of phase $\theta$ in
$(M,J,g,\omega,\Omega)$ are defined as submanifolds calibrated by ${\rm
Re}(e^{\sqrt{-1}\theta}\Omega)$.

The complexification of $C^{\omega}$-Riemannian manifold is defined as
follows, where $C^{\omega}$ means the real analyticity. Let $(M,g)$ be a
$C^{\omega}$-Riemannian manifold, $TM$ be the tangent bundle of $M$ and
$J_{A}$ be the adapted complex structure (associated to $g$) defined on a
tubular neighborhood $U_{A}$ of the zero section of $TM$, where we note that
$J_{A}$ is defined on the whole of $TM$ (i.e., $U_{A}=TM$) when $(M,g)$ is of
non-negative curvature. See [22] and [10, 11] about the definition of the
adapted complex structure. The complex manifold $(U_{A},J_{A})$ is regarded as
the complexification of $(M,g)$ under the identification of the zero-section
of $TM$ with $M$. Let $G/K$ be a symmetric space of compact type. Since $G/K$
is a $C^{\omega}$-Riemannian manifold of non-negative curvature, the adapted
complex structure $J_{A}$ of $G/K$ is defined on the whole of the tangent
bundle $T(G/K)$. As above, $(T(G/K),J_{A})$ is regarded as the
complexification of $G/K$. We can also define the complexification of $G/K$ as
follows. Let $G^{\mathbb{C}}$ and $K^{\mathbb{C}}$ be the complexifications of
$G$ and $K$, respectively. Denote by
$\mathfrak{g},\mathfrak{k},\mathfrak{g}^{\mathbb{C}}$ and
$\mathfrak{k}^{\mathbb{C}}$ be the Lie algebras of $G,K,G^{\mathbb{C}}$ and
$K^{\mathbb{C}}$, respectively. Let
$\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{p}$ and
$\mathfrak{g}^{\mathbb{C}}=\mathfrak{k}^{\mathbb{C}}\oplus\mathfrak{p}^{\mathbb{C}}$
be the canonical decompositions associated to the semi-simple symmetric pairs
$(G,K)$ and $(G^{\mathbb{C}},K^{\mathbb{C}})$, respectively. Here we note that
$\mathfrak{p}$ and $\mathfrak{p}^{\mathbb{C}}$ are identified with the tangent
spaces $T_{eK}(G/K)$ and $T_{eK^{\mathbb{C}}}(G^{\mathbb{C}}/K^{\mathbb{C}})$,
respectively, where $e$ is the identity element of $G^{\mathbb{C}}$. For the
simplicity, set $o:=eK^{\mathbb{C}}(=eK)$. Define a complex linear
transformation ${\bf j}$ of $\mathfrak{p}^{\mathbb{C}}$ by ${\bf
j}(v):=\sqrt{-1}v\,\,\,\,(v\in\mathfrak{p}^{\mathbb{C}})$. Since ${\bf j}$ is
${\rm Ad}_{G^{\mathbb{C}}}(K^{\mathbb{C}})$-invariant, we can define
$G^{\mathbb{C}}$-invariant complex structure ${\bf J}$ of
$G^{\mathbb{C}}/K^{\mathbb{C}}$ satisfying ${\bf J}_{o}={\bf j}$ uniquely,
where ${\rm Ad}_{G^{\mathbb{C}}}$ denotes the adjoint representation of
$G^{\mathbb{C}}$. The complex manifold $(G^{\mathbb{C}}/K^{\mathbb{C}},{\bf
J})$ is regarded as another complexification of $G/K$ under the identification
of the orbit $G\cdot o$ of the subaction $G\curvearrowright
G^{\mathbb{C}}/K^{\mathbb{C}}$ of the natural action
$G^{\mathbb{C}}\curvearrowright G^{\mathbb{C}}/K^{\mathbb{C}}$ with $G/K$. We
can define a natural holomorphic diffeomorphism between two complexifications
$(T(G/K),J_{A})$ and $(G^{\mathbb{C}}/K^{\mathbb{C}},{\bf J})$ of $G/K$ as
follows. Let $B$ be the Killing form of $\mathfrak{g}$ and set $B_{A}:=-2{\rm
Re}\,B^{\mathbb{C}}$. Since $B_{A}$ is ${\rm
Ad}_{G^{\mathbb{C}}}(K^{\mathbb{C}})$-invariant, we can define
$G^{\mathbb{C}}$-invariant pseudo-Riemannian metric $\beta_{A}$ of
$G^{\mathbb{C}}/K^{\mathbb{C}}$ satisfying $(\beta_{A})_{o}=B_{A}$ uniquely.
The pseudo-Riemannian manifold $(G^{\mathbb{C}}/K^{\mathbb{C}},\beta_{A})$ is
one of semi-simple pseudo-Riemannian symmetric spaces. Also, the triple
$(G^{\mathbb{C}}/K^{\mathbb{C}},J_{A},\beta_{A})$ gives an anti-K$\ddot{\rm
a}$hler manifold. See [15] about the definition of the anti-K$\ddot{\rm
a}$hler manifold. Denote by ${\rm Exp}_{p}$ the exponential map of
$(G^{\mathbb{C}}/K^{\mathbb{C}},\beta_{A})$ at $p\in
G^{\mathbb{C}}/K^{\mathbb{C}}$. The natural bijection
$\displaystyle{\Psi:(T(G/K),J_{A})\mathop{\longrightarrow}_{\cong}(G^{\mathbb{C}}/K^{\mathbb{C}},{\bf
J})}$ is given by

$\Psi(v):={\rm Exp}_{p}({\bf J}_{p}(v))\quad\,\,(p\in G/K,\,\,v\in
T_{p}(G/K))$

(see Figure 1), where $v$ in the right-hand side is regarded as a tangent
vector of the submanifold $G\cdot o(\approx G/K)$ in
$G^{\mathbb{C}}/K^{\mathbb{C}}$. It is shown that $\Psi$ is a holomorphic
diffeomorphism between $(T(G/K),J_{A})$ and
$(G^{\mathbb{C}}/K^{\mathbb{C}},{\bf J})$ (see Section 2 about the proof of
this fact). Thus these two complexifications of $G/K$ are identified through
$\Psi$.

In 1993, M. B. Stenzel ([21]) showed that there exists a $G$-invariant
complete Ricci-flat metric on the cotangent bundle of a rank one symmetric
space $G/K$ of compact type. In 2004, R. Bielawski ([4]) investigated the
existence of $G$-invariant metrics with the prescribed Ricci tensor on the
complexification $G^{\mathbb{C}}/K^{\mathbb{C}}$ of a general rank symmetric
space $G/K$ of compact type (see [2] also), where $G,\,K$ and $G^{\mathbb{C}}$
are assumed to be connected. Note that, in [4], it is stated that the
$G$-invariant metrics with the prescribed Ricci tensor are defined on the
whole of $G^{\mathbb{C}}/K^{\mathbb{C}}$ but it seems that there is a gap in
the process of the proof, where we note that there is a gap in the process of
the proof of the main theorem in [2] also. In 2019, the author ([18])
investigated the existence of Calabi-Yau structures on the complexification
$G^{\mathbb{C}}/K^{\mathbb{C}}$ of a general rank symmetric space $G/K$ of
compact type but there are some gaps in the proof of some lemmas in the paper.
In this paper, we close the gaps and prove the following fact in the case
where $G/K$ is of rank two.

Theorem A. If $G/K$ is a rank two symmetric space of compact type, then there
exists a $G$-invariant $C^{\infty}$-Calabi-Yau structure on the
complexification $G^{\mathbb{C}}/K^{\mathbb{C}}$ of $G/K$.

$o$$p$$G\cdot o=G/K$$\Psi(T_{p}(G\cdot o))$${\bf
J}_{p}(v)$$v$$G^{\mathbb{C}}/K^{\mathbb{C}}$$\Psi(v)={\rm Exp}_{p}({\bf
J}_{p}(v))$$\Psi$$0$-section(=G/K)$T(G/K)$$v$$T_{p}(G/K)$$p$

Figure 1 : The identification of $(T(G/K),J_{A})$ and
$(G^{\mathbb{C}}/K^{\mathbb{C}},{\bf J})$

##  2 Basic notions and facts

In this section, we first recall the adapted complex structure of the tangent
bundle of complete $C^{\omega}$-Riemanninan manifold $(M,g)$, which was
introduced by R. Sz$\ddot{\rm o}$ke ([22]) and V. Guillemin-M. B. Stenzel
([10, 11]), and show that $\Psi$ stated in Introduction is a holomorphic
diffeomorphism. For each geodesic $\gamma:\mathbb{R}\to M$ in $(M,g)$, we
define $\gamma_{\ast}:\mathbb{C}\to TM$ by

$\gamma_{\ast}(s+t\sqrt{-1}):=t\gamma^{\prime}(s)\quad\,\,(s+t\sqrt{-1}\in\mathbb{C}),$

where $TM$ is the tangent bunndle of $M$ and $\gamma^{\prime}(s)$ is the
velocity vector of $\gamma$ at $s$. Then V. Guillemin and M. B. Stenzel showed
that there exists a unique complex structure $J_{A}$ of $TM$ such that, for
any geodesic $\gamma:\mathbb{R}\to M$ in $(M,g)$,
$\gamma_{\ast}:\mathbb{C}\to(TM,J_{A})$ is holomorphic. This complex structure
is called the adapted complex structure of $TM$ associated to $g$.

Proposition 2.1. Let $G/K$ be a symmetric space of compact type and
$\Psi:T(G/K)\to G^{\mathbb{C}}/K^{\mathbb{C}}$ be the map stated in
Introduction. Then the map $\Psi$ is a holomorphic diffeomorphism of
$(T(G/K),J_{A})$ onto $(G^{\mathbb{C}}/K^{\mathbb{C}},{\bf J})$.

Proof. It is clear that $\Psi$ is a diffeomorphism. We shall show that $\Psi$
is holomorphic. Let $\gamma:\mathbb{R}\to G/K$ be a geodesic in $G/K$ and set
$\gamma^{\mathbb{C}}:=\Psi\circ\gamma_{\ast}$. Then we have

$\gamma^{\mathbb{C}}(s+t\sqrt{-1})={\rm Exp}_{\gamma(s)}(t{\bf
J}_{\gamma(s)}(\gamma^{\prime}(s)))$

(see Figure 2), which is the complexification of the geodesic $\gamma$ in
$G/K(\approx G\cdot o\subset G^{\mathbb{C}}/K^{\mathbb{C}}))$ in the sense of
[16, 17]. By using Proposition 3.1 of [17], we can show that
$\gamma^{\mathbb{C}}:\mathbb{C}\to(G^{\mathbb{C}}/K^{\mathbb{C}},{\bf J})$ is
holomorphic. This fact together with the arbitrariness of $\gamma$ implies
that $\Psi$ is holomorphic. ∎

$\Psi$$\mathbb{C}$$T(G/K)$$G^{\mathbb{C}}/K^{\mathbb{C}}$$1$$2\sqrt{-1}$$\sqrt{-1}$$s$$t\sqrt{-1}$$\gamma_{\ast}$$v$$\gamma^{\prime}(s)$$2\gamma^{\prime}(s)$$2v$$v$${\bf
J}_{\gamma(0)}(v)$${\rm Exp}_{\gamma(0)}({\bf J}_{\gamma(0)}(2v))$${\rm
Exp}_{\gamma(0)}({\bf J}_{\gamma(0)}(v))$$0$-section$G\cdot
o(=G/K)$$\gamma_{v}^{\mathbb{C}}$

Figure 2 : The image of $\gamma_{\ast}$

Remark 2.1. In [18], this fact was stated but the proof was not given.

Let $G/K$ be a symmetric space of compact type and
$(G^{\mathbb{C}}/K^{\mathbb{C}},\,{\bf J})$ be the complexification of $G/K$.
Let $\psi:G^{\mathbb{C}}/K^{\mathbb{C}}\to\mathbb{R}$ be a strictly
plurisubharmonic function over $G^{\mathbb{C}}/K^{\mathbb{C}}$, where we note
that “strictly plurisubharmonicity” means that the Hermitian matrix
$\displaystyle{\left(\frac{\partial^{2}\psi}{\partial
z_{i}\partial\bar{z}_{j}}\right)}$ is positive (or equivalently,
$(\partial\overline{\partial}\psi)(\bm{Z},\overline{\bm{Z}})>0$ holds for any
nonzero $(1,0)$-vector $\bm{Z}$). Then
$\omega_{\psi}:=\sqrt{-1}\partial\overline{\partial}\psi$ is a real non-
degenerate closed $2$-form on $G^{\mathbb{C}}/K^{\mathbb{C}}$ and the
symmetric $(0,2)$-tensor field $\beta_{\psi}$ associated to ${\bf J}$ and
$\omega_{\psi}$ is positive definite. Hence $({\bf J},\beta_{\psi})$ gives a
K$\ddot{\rm a}$hler structure of $G^{\mathbb{C}}/K^{\mathbb{C}}$. Thus, from
each strictly plurisubharmonic function over $G^{\mathbb{C}}/K^{\mathbb{C}}$,
we can construct a K$\ddot{\rm a}$hler structure of
$G^{\mathbb{C}}/K^{\mathbb{C}}$. Denote by ${\rm Exp}_{p}$ the exponential map
of the anti-K$\ddot{\rm a}$hler manifold
$(G^{\mathbb{C}}/K^{\mathbb{C}},\beta_{A})$ at $p(\in
G^{\mathbb{C}}/K^{\mathbb{C}})$ and $\exp$ the exponentional map of the Lie
group $G^{\mathbb{C}}$. Set
$\mathfrak{g}^{d}:=\mathfrak{k}\oplus\sqrt{-1}\mathfrak{p}(\subset\mathfrak{g}^{\mathbb{C}})$
and $G^{d}=\exp(\mathfrak{g}^{d})$. Denote by $\beta_{G/K}$ the $G$-invariant
(Riemannian) metric on $G/K$ induced from
$-B|_{\mathfrak{p}\times\mathfrak{p}}$ and $\beta_{G^{d}/K}$ the
$G^{d}$-invariant negative definite metric on $G^{d}/K$ induced from $-({\rm
Re}\,B^{\mathbb{C}})|_{\sqrt{-1}\mathfrak{p}\times\sqrt{-1}\mathfrak{p}}$. We
may assume that the metric of $G/K$ is equal to $\beta_{G/K}$ by
homothetically transforming the metric of $G/K$ if necessary. On the other
hand, the Riemannian manifold $(G^{d}/K,-\beta_{G^{d}/K})$ is a (Riemannian)
symmetric space of non-compact type. The orbit $G\cdot o$ is isometric to
$(G/K,\beta_{G/K})$ and the normal umbrella ${\rm
Exp}_{o}(T^{\perp}_{o}(G\cdot o))(=G^{d}\cdot o)$ is isometric to
$(G^{d}/K,\beta_{G^{d}/K})$. The complexification $\mathfrak{p}^{\mathbb{C}}$
of $\mathfrak{p}$ is identified with $T_{o}(G^{\mathbb{C}}/K^{\mathbb{C}})$
and $\sqrt{-1}\mathfrak{p}$ is identified with $T_{o}({\rm
Exp}_{o}(T^{\perp}_{o}(G\cdot o)))$. Let $\mathfrak{a}$ be a maximal abelian
subspace of $\mathfrak{p}$, where we note that ${\rm dim}\,\mathfrak{a}=r$.
Denote by $W$ the Weyl group of $G^{d}/K$ with respect to
$\sqrt{-1}\mathfrak{a}$. This group acts on $\sqrt{-1}\mathfrak{a}$. Let
$C(\subset\sqrt{-1}\mathfrak{a})$ be a Weyl domain (i.e., a fundamental domain
of the action $W\curvearrowright\sqrt{-1}\mathfrak{a}$). Then we have
$G\cdot{\rm Exp}_{o}(\overline{C})=G^{\mathbb{C}}/K^{\mathbb{C}}$, where
$\overline{C}$ is the closure of $C$. For a $W$-invariant connected open
neighborhood $D$ of $0$ in $\sqrt{-1}\mathfrak{a}$, we define a neighborhood
$U_{1}(D)$ of $o$ in $\Sigma:={\rm Exp}_{o}(\sqrt{-1}\mathfrak{a})$ by
$U_{1}(D):={\rm Exp}_{o}(D)$, a neighborhood $U_{2}(D)$ of $o$ in $G^{d}/K$ by
$U_{2}(D):=K\cdot U_{1}(D)$ and a tubular neighborhood $U_{3}(D)$ of $G\cdot
o$ in $G^{\mathbb{C}}/K^{\mathbb{C}}$ by $U_{3}(D):=G\cdot U_{1}(D)$ (see
Figure 3). Denote by ${\rm Conv}_{W}^{+}(D)$ the space of all $W$-invariant
strictly convex ($C^{\infty}$-)functions over $D$, ${\rm
Conv}_{K}^{+}(U_{2}(D))$ the space of all $K$-invariant strictly convex
($C^{\infty}$-)functions over $U_{2}(D)$ and $PH_{G}^{+}(U_{3}(D))$ the space
of all $G$-invariant strictly plurisubharmonic ($C^{\infty}$-)functions over
$U_{3}(D)$. The restriction map (which is denoted by $\mathcal{R}_{32}^{D}$)
from $U_{3}(D)$ to $U_{2}(D)$ gives an isomorphism of $PH_{G}^{+}(U_{3}(D))$
onto ${\rm Conv}_{K}^{+}(U_{2}(D))$ and the composition of the restriction map
(which is denoted by $\mathcal{R}_{31}^{D}$) from $U_{3}(D)$ to $U_{1}(D)$
with ${\rm Exp}_{o}$ gives an isomorphism of $PH_{G}^{+}(U_{3}(D))$ onto ${\rm
Conv}_{W}^{+}(D)$ (see [1]). Hence we suffice to construct $W$-invariant
strictly convex functions over $D$ or $K$-invariant strictly convex functions
over $U_{2}(D)$ to construct strictly plurisubharmonic functions over
$U_{3}(D)$. Let $\psi$ be a $G$-invariant strictly plurisubharmonic
($C^{\infty}$-)function over $U_{3}(D)$. Set
$\bar{\psi}:=\mathcal{R}_{32}^{D}(\psi)$ and
$\bar{\bar{\psi}}:=\mathcal{R}_{31}^{D}(\psi)\circ{\rm Exp}_{o}$. Conversely,
for a $W$-invariant strictly convex ($C^{\infty}$-)function $\rho$ over $D$,
denote by $\rho^{h}$ the $G$-invariant strictly plurisubharmonic
($C^{\infty}$-)function $\psi$ over $U_{3}(D)$ with $\bar{\bar{\psi}}=\rho$.
Similarly, for a $K$-invariant strictly

$G^{d}\cdot o$$U_{2}(D)$${\rm
Exp}_{o}(\sqrt{-1}\mathfrak{a})$$U_{1}(D)$$U_{3}(D)$$G\cdot
o$$G^{\mathbb{C}}/K^{\mathbb{C}}$

Figure 3 : $G$-invariant tubular neighborhood of $G\cdot o$

convex ($C^{\infty}$-)function $\sigma$ over $U_{2}(D)$, set
$\overline{\sigma}:=\mathcal{R}_{21}^{D}(\sigma)\circ{\rm Exp}_{o}$ and denote
by $\sigma^{h}$ the $G$-invariant strictly plurisubharmonic
($C^{\infty}$-)function $\psi$ over $U_{3}(D)$ with $\bar{\psi}=\sigma$. Also,
for a $W$-invariant strictly convex ($C^{\infty}$-)function $\rho$ over $D$,
denote by $\rho^{d}$ the $K$-invariant strictly convex $C^{\infty}$-function
$\sigma$ over $U_{2}(D)$ with $\overline{\sigma}=\rho$. Denote by $Ric_{\psi}$
the Ricci form of $\beta_{\psi}$. It is known that $Ric_{\psi}$ is described
as

$Ric_{\psi}=-\sqrt{-1}\partial\overline{\partial}\log\,{\rm
det}\left(\frac{\partial^{2}\psi}{\partial z_{i}\partial\bar{z}_{j}}\right)$
$None$

(see P158 of [14]), where $(z_{1},\cdots,z_{n})$ is any complex coordinates of
$G^{\mathbb{C}}/K^{\mathbb{C}}$. Note that the right-hand side of $(2.1)$ is
independent of the choice of the complex coordinates $(z_{1},\cdots,z_{n})$ of
$G^{\mathbb{C}}/K^{\mathbb{C}}$.

##  3 Proof of Theorem A

In this section, we shall prove Theorem A stated in Introduction. First we
define a natural $G^{\mathbb{C}}$-invariant non-vanishing holomorphic
$(n,0)$-form on $G^{\mathbb{C}}/K^{\mathbb{C}}$. Take an orthonormal basis
$(\bm{e}_{1},\cdots,\bm{e}_{n})$ of $\mathfrak{p}$ with respect to $-B$ and
let $(\theta^{1},\cdots,\theta^{n})$ be the dual basis of
$(\bm{e}_{1},\cdots,\bm{e}_{n})$. Also, let $(\theta^{i})^{\mathbb{C}}$
($i=1,\cdots,n$) be the complexification of $\theta^{i}$. Since
$(\theta^{1})^{\mathbb{C}}\wedge\cdots\wedge(\theta^{n})^{\mathbb{C}}$ is
${\rm
Ad}_{G^{\mathbb{C}}}(K^{\mathbb{C}})|_{\mathfrak{p}^{\mathbb{C}}}$-invariant,
we obtain the $G^{\mathbb{C}}$-invariant non-vanishing holomorphic
$(n,0)$-form $\Omega$ on $G^{\mathbb{C}}/K^{\mathbb{C}}$ with
$\Omega_{eK^{\mathbb{C}}}=(\theta^{1})^{\mathbb{C}}\wedge\cdots\wedge(\theta^{n})^{\mathbb{C}}$.

Let ${\bf J}$ be the $G^{\mathbb{C}}$-invariant complex structure with ${\bf
J}_{eK}={\bf j}(=\sqrt{-1}\,{\rm id}_{\mathfrak{p}}$). Take any point
$p_{0}:={\rm Exp}_{o}(\bm{Z}_{0})$ ($\bm{Z}_{0}\in\sqrt{-1}\mathfrak{p}$) and
an orthonormal basis $(\bm{e}_{1},\cdots,\bm{e}_{n})$ of
$\sqrt{-1}\mathfrak{p}$ with respect to $\beta_{G^{d}/K}$ satisfying
$\bm{e}_{i}\in\sqrt{-1}\mathfrak{a}$ ($i=1,\cdots,r$), where ${\rm Exp}_{o}$
is the exponential map of $(G^{\mathbb{C}}/K^{\mathbb{C}},\beta_{A})$ at $o$.
Let $\widetilde{U}_{o}$ be a sufficiently small neighborhood of the origin $0$
in $\mathfrak{p}^{\mathbb{C}}$. Define a local complex coordinates
$\varphi_{o}=(z^{o}_{1},\cdots,z^{o}_{n})$ on $U_{o}:={\rm
Exp}_{o}(\widetilde{U}_{o})$ by

$({\rm
Exp}_{o}|_{\widetilde{U}_{o}})^{-1}(p)=\sum_{i=1}^{n}z_{i}^{o}(p)\bm{e}_{i}\quad\,\,(p\in
U_{o}).$

By Proposition 3.1 of [17], $\varphi_{o}$ gives a local complex coordinates of
the complex manifold $(G^{\mathbb{C}}/K^{\mathbb{C}},{\bf J})$. Set
$U_{p_{0}}:=\exp(\bm{Z}_{0})(U_{o})$, where $\exp$ is the exponential map of
the Lie group $G^{\mathbb{C}}$. Define a local complex coordinates
$\varphi^{p_{0}}=(z^{p_{0}}_{1},\cdots,z^{p_{0}}_{n})$ on $U_{p_{0}}$ by
$z^{p_{0}}_{i}=z^{o}_{i}\circ\exp(\bm{Z}_{0})^{-1}$ ($i=1,\cdots,n$). Let
$z^{o}_{i}=x^{o}_{i}+\sqrt{-1}y^{o}_{i}$ and
$z^{p_{0}}_{i}=x^{p_{0}}_{i}+\sqrt{-1}y^{p_{0}}_{i}$ ($i=1,\cdots,n$). We call
such a local complex coordinates
$\varphi^{p_{0}}=(z^{p_{0}}_{1},\cdots,z^{p_{0}}_{n})$ a normal complex
coordinates about a point $p_{0}$ of the real form $G^{d}\cdot o(=G^{d}/K)$
associated to $(\bm{e}_{1},\cdots,\bm{e}_{n})$. Set
$\bm{e}^{p_{0}}_{i}:=\exp_{G^{d}}(\bm{Z}_{0})_{\ast o}(\bm{e}_{i})$. Then we
note that the following relation holds:

${\rm
Exp}_{p_{0}}^{-1}(p)=\sum_{i=1}^{n}\left(x^{p_{0}}_{i}(p)\bm{e}^{p_{0}}_{i}+y^{p_{0}}_{i}(p)J_{p_{0}}(\bm{e}^{p_{0}}_{i})\right)\quad(p\in
U_{p_{0}}).$

For the simplicity, we denote $\sqrt{-1}\mathfrak{p}$ and
$\sqrt{-1}\mathfrak{a}$ by $\mathfrak{p}^{d}$ and $\mathfrak{a}^{d}$,
respectively. For $\lambda\in(\mathfrak{a}^{d})^{\ast}$, we define
$\mathfrak{p}^{d}_{\lambda}$ and $\mathfrak{k}_{\lambda}$ by

$\mathfrak{k}_{\lambda}:=\\{v\in\mathfrak{k}\,|\,{\rm
ad}(\bm{Z})^{2}(v)=\lambda(\bm{Z})^{2}v\,\,(\forall\,\bm{Z}\in\mathfrak{a}^{d})\\}$

and

$\mathfrak{p}^{d}_{\lambda}:=\\{v\in\mathfrak{p}^{d}\,|\,{\rm
ad}(\bm{Z})^{2}(v)=\lambda(\bm{Z})^{2}v\,\,(\forall\,\bm{Z}\in\mathfrak{a}^{d})\\}.$

Also, we define $\triangle(\subset(\mathfrak{a}^{d})^{\ast})$ by

$\triangle:=\\{\lambda\in(\mathfrak{a}^{d})^{\ast}\setminus\\{0\\}\,|\,\mathfrak{p}^{d}_{\lambda}\not=\\{0\\}\,\,\\},$

which is the root system. Let $\triangle_{+}$ be the positive root subsystem
of $\triangle$ with respect to some lexicographic ordering of
$(\mathfrak{a}^{d})^{\ast}$. Then we have the following root space
decompositions

$\mathfrak{k}=\mathfrak{z}_{\mathfrak{k}}(\mathfrak{a}^{d})\oplus\left(\mathop{\oplus}_{\lambda\in\triangle_{+}}\mathfrak{k}_{\lambda}\right)\quad{\rm
and}\quad\mathfrak{p}^{d}=\mathfrak{a}^{d}\oplus\left(\mathop{\oplus}_{\lambda\in\triangle_{+}}\mathfrak{p}^{d}_{\lambda}\right),$

where $\mathfrak{z}_{\mathfrak{k}}(\mathfrak{a}^{d})$ is the cetralizer of
$\mathfrak{a}^{d}$ in $\mathfrak{k}$. Set $m_{\lambda}:={\rm
dim}\,\mathfrak{p}^{d}_{\lambda}$ ($\lambda\in\triangle_{+}$). Note that the
first root space deomposition is the simultaneously eigenspace decomposition
of the commutative family

$\\{{\rm
ad}(\bm{Z})^{2}:\mathfrak{k}\to\mathfrak{k}\,|\,\bm{Z}\in\mathfrak{a}^{d}\\}$

of the symmetric transformations of $\mathfrak{k}$ and that the second root
space decomposition is the simultaneously eigenspace decomposition of the
commutative family

$\\{{\rm
ad}(\bm{Z})^{2}:\mathfrak{p}^{d}\to\mathfrak{p}^{d}\,|\,\bm{Z}\in\mathfrak{a}^{d}\\}$

of symmetric transformations of $\mathfrak{p}^{d}$, where ${\rm ad}$ is the
adjoint representation of $\mathfrak{g}$.

Let $\mathcal{C}(\subset\mathfrak{a}^{d})$ be a Weyl domain (i.e., a
fundamental domain of the Weyl group action
$W\curvearrowright\mathfrak{a}^{d}$). The Wyle domain $\mathcal{C}$ is given
by

$\mathcal{C}:=\\{{\bm{v}}\in\mathfrak{a}^{d}\,|\,\lambda({\bm{v}})>0\,\,(\forall\,\lambda\in\triangle_{+})\\}.$

Points of $W\cdot\mathcal{C}$ and $G\cdot{\rm Exp}_{o}(W\cdot\mathcal{C})$ are
called regular points. We call $G\cdot{\rm Exp}_{o}(W\cdot\mathcal{C})$ the
regular point set of $G^{\mathbb{C}}/K^{\mathbb{C}}$ and denote it by
$(G^{\mathbb{C}}/K^{\mathbb{C}})_{reg}$. Also, we call $K\cdot{\rm
Exp}_{o}(W\cdot\mathcal{C})$ the regular point set of $G^{d}/K$ and denote it
by $(G^{d}/K)_{reg}$. Note that $(G^{\mathbb{C}}/K^{\mathbb{C}})_{reg}$ (resp.
$(G^{d}/K)_{reg}$) is an open dense subset of $G^{\mathbb{C}}/K^{\mathbb{C}}$
(resp. $G^{d}/K$).

Take a regular point $p:={\rm Exp}_{o}(\bm{Z})$ ($\bm{Z}\in
W\cdot\mathcal{C}$) of the isotropy group action $K\curvearrowright G^{d}/K$.
Then we have

$(\exp\,\bm{Z})_{\ast}^{-1}(T_{p}(K\cdot
p))=\mathop{\oplus}_{\lambda\in\triangle_{+}}\mathfrak{p}^{d}_{\lambda}$
$None$

and

$(\exp\,\bm{Z})_{\ast}^{-1}(T_{p}^{\perp}(K\cdot p))=\mathfrak{a}^{d},$

where $T_{p}^{\perp}(K\cdot p)$ is the normal space of $K\cdot p$ in
$G^{d}\cdot o(=G^{d}/K)$. Denote by $A$ and $h$ the shape tensor and the
second fundamental form of the principal orbit $K\cdot p$ (which is a
submanifold in $G^{d}/K$), respectively. For any
${\bm{v}}\in\mathfrak{a}^{d}$, we have

$(A_{p})_{(\exp\,\bm{Z})_{\ast}({\bm{v}})}|_{(\exp\,\bm{Z})_{\ast}(\mathfrak{p}^{d}_{\lambda})}=-\frac{\lambda({\bm{v}})}{\tanh\lambda(\bm{Z})}\,{\rm
id}$ $None$

(see [23]).

Both Lie groups $G$ and $K^{\mathbb{C}}$ are symmetric subgroups of
$G^{\mathbb{C}}$, that is, the actions $G\curvearrowright
G^{\mathbb{C}}/K^{\mathbb{C}}$ and $K^{\mathbb{C}}\curvearrowright
G^{\mathbb{C}}/K^{\mathbb{C}}$ are Hermann type action, where we note that the
terminology of “Hermann type action” was used in [16]. Let $\theta_{G}$ and
$\theta_{K^{\mathbb{C}}}$ be the involutions of $G^{\mathbb{C}}$ with $({\rm
Fix}\,\theta_{G})_{0}\subset G\subset{\rm Fix}\,\theta_{G}$ and $({\rm
Fix}\,\theta_{K^{\mathbb{C}}})_{0}\subset K^{\mathbb{C}}\subset{\rm
Fix}\,\theta_{K^{\mathbb{C}}}$, respectively. Denote by the same symbols the
differentials of $\theta_{G}$ and $\theta_{K^{\mathbb{C}}}$ at $e$,
respectively. Note that $\theta_{G}$ is the Cartan involution of
$G^{\mathbb{C}}$. The eigenspace decompositions of $\theta_{G}$ and
$\theta_{K^{\mathbb{C}}}$ are as
$\mathfrak{g}^{\mathbb{C}}=\mathfrak{g}\oplus\sqrt{-1}\mathfrak{g}$ and
$\mathfrak{g}^{\mathbb{C}}=\mathfrak{k}^{\mathbb{C}}\oplus\mathfrak{p}^{\mathbb{C}}$,
respectively. The root space decomposition of
$\mathfrak{p}^{\mathbb{C}}(=T_{o}(G^{\mathbb{C}}/K^{\mathbb{C}})$ for a
maximal abelian subspace $\mathfrak{a}^{d}$ of
$\mathfrak{p}^{\mathbb{C}}\cap\sqrt{-1}\mathfrak{g}=\mathfrak{p}^{d}$ is given
by

$\mathfrak{p}^{\mathbb{C}}=\mathfrak{a}^{\mathbb{C}}\oplus\left(\mathop{\oplus}_{\lambda\in\triangle_{+}}\mathfrak{p}_{\lambda}^{\mathbb{C}}\right).$

Note that this root space decomposition is the simultaneously eigenspace
decomposition of the commutative family

$\\{{\rm
ad}({\bm{v}})^{2}:\mathfrak{p}^{\mathbb{C}}\to\mathfrak{p}^{\mathbb{C}}\,|\,{\bm{v}}\in\mathfrak{a}^{d}\\}$

of symmetric transformations of $\mathfrak{p}^{\mathbb{C}}$, where ${\rm ad}$
is the adjoint representation of $\mathfrak{g}^{\mathbb{C}}$. Also, the
simultaneously eigenspace decomposition of the extended commutative family

$\\{{\rm
ad}({\bm{v}})^{2}:\mathfrak{p}^{\mathbb{C}}\to\mathfrak{p}^{\mathbb{C}}\,|\,{\bm{v}}\in\mathfrak{a}^{d}\\}\cup\\{\theta_{G}|_{\mathfrak{p}^{\mathbb{C}}}\\}$

of symmetric transformations of $\mathfrak{p}^{\mathbb{C}}$ is given by

$\mathfrak{p}^{\mathbb{C}}=\mathfrak{a}\oplus\mathfrak{a}^{d}\oplus\left(\mathop{\oplus}_{\lambda\in\triangle_{+}}\mathfrak{p}_{\lambda}\right)\oplus\left(\mathop{\oplus}_{\lambda\in\triangle_{+}}\mathfrak{p}^{d}_{\lambda}\right).$
$None$

Fix a regular point $p:={\rm Exp}_{o}(\bm{Z})$ ($\bm{Z}\in W\cdot\mathcal{C}$)
of the action $G\curvearrowright G^{\mathbb{C}}/K^{\mathbb{C}}$. Then we have

$(\exp\,\bm{Z})_{\ast}^{-1}(T_{p}(G\cdot
p))=\mathfrak{a}\oplus\left(\mathop{\oplus}_{\lambda\in\triangle_{+}}\mathfrak{p}_{\lambda}\right)\oplus\left(\mathop{\oplus}_{\lambda\in\triangle_{+}}\mathfrak{p}^{d}_{\lambda}\right)$
$None$

and

$(\exp\,\bm{Z})_{\ast}^{-1}(T_{p}^{\perp}(G\cdot p))=\mathfrak{a}^{d}.$

Denote by $\widehat{A}$ and $\widehat{h}$ the shape tensor and the second
fundamental form of the regular orbit $G\cdot p$ (which is a submanifold in
$G^{\mathbb{C}}/K^{\mathbb{C}}$), respectively. Then, according to $(6.1)$ and
$(6.2)$ in [16], we have

$(\widehat{A}_{p})_{(\exp\,\bm{Z})_{\ast}({\bm{v}})}|_{(\exp\,\bm{Z})_{\ast}(\mathfrak{p}^{d}_{\lambda})}=-\frac{\lambda({\bm{v}})}{\tanh\lambda(\bm{Z})}\,{\rm
id}\quad\,\,(\lambda\in\triangle_{+})$ $None$

and

$(\widehat{A}_{p})_{(\exp\,\bm{Z})_{\ast}({\bm{v}})}|_{(\exp\,\bm{Z})_{\ast}(\mathfrak{p}_{\lambda})}=-\lambda({\bm{v}})\tanh\lambda(\bm{Z})\,{\rm
id}\quad\,\,(\lambda\in\triangle_{+})$ $None$

for ${\bm{v}}\in(\exp\,\bm{Z})_{\ast}^{-1}(T_{p}^{\perp}(G\cdot o))$.

Take an orthonormal base $(\bm{e}^{0}_{i})_{i=1}^{r}$ of $\mathfrak{a}^{d}$
and an orthonormal base $(\bm{e}^{\lambda}_{i})_{i=1}^{m_{\lambda}}$ of
$\mathfrak{p}^{d}_{\lambda}$ with respect to $\beta_{A}$. Let
$\triangle_{+}=\\{\lambda_{1},\cdots,\lambda_{l}\\}$
($\lambda_{1}<\lambda_{2}<\cdots<\lambda_{l}$), where $<$ is a lexicographical
ordering of $\mathfrak{a}^{d}$ with respect to a basis of $\mathfrak{a}^{d}$.
Define $(\bm{e}_{1},\cdots,\bm{e}_{n})$ by

$\bm{e}_{i}:=\left\\{\begin{array}[]{ll}\bm{e}^{0}_{i}&(1\leq i\leq m_{0})\\\
\bm{e}^{\lambda_{1}}_{i}&(m_{0}+1\leq i\leq m_{0}+m_{\lambda_{1}})\\\
\bm{e}^{\lambda_{2}}_{i}&(m_{0}+m_{\lambda_{1}}+1\leq i\leq
m_{0}+m_{\lambda_{1}}+m_{\lambda_{2}})\\\
\vdots&\qquad\qquad\qquad\qquad\,\,\vdots\\\
\bm{e}^{\lambda_{l}}_{i}&(m_{0}+m_{\lambda_{1}}+\cdots m_{\lambda_{l-1}}+1\leq
i\leq m_{0}+m_{\lambda_{1}}+\cdots+m_{\lambda_{l}})\end{array}\right.$ $None$

and $(\bm{e}^{p}_{1},\cdots,\bm{e}^{p}_{n})$ by
$\bm{e}^{p}_{i}:=(\exp\,\bm{Z})_{\ast}(\bm{e}_{i})$ ($i=1,\cdots,n$). For the
convenience, set $m_{0}:=r$, $\mathfrak{p}^{d}_{0}:=\mathfrak{a}^{d}$ and
$\lambda_{0}:=0$.

Lemma 3.1. Assume that $p\in(G^{d}/K)_{\rm reg}$ (i.e.,
$\bm{Z}\in\mathcal{C}$). Let $(U,\varphi=(z_{1},\cdots,z_{n}))$ be a normal
complex coordinates about a point $p$ of the real form $G^{d}\cdot
o(=G^{d}/K)$ associated to $(\bm{e}^{p}_{1},\cdots,\bm{e}^{p}_{n})$. Then, for
any $\rho\in{\rm Conv}_{W}^{+}(D)$, we have

$\left.\frac{\partial^{2}\rho^{h}}{\partial
z_{i}\partial\overline{z}_{j}}\right|_{p}=\left\\{\begin{array}[]{ll}\displaystyle{\frac{1}{4}\,(\nabla^{d}d\rho^{d})_{p}((\bm{e}_{i}^{0})^{p},(\bm{e}_{i}^{0})^{p})}&(i=j\in\\{1,\cdots,m_{0}\\})\\\
\displaystyle{\frac{1}{4}\,(\,\,(\nabla^{d}d\rho^{d})_{p}((\bm{e}_{i-m_{0}}^{\lambda_{1}})^{p},(\bm{e}_{i-m_{0}}^{\lambda_{1}})^{p})}&\\\
\displaystyle{\qquad+\tanh\lambda_{1}(\bm{Z})\cdot\lambda_{1}(({\rm
grad}\,\rho)_{\bm{Z}})\,\,)}&(i=j\in\\{m_{0}+1,\cdots,m_{0}+m_{\lambda_{1}}\\})\\\
\qquad\qquad\qquad\qquad\,\,\vdots&\qquad\qquad\qquad\qquad\,\,\vdots\\\
\displaystyle{\frac{1}{4}\,(\,\,(\nabla^{d}d\rho^{d})_{p}((\bm{e}^{\lambda_{l}}_{i-\sum_{a=0}^{l-1}m_{\lambda_{a}}})^{p},(\bm{e}^{\lambda_{l}}_{i-\sum_{a=0}^{l-1}m_{\lambda_{a}}})^{p})}&(i=j\in\\{m_{0}+\cdots+m_{\lambda_{l-1}}+1,\\\
\displaystyle{\qquad+\tanh\lambda_{l}(\bm{Z})\cdot\lambda_{l}(({\rm
grad}\,\rho)_{\bm{Z}})\,\,)}&\qquad\qquad\,\,\,\cdots,m_{0}+\cdots+m_{\lambda_{l}}\\})\\\
\displaystyle{\frac{1}{4}\,(\nabla^{d}d\rho^{d})_{p}(\bm{e}_{i}^{p},\bm{e}_{j}^{p})}&(i\not=j).\end{array}\right.$

where $\lambda_{a}^{\sharp}$ denotes the element of
$(\exp\,\bm{Z})_{\ast}^{-1}(T_{p}^{\perp}(G\cdot p))$ such that
$\beta_{A}(\lambda_{a}^{\sharp},{\bm{v}})=\lambda_{a}({\bm{v}})$ holds for any
${\bm{v}}\in(\exp\,\bm{Z})_{\ast}^{-1}(T_{p}^{\perp}(G\cdot o))$.

Proof. First we note that $({\bf J}\bm{e}^{p}_{1},\cdots,{\bf
J}\bm{e}^{p}_{n})$ be an orthonormal basis of
$T_{p}^{\perp}(G^{d}/K)\,\,(\subset$
$T_{p}(G\cdot p))$. Let $\gamma_{i}$ ($i=1,\cdots,n$) be the geodesic in
$G^{d}/K$ with $\gamma_{i}^{\prime}(0)=e^{p}_{i}$ and $\widehat{\gamma}_{i}$
($i=1,\cdots,n$) be the geodesic in $G\cdot p$ with
$\widehat{\gamma}_{i}^{\prime}(0)={\bf J}e^{p}_{i}$. Then, from the
$G$-invariance of $\rho^{h}$, we can show that

$\displaystyle(\nabla d\rho^{h})_{p}({\bf J}\bm{e}^{p}_{i},{\bf
J}\bm{e}^{p}_{i})$
$\displaystyle=\left.\frac{d^{2}(\rho^{h}\circ\widehat{\gamma}_{i})}{ds^{2}}\right|_{s=0}-(d\rho^{h})_{p}(\nabla_{\widehat{\gamma}_{i}^{\prime}(0)}\widehat{\gamma}_{i}^{\prime})$
$\displaystyle=-(d\rho^{d})_{p}(\widehat{h}_{p}({\bf J}\bm{e}^{p}_{i},{\bf
J}\bm{e}^{p}_{i})).$

Similarly, we can show

$(\nabla d\rho^{h})_{p}({\bf J}e^{p}_{i}+{\bf J}e^{p}_{j},{\bf
J}e^{p}_{i}+{\bf J}e^{p}_{j})=-(d\rho^{d})_{p}(\widehat{h}_{p}({\bf
J}\bm{e}^{p}_{i}+{\bf J}\bm{e}^{p}_{j},{\bf J}\bm{e}^{p}_{i}+{\bf
J}\bm{e}^{p}_{j})).$

Hence, from the symmetricnesses of $(\nabla d\rho^{h})_{p}$ and
$\widehat{h}_{p}$, we have

$(\nabla d\rho^{h})_{p}({\bf J}e^{p}_{i},{\bf
J}e^{p}_{j})=-(d\rho^{d})_{p}(\widehat{h}_{p}({\bf J}\bm{e}^{p}_{i},{\bf
J}\bm{e}^{p}_{j}))\quad\,\,(1\leq i,j\leq n).$ $None$

Hence, from $(3.6)$, we obtain

$(\nabla d\rho^{h})_{p}({\bf J}(e^{\lambda_{a}}_{i})^{p},\,{\bf
J}(e^{\lambda_{b}}_{j})^{p})=\delta_{ab}\delta_{ij}\,\tanh\lambda(\bm{Z})\,(d\rho^{d})_{p}((\exp\,\bm{Z})_{\ast}(\lambda_{a}^{\sharp})).$
$None$

Let $\gamma_{j}$ be the geodesic in $G^{d}/K$ with
$\gamma_{j}^{\prime}(0)=\bm{e}^{p}_{j}$, $\widetilde{{\bf J}\bm{e}_{i}^{p}}$
be the parallel normal vector field of $G^{d}\cdot o$ along $\gamma_{j}$ with
$(\widetilde{{\bf J}\bm{e}^{p}_{i}})_{0}={\bf J}\bm{e}^{p}_{i}$. Also, let
$\widehat{\gamma}^{t}$ be the geodesic in $G\cdot\gamma_{j}(t)$ with
$(\widehat{\gamma}^{t})^{\prime}(0)=(\widetilde{{\bf J}\bm{e}^{p}_{i}})_{t}$.
Define a geodesic variation $\delta$ by $\delta(s,t):=\widehat{\gamma}^{t}(s)$
(see Figure 4). Then we have

$(\nabla d\rho^{h})_{p}({\bf
J}\bm{e}^{p}_{i},\,\bm{e}^{p}_{j})=\left.\frac{\partial^{2}(\rho^{h}\circ\delta)}{\partial
s\partial
t}\right|_{s=t=0}-(d\rho^{h})_{p}(\nabla_{\bm{e}^{p}_{j}}\widetilde{{\bf
J}\bm{e}^{p}_{i}})={\bf 0}$ $None$

$G^{d}\cdot o$${\rm Exp}_{o}(\mathfrak{a}^{d})$$G\cdot
o$$G^{\mathbb{C}}/K^{\mathbb{C}}$This inside tube is $G\cdot
p$.$o$$\gamma_{j}$$\delta$$K\cdot p$$p$’s $:\,\widetilde{{\bf
J}\bm{e}_{i}^{p}}$$\bm{e}^{p}_{j}$This outside tube is
$G\cdot\gamma_{j}(3/4)$.

Figure 4 $\,:\,$ Geodesic variation $\delta$

because $\rho^{h}$ is $G$-invariant and $G^{d}\cdot o$ is totally geodesic in
$G^{\mathbb{C}}/K^{\mathbb{C}}$. From $(3.8)$ and $(3.10)$, we obtain

$\displaystyle\left.\frac{\partial^{2}\rho^{h}}{\partial
z_{i}\partial\overline{z}_{j}}\right|_{p}=$ $\displaystyle(\nabla
d\rho^{h})_{p}^{\mathbb{C}}\left(\frac{1}{2}(\bm{e}_{i}^{p}-\sqrt{-1}{\bf
J}\bm{e}_{i}^{p}),\,\frac{1}{2}(\bm{e}_{j}^{p}+\sqrt{-1}{\bf
J}\bm{e}_{j}^{p})\right)$ $\displaystyle=$
$\displaystyle\frac{1}{4}\\{\,(\nabla
d\rho^{h})_{p}(\bm{e}_{i}^{p},\bm{e}_{j}^{p})+(\nabla d\rho^{h})_{p}({\bf
J}\bm{e}_{i}^{p},{\bf J}\bm{e}_{j}^{p})$ $\displaystyle-\sqrt{-1}(\nabla
d\rho^{h})_{p}({\bf J}\bm{e}_{i}^{p},\bm{e}_{j}^{p})+\sqrt{-1}(\nabla
d\rho^{h})_{p}(\bm{e}_{i}^{p},{\bf J}\bm{e}_{j}^{p})\,\\}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\,\left((\nabla^{d}d\rho^{d})_{p}(\bm{e}_{i}^{p},\bm{e}_{j}^{p})-(d\rho^{d})_{p}(\widehat{h}_{p}({\bf
J}\bm{e}^{p}_{i},{\bf J}\bm{e}^{p}_{j})\,\right)$ $\displaystyle=$
$\displaystyle\frac{1}{4}\,\left((\nabla^{d}d\rho^{d})_{p}(\bm{e}_{i}^{p},\bm{e}_{j}^{p})-(d\rho)_{\bm{Z}}(\widehat{h}_{p}({\bf
J}\bm{e}^{p}_{i},{\bf J}\bm{e}^{p}_{j}))\,\right),$

where $\widehat{h}_{p}({\bf J}\bm{e}^{p}_{i},{\bf J}\bm{e}^{p}_{j})$ is
regarded as an element of $T_{\bm{Z}}\mathfrak{a}^{d}$ under the
identification of $T_{p}^{\perp}(G\cdot p)=T_{p}{\rm
Exp}_{o}(\mathfrak{a}^{d})$ and $T_{\bm{Z}}\mathfrak{a}^{d}$. From this
relation and $(3.6)$, we can derive the desired relation. ∎

Also, we can show the following fact.

Lemma 3.2. Assume that $p\in(G^{d}/K)_{\rm reg}$ (i.e.,
$\bm{Z}\in\mathcal{C}$). Then, for any $\rho\in{\rm Conv}_{W}^{+}(D)$, we have

$(\nabla^{d}d\rho^{d})_{p}((\bm{e}^{0}_{i})^{p},(\bm{e}^{0}_{j})^{p})=(\nabla^{0}d\rho)_{\bm{Z}}(\bm{e}^{0}_{i},\bm{e}^{0}_{j})={\rm
det}\left(\left.\frac{\partial^{2}\rho}{\partial x_{i}\partial
x_{j}}\right|_{Z}\right),$ $None$

where $(x_{1},\cdots,x_{r})$ is the Euclidean coordinate of $\mathfrak{a}^{d}$
and

$(\nabla^{d}d\rho^{d})_{p}((\bm{e}^{\lambda_{a}}_{i})^{p},(\bm{e}^{\lambda_{b}}_{j})^{p})=\frac{1}{\tanh\lambda_{a}(\bm{Z})}\cdot\lambda_{a}(({\rm
grad}\,\rho)_{\bm{Z}})\delta_{ab}\delta_{ij}.$ $None$

Also, we have

$(\nabla^{d}d\rho^{d})_{p}((\bm{e}^{\lambda_{a}}_{i})^{p},(\bm{e}^{0}_{j})^{p})=0.$
$None$

Proof. Since $p\in(G^{d}/K)_{\rm reg}$, the normal space $T_{p}^{\perp}\Sigma$
of the submanifold $\Sigma$ in $G^{d}/K$ at $p$ is equal to $T_{p}(K\cdot p)$.
Let $\gamma_{i}$ ($i=1,\cdots,r$) be the geodesic in $\Sigma$ with
$\gamma_{i}^{\prime}(0)=(\bm{e}_{i}^{0})^{p}$. Then, since ${\gamma}_{i}$
($i=1,\cdots,r$) is a geodesic in $G^{d}/K$, we have

$\displaystyle(\nabla^{d}d\rho^{d})_{p}((\bm{e}_{i}^{0})^{p},(\bm{e}_{i}^{0})^{p})$
$\displaystyle=\left.\frac{d^{2}(\rho^{d}\circ{\gamma}_{i})}{ds^{2}}\right|_{s=0}-d\rho^{d}_{p}(\nabla^{d}_{\gamma_{i}^{\prime}(0)}\gamma_{i}^{\prime})$
$\displaystyle=\left.\frac{d^{2}(\rho^{d}\circ{\gamma}_{i})}{ds^{2}}\right|_{s=0}-d\rho_{\bm{Z}}(\nabla^{0}_{\gamma_{i}^{\prime}(0)}\gamma_{i}^{\prime})$
$\displaystyle=(\nabla^{0}d\rho)_{\bm{Z}}(\bm{e}_{i}^{0},\bm{e}_{i}^{0}).$

Similarly, we can show that

$(\nabla^{d}d\rho^{d})_{p}((\bm{e}_{i}^{0})^{p}+(\bm{e}_{j}^{0})^{p},(\bm{e}_{i}^{0})^{p}+(\bm{e}_{j}^{0})^{p})=(\nabla^{0}d\rho)_{\bm{Z}}(\bm{e}_{i}^{0}+\bm{e}_{j}^{0},\bm{e}_{i}^{0}+\bm{e}_{j}^{0}).$

Hence, from the symmetricnesses of $(\nabla^{d}d\rho^{d})_{p}$ and
$(\nabla^{0}d\rho)_{\bm{Z}}$, we have

$(\nabla^{d}d\rho^{d})_{p}((\bm{e}_{i}^{0})^{p},(\bm{e}_{j}^{0})^{p})=(\nabla^{0}d\rho)_{Z}(\bm{e}_{i}^{0},\bm{e}_{j}^{0}).$

Thus we obtain the desired relation $(3.11)$. Take ${\bm{w}}\in T_{p}(K\cdot
p)$ and let $\widehat{\gamma}$ be the geodesic in $K\cdot p$ with
$\widehat{\gamma}^{\prime}(0)={\bm{w}}$. Then, since $\rho^{d}$ is
$K$-invariant, we have

$\displaystyle(\nabla^{d}d\rho^{d})_{p}({\bm{w}},{\bm{w}})=\left.\frac{d^{2}(\rho^{d}\circ\widehat{\gamma})}{ds^{2}}\right|_{s=0}-d\rho^{d}_{p}(h_{p}({\bm{w}},{\bm{w}}))=-d\rho_{\bm{Z}}(h_{p}({\bm{w}},{\bm{w}})).$

Hence, from the symmetricnesses of $(\nabla^{d}d\rho^{d})_{p}$ and
$d\rho_{\bm{Z}}(h_{p}(\cdot,\cdot))$, we have

$(\nabla^{d}d\rho^{d})_{p}({\bm{w}}_{1},{\bm{w}}_{2})=-d\rho_{\bm{Z}}(h_{p}({\bm{w}}_{1},{\bm{w}}_{2}))$

for any ${\bm{w}}_{1},{\bm{w}}_{2}\in T_{p}(K\cdot p)$. Furthermore, from
$(3.2)$, we obtain the desired relation $(3.12)$.

Let $\gamma_{i}$ be the geodesic in $K\cdot p$ with
$\gamma_{i}^{\prime}(0)=(\bm{e}^{\lambda_{a}}_{i})^{p}$. This geodesic
$\widehat{\gamma}_{i}$ is given as $\widehat{\gamma}_{i}(t):=\exp(tw)(p)$ for
some $w\in\mathfrak{k}$. Define a vector field $\widetilde{\bm{e}_{j}^{0}}$
along $\widehat{\gamma}_{i}$ by
$\widetilde{\bm{e}_{j}^{0}}(t):=\exp(tw)_{\ast}(\bm{e}_{j}^{0})$. It is shown
that $\widetilde{\bm{e}_{j}^{0}}$ is a $K$-equivariant
$\nabla^{\perp}$-parallel normal vector field of $K\cdot p$ along $\gamma_{i}$
because $K$-action on $G^{d}/K$ is hyperpolar, where $\nabla^{\perp}$ denotes
the normal connection of the submanifold $K\cdot p$ in $G^{d}/K$. Since
$\rho^{d}$ is $K$-invariant and $\widetilde{\bm{e}_{j}^{0}}$ is
$\nabla^{\perp}$-parallel, we have

$(\nabla^{d}d\rho^{d})_{p}((\bm{e}_{i}^{\lambda_{a}})^{p},(\bm{e}_{j}^{0})^{p})=\left.\frac{d\widetilde{\bm{e}_{j}^{0}}(t)(\rho^{d})}{dt}\right|_{t=0}-(\nabla^{d}_{\gamma^{\prime}_{i}(0)}\widetilde{\bm{e}_{j}^{0}})(\rho^{d})=((A_{p})_{(\bm{e}_{j}^{0})^{p}}((\bm{e}_{i}^{\lambda_{a}})^{p})(\rho^{d})=0.$

∎

Define a non-linear differential operator $\mathcal{D}:{\rm
Cong}_{W}^{+}(\mathfrak{a}^{d})\to C^{\infty}(W\cdot\mathcal{C})$ of order one
by

$\mathcal{D}(\rho):=\mathop{\Pi}_{\lambda\in\triangle_{+}}\left(\frac{2(\lambda\circ{\rm
grad}\,\rho)|_{W\cdot\mathcal{C}}}{\tanh(2\lambda|_{W\cdot\mathcal{C}})}\right)^{m_{\lambda}}\quad(\rho\in{\rm
Cong}_{W}^{+}(\mathfrak{a}^{d})),$ $None$

where ${\rm grad}\,\rho$ is the gradient vector field of $\rho$ with respect
to $\beta_{0}$. It is easy to show that $\mathcal{D}({\rm
Cong}_{W}^{+}(\mathfrak{a}^{d}))\subset C^{\infty}_{W}(W\cdot\mathcal{C})$
holds.

From Lemmas 3.1 and 3.2, we obtain the following fact.

Proposition 3.3. For $p:={\rm Exp}_{o}(\bm{Z})$ ($\bm{Z}\in\mathcal{C}$), the
following relations hold:

${\rm det}\left(\left.\frac{\partial^{2}\rho^{h}}{\partial
z^{p}_{i}\partial\overline{z}^{p}_{j}}\right|_{p}\right)=\frac{1}{4^{n}}\cdot{\rm
det}\left(\left.\frac{\partial^{2}\rho}{\partial x_{i}\partial
x_{j}}\right|_{Z}\right)\cdot\mathcal{D}(\rho)(\bm{Z}),$ $None$

where $(x_{1},\cdots,x_{r})$ is the Euclidean coordinate of
$\mathfrak{a}^{d}$.

Proof. The relations in Lemmas 3.1 and 3.2, we can derive

$\left.\frac{\partial^{2}\rho^{h}}{\partial
z_{i}\partial\overline{z}_{j}}\right|_{p}=\left\\{\begin{array}[]{ll}\displaystyle{\frac{1}{4}\cdot{\rm
det}\left(\left.\frac{\partial^{2}\rho}{\partial x_{i}\partial
x_{j}}\right|_{Z}\right)}&(i,j\in\\{1,\cdots,m_{0}\\})\\\
\displaystyle{\frac{1}{2}\cdot\frac{\lambda_{1}(({\rm
grad}\,\rho)_{\bm{Z}})}{\tanh\,2\lambda_{1}(\bm{Z})}\cdot\delta_{ij}}&(i,j\in\\{m_{0}+1,\cdots,m_{0}+m_{\lambda_{1}}\\})\\\
\qquad\qquad\,\,\vdots&\qquad\qquad\qquad\vdots\\\
\displaystyle{\frac{1}{2}\cdot\frac{\lambda_{l}(({\rm
grad}\,\rho)_{\bm{Z}})}{\tanh\,2\lambda_{l}(\bm{Z})}\cdot\delta_{ij}}&(i,j\in\\{m_{0}+\cdots+m_{\lambda_{l-1}}+1,\\\
&\qquad\qquad\,\cdots,m_{0}+\cdots+m_{\lambda_{l}}\\}).\end{array}\right.$
$None$

From this relation, we can derive the desired relation. ∎

Remark 3.1. The relation $(3.15)$ and $(3.16)$ corresponds to the relations in
Corollary 1.3 and Theorem 1.2 of [9] in the case where
$G^{\mathbb{C}}/K^{\mathbb{C}}$ is the complexification of a compact semi-
simle Lie group (i.e., $G^{\mathbb{C}}/K^{\mathbb{C}}=(G^{\mathbb{C}}\times
G^{\mathbb{C}})/\triangle G^{\mathbb{C}}$) but the relations $(3.15)$ and
$(3.16)$ in the case differ from the relations in [9] somewhat. Also, the
above calculation method of $(3.15)$ and $(3.16)$ differs from the calculation
method in [9] completely.

Let $(x_{1},\cdots,x_{r})$ be the Euclidean coordinate of $\mathfrak{a}^{d}$.
From $(2.1)$ and $(3.15)$, we see that $\beta_{\rho^{h}}$ is Ricci-flat if and
only if the following relation

$\mathop{\Pi}_{\lambda\in\triangle_{+}}2^{m_{\lambda}}(\lambda\circ{\rm
grad}\,\rho)^{m_{\lambda}}\cdot{\rm det}\left(\frac{\partial^{2}\rho}{\partial
x_{i}\partial
x_{j}}\right)=c\cdot\mathop{\Pi}_{\lambda\in\triangle_{+}}\tanh^{m_{\lambda}}(2\lambda)$
$None$

holds for some nonzero constant $c$.

In 1990, L. A. Caffarelli proved the following fact (see Theorem 1-(b) in
[7]).

Proposition 3.4([7]). Let $D$ be a convex bounded domain of $\mathbb{R}^{r}$
with $B^{r}(\varepsilon)\subset\partial D\subset B^{r}(r\varepsilon)$
($\varepsilon>0$) and $f$ be a strictly positive $C^{0,\alpha}$-continuous
function on $D$ ($\alpha>0$), where $B^{r}(\cdot)$ denotes the Euclidean ball
of radius $(\cdot)$ centered at $(0,\cdots,0)$. If $\rho:D\to\mathbb{R}$ be a
convex viscocity solution of

${\rm det}\left(\frac{\partial^{2}\rho}{\partial x_{i}\partial
x_{j}}\right)=f\quad\,\,\,\,({\rm on}\,\,D)$

($(x_{1},\cdots,x_{r})\,:\,$ the Euclidean coordinate of $\mathbb{R}^{r}$)
satisfying $\rho|_{\partial D}=0$, then $\rho$ is of class $C^{2,\alpha}$ on
$B^{r}(\frac{\varepsilon}{2})$.

In 2003, R. Bielawski proved the following fact (see Theorems A.1 and
Corollary A.3 in [2]).

Proposition 3.5([2]). Let $W\curvearrowright\mathbb{R}^{r}$ be a finite
irreducible reflection group and $F_{1},F_{2}$ be non-negative locally bounded
$W$-invariant measurable functions on $\mathbb{R}^{r}$. Assume that
$\displaystyle{\int_{\mathbb{R}^{r}}F_{1}(x_{1},\cdots,x_{r})\,dx_{1}\cdots
dx_{r}=\infty}$ holds. Then the following statements (i) and (ii) hold.

(i) There exists a $W$-invariant strictly convex global weak solution of the
following Monge-Amp$\acute{\rm e}$re equation:

$(F_{1}\circ{\rm grad}\,\rho)\cdot{\rm
det}\left(\frac{\partial^{2}\rho}{\partial x_{i}\partial x_{j}}\right)=F_{2},$
$None$

where the gradient vector field ${\rm grad}\,\rho$ is regarded as a (multi-
valued) map of $\mathbb{R}^{r}$ to oneself. This weak solution $\rho$ is
Lipschitz continuous.

(ii) If $F_{1},F_{2}$ are positive functions of class $C^{k,\alpha}$ ($k\geq
0,\,\,\alpha>0$), then there exists a $W$-invariant strictly convex global
$C^{k+2,\alpha}$-solution of $(3.18)$. According to the standard discussion,
the $W$-invariant strictly convex global $C^{k+2,\alpha}$-solution is of class
$C^{\infty}$.

Remark 3.2. (i) A $W$-invariant strictly convex global weak solution of
$(3.9)$ means a convex function $\rho$ on $\mathbb{R}^{r}$ such that

$\int_{B}F_{2}(x_{1},\cdots,x_{r})\,dx_{1}\cdots dx_{r}=\int_{({\rm
grad}\,\rho)(B)}F_{1}(x_{1},\cdots,x_{r})\,dx_{1}\cdots dx_{r}$ $None$

holds for any Borel set $B$ of $\mathbb{R}^{r}$.

(ii) The gradient vector field ${\rm grad}\,\rho$ is defined by

$\displaystyle({\rm grad}\,\rho)_{x}:=\\{$
$\displaystyle(d\nu_{P})_{x}((\frac{\partial}{\partial
x_{1}})_{x}),\cdots,(d\nu_{P})_{x}((\frac{\partial}{\partial x_{r}})_{x}))$
$\displaystyle\,|\,P\,\,{\rm support}\,\,{\rm hyperplane}\,\,{\rm of}\,\,{\rm
the}\,\,{\rm graph}\,\,{\rm of}\,\,\rho\,\,{\rm at}\,\,x\\},$

where $\nu_{P}$ is the affine function whose graph is equal to $P$.

Define functions $\widehat{F}_{1}$ and $\widehat{F}_{2}$ on $\mathfrak{a}^{d}$
by

$\widehat{F}_{1}(\bm{Z}):=\mathop{\Pi}_{\lambda\in\triangle_{+}}2^{m_{\lambda}}|\lambda(\bm{Z})|^{m_{\lambda}},\quad\widehat{F}_{2}(\bm{Z}):=c\cdot\mathop{\Pi}_{\lambda\in\triangle_{+}}|\tanh(2\lambda(\bm{Z}))|^{m_{\lambda}}\quad(\bm{Z}\in\mathfrak{a}^{d}).$

Then the relation $(3.17)$ is rewitten as

$(\widehat{F}_{1}\circ{\rm grad}\,\rho)\cdot{\rm
det}\left(\frac{\partial^{2}\rho}{\partial x_{i}\partial
x_{j}}\right)=\widehat{F}_{2}.$ $None$

It is clear that $\widehat{F}_{i}$ ($i=1,2$) are locally bounded and non-
negative $W$-invariant continuous (hence measurable) functions on
$\mathfrak{a}^{d}$ and that $\widehat{F}_{1}$ satisfies
$\displaystyle{\int_{\mathfrak{a}^{d}}\widehat{F}_{1}(x_{1},\cdots,x_{r})\,dx_{1}\cdots
dx_{r}=\infty}$. Hence we can apply the statement (i) of Proposition 3.5 to
$(3.20)$ and show the existence of a $W$-invariant strictly convex global weak
solution of $(3.20)$. Regrettably, $\widehat{F}_{i}$ ($i=1,2$) are not
positive. In fact, they are equal to zero along $\mathfrak{a}^{d}\setminus
W\cdot\mathcal{C}$. Also, they are not necessarily of class $C^{1}$ along
$\mathfrak{a}^{d}\setminus W\cdot\mathcal{C}$. Hence we cannot apply the
statement (ii) of Proposition 3.5 to $(3.20)$. However, we can derive the
statement of Theorem A (stated in Introduction) in the case where $G/K$ is of
rank two. We prove Theorem A by using Propositions 3.3, 3.4 and 3.5.

Proof of Theorem A. As above, since $\widehat{F}_{i}$ ($i=1,2$) are locally
bounded and non-negative $W$-invariant continuous (hence measurable) functions
on $\mathfrak{a}^{d}$ and that $\widehat{F}_{1}$ satisfies
$\displaystyle{\int_{\mathfrak{a}^{d}}\widehat{F}_{1}(x_{1},x_{2})}$
$\,dx_{1}dx_{2}=\infty$, it follows from (i) of Proposition 3.5 that there
exists a $W$-invariant strictly convex weak solution of $(3.20)$ defined on
the whole of $\mathfrak{a}^{d}$. Let $\rho$ be the weak solution of $(3.20)$.
Also, let $\Pi=\\{\lambda_{1},\lambda_{2}\\}\,(\subset\triangle_{+})$ be the
fundamental root system. Then the Weyl domain $\mathcal{C}$ is given by

$\mathcal{C}=\\{\bm{Z}\,|\,\lambda_{b}(\bm{Z})>0\,\,\,\,(b=1,2)\\}.$

Since $\rho$ satisfies $(3.20)$, we have

$\begin{array}[]{l}\displaystyle{{\rm
det}\left(\left.\frac{\partial^{2}\rho}{\partial x_{i}\partial
x_{j}}\right|_{\bm{Z}}\right)=\frac{\widehat{F}_{2}(\bm{Z})}{\widehat{F}_{1}(({\rm
grad}\,\rho)(\bm{Z}))}=c\cdot\mathop{\Pi}_{\lambda\in\triangle_{+}}\left(\frac{\tanh(2\lambda(\bm{Z}))}{2\lambda(({\rm
grad}\,\rho)(\bm{Z}))}\right)^{m_{\lambda}}}\\\ \hskip
256.0748pt\displaystyle{(\bm{Z}\in W\cdot\mathcal{C}),}\end{array}$ $None$

Here we note that, for any $\bm{Z}\in W\cdot\mathcal{C}$, $\lambda(({\rm
grad}\,\rho)(\bm{Z}))$’s ($\lambda\in\triangle_{+}$) are not equal to zero In
fact, $({\rm grad}\,\rho)(\bm{Z})$ belongs to $W\cdot\mathcal{C}$ because
$\rho$ is $W$-invariant and convex (see Figure 5), where we note that

$W\cdot\mathcal{C}=\mathfrak{a}^{d}\setminus\left(\mathop{\cup}_{\lambda\in\triangle_{+}}\lambda^{-1}(0)\right).$

On the other hand, if $\bm{Z}\in(\lambda^{o})^{-1}(0)\setminus\\{{\bf
0}\\}\,(\subset\mathfrak{a}^{d}\setminus W\cdot\mathcal{C})$ for some
$\lambda^{o}\in\triangle_{+}$, then $({\rm grad}\,\rho)(\bm{Z})$ is tangent to
$(\lambda^{o})^{-1}(0)$ and hence it is expressed as $({\rm
grad}\,\rho)(\bm{Z})=a_{\bm{Z}}\bm{Z}$ for some real constant $a_{\bm{Z}}$
(see Figure 5). Also, since $\rho$ is a strictly convex function of the
minimum point ${\bf 0}(\in\mathfrak{a}^{d})$, $a_{\bm{Z}}$ is positive for any
$\bm{Z}\in(\lambda^{o})^{-1}(0)\setminus\\{{\bf 0}\\}$. Define a function
$\widehat{F}$ on $W\cdot\mathcal{C}$ by

$\widehat{F}(\bm{Z}):=\frac{\widehat{F}_{2}(\bm{Z})}{\widehat{F}_{1}(({\rm
grad}\,\rho)(\bm{Z}))}\quad\,\,(\bm{Z}\in W\cdot\mathcal{C}).$

Since $\rho$ is a $W$-invariant strictly convex function on $\mathfrak{a}^{d}$
and
$\displaystyle{\int_{\mathfrak{a}^{d}}\widehat{F}_{2}(x_{1},x_{2})\,dx_{1}dx_{2}>0}$,
it is shown that $\rho$ is a proper function (see Proposition 1.2 of [3]).
Hence ${\rm grad}\,\rho:\mathfrak{a}^{d}\to\mathfrak{a}^{d}$ is H$\ddot{\rm
o}$lder continuous. Similarly, by using
$\displaystyle{\int_{\mathfrak{a}^{d}}\widehat{F}_{1}(x_{1},x_{2})\,dx_{1}dx_{2}>0}$
instead of
$\displaystyle{\int_{\mathfrak{a}^{d}}\widehat{F}_{2}(x_{1},x_{2})\,dx_{1}dx_{2}>0}$,
it is shown that $({\rm grad}\,\rho)^{-1}:\mathfrak{a}^{d}\to\mathfrak{a}^{d}$
is H$\ddot{\rm o}$lder continuous. Define a function $\eta$ on
$\mathfrak{a}^{d}\setminus\\{{\bf 0}\\}$ by

$\eta(\bm{Z}):=\frac{\|({\rm
grad}\,\rho)(\bm{Z})\|}{\|\bm{Z}\|}\quad\,\,(\bm{Z}\in\mathfrak{a}^{d}\setminus\\{{\bf
0}\\}).$

Note that $\eta(\bm{Z})$ is equal to the above $a_{\bm{Z}}$. Since ${\rm
grad}\,\rho$ and $({\rm grad}\,\rho)^{-1}$ are H$\ddot{\rm o}$lder continuous,
$\eta$ and $\displaystyle{\frac{1}{\eta}}$ are continuous and bounded on
$\mathfrak{a}^{d}\setminus\\{{\bf 0}\\}$, where we used also $({\rm
grad}\,\rho)({\bf 0})={\bf 0}$. Hence there exist positive constants $C_{i}$
($i=1,2$) satisfying

$C_{1}\leq\eta(\bm{Z})\leq
C_{2}\quad\,\,(\bm{Z}\in\mathfrak{a}^{d}\setminus\\{{\bf 0}\\}).$ $None$

$x_{2}$$x_{1}$$\lambda_{1}^{-1}(0)$$\lambda_{2}^{-1}(0)$’s are ${\rm
grad}\,\rho|_{\rho^{-1}(1)}$.$\mathcal{C}$$(\lambda_{1}+\lambda_{2})^{-1}(0)$$\rho^{-1}(1)$

Figure 5 $\,:\,$ $({\rm grad}\,\rho)|_{\rho^{-1}(1)}$ in the case where
$\triangle$ is of type $(\mathfrak{a}_{2})$

Take $\bm{Z}\in(\lambda^{o})^{-1}(0)\setminus\\{{\bf 0}\\}$. Let
$\\{\bm{Z}_{k}\\}_{k=1}^{\infty}$ be a sequence in $W\cdot\mathcal{C}$
converging to $\bm{Z}$. Then we have

$\lim_{k\to\infty}\frac{\tanh(2\lambda^{o}(\bm{Z}_{k}))}{2\lambda^{o}(({\rm
grad}\,\rho)(\bm{Z}_{k}))}=\frac{1}{\eta(\bm{Z})}.$ $None$

Hence we obtain

$\lim_{k\to\infty}\widehat{F}(\bm{Z}_{k})=c\cdot\left(\frac{1}{\eta(\bm{Z})}\right)^{m_{\lambda^{o}}}\cdot\mathop{\Pi}_{\lambda\in\triangle_{+}\setminus\\{\lambda^{o}\\}}\left(\frac{\tanh(2\lambda(\bm{Z}))}{2\lambda(({\rm
grad}\,\rho)(\bm{Z}))}\right)^{m_{\lambda}}.$ $None$

This fact together with the arbitrariness of $\lambda^{o}$ implies that
$\widehat{F}$ extends to a $C^{0,\alpha}$-function ($\alpha>0$) on
$\mathfrak{a}^{d}\setminus\\{{\bf 0}\\}$. Denote by $\widehat{F}^{e}$ this
extended function. As above, since

$({\rm
grad}\,\rho)(\bm{Z})\left\\{\begin{array}[]{ll}=\eta(\bm{Z})\bm{Z}&(\bm{Z}\in\mathfrak{a}^{d}\setminus(W\cdot\mathcal{C}\cup\\{{\bf
0}\\}))\\\ \in W\cdot\mathcal{C}&(\bm{Z}\in
W\cdot\mathcal{C})\end{array}\right.$

and $(3.22)$ hold, we can show

$\mathop{\inf}_{\bm{Z}\in W\cdot\mathcal{C}}\left|\frac{\lambda(({\rm
grad}\,\rho)(\bm{Z}))}{\lambda(\bm{Z})}\right|>0\quad(\lambda\in\triangle_{+})$
$None$

and

$\mathop{\sup}_{\bm{Z}\in W\cdot\mathcal{C}}\left|\frac{\lambda(({\rm
grad}\,\rho)(\bm{Z}))}{\lambda(\bm{Z})}\right|<\infty\quad(\lambda\in\triangle_{+}).$
$None$

From $(3.21),\,(3.24),\,(3.25)$ and $(3.26)$, we can show

$\sup_{\mathfrak{a}^{d}\setminus\\{{\bf
0}\\}}\,\widehat{F}^{e}<\infty\quad\,\,{\rm
and}\quad\,\,\inf_{\mathfrak{a}^{d}\setminus\\{{\bf
0}\\}}\,\widehat{F}^{e}>0,$ $None$

By using this fact, we shall show that $\widehat{F}^{e}$ extends a
$C^{0,\alpha}$-function ($\alpha>0$) on the whole of $\mathfrak{a}^{d}$.
Suppose that $\widehat{F}^{e}$ does not extend a $C^{0,\alpha}$-function on
the whole of $\mathfrak{a}^{d}$, that is, there does not exist
$\lim\limits_{\bm{Z}\to{\bf 0}}\widehat{F}^{e}(\bm{Z})$. Then, by using
$(3.27)$, we can show that

$\lim_{\varepsilon\to 0}\left(\mathop{\sup}_{\bm{Z}\in
S^{2}(\varepsilon)}\widehat{F}^{e}(\bm{Z})-\mathop{\inf}_{\bm{Z}\in
S^{2}(\varepsilon)}\widehat{F}^{e}(\bm{Z})\right)\not=0,$

where $S^{2}(\varepsilon)$ denotes the sphere of radius $\varepsilon$ centered
at ${\bf 0}$ in $\mathfrak{a}^{d}$. This contradicts the fact that
$\widehat{F}^{e}$ is of class $C^{0,\alpha}$. Hence
$\lim\limits_{\bm{Z}\to{\bf 0}}\widehat{F}^{e}(\bm{Z})$ exists. This fact
implies that $\widehat{F}^{e}$ extends $C^{0,\alpha}$-function the whole of
$\mathfrak{a}^{d}$. Denote by $\widehat{F}^{ee}$ this extended function. Take
any positive constant $b$. Set $D_{b}:=\rho^{-1}([0,b])$ and
$\rho_{b}:=\rho-b$. By operating a suitable affine transformation of
$\mathfrak{a}^{d}$, $B^{r}(\varepsilon_{b})\subset\partial D_{b}\subset
B^{r}(r\varepsilon_{b})$ holds for some $\varepsilon_{b}>0$. Note that
$\varepsilon_{b}\to\infty$ as $b\to\infty$. Since $\rho_{b}$ satisfies

${\rm det}\left(\frac{\partial^{2}\rho_{b}}{\partial x_{i}\partial
x_{j}}\right)=\frac{\widehat{F}_{2}}{\widehat{F}_{1}({\rm grad}\,\rho_{b})}$
$None$

on $D_{b}$ and $\rho_{b}|_{\partial D_{b}}=0$, it follows from $(3.22)$ and
Proposition 3.4 that $\rho_{b}$ (hence $\rho$) is of class $C^{2,\alpha}$
($\alpha>0$) on $B^{r}(\frac{\varepsilon_{b}}{2})$. Since
$\varepsilon_{b}\to\infty$ as $b\to\infty$, we see that $\rho$ is of class
$C^{2,\alpha}$ on the whole of $\mathfrak{a}^{d}$. Hence, by standard
discussion, the higher regularity ($C^{\infty}$-property) of $\rho$ is shown.
Then $\omega_{\rho^{h}}$ is of class $C^{\infty}$ on the whole of
$G^{\mathbb{C}}/K^{\mathbb{C}}$. According to Proposition 3.3, for the normal
complex coordinate $(z^{p}_{1},\cdots,z^{p}_{n})$ about $p={\rm
Exp}_{o}(\bm{Z})$ ($\bm{Z}\in W\cdot\mathcal{C}$), we have the following
expression of $(\omega_{\rho^{h}})_{p}^{n}$:

$\displaystyle(\omega_{\rho^{h}})_{p}^{n}=$
$\displaystyle(-1)^{\frac{n(n-1)}{2}}\sqrt{-1}^{n}\cdot n!\cdot{\rm
det}\left(\left.\frac{\partial^{2}\rho^{h}}{\partial
z^{p}_{i}\partial\bar{z}^{p}_{j}}\right|_{p}\right)$
$\displaystyle\times(dz^{p}_{1}\wedge\cdots\wedge dz^{p}_{n}\wedge
d\bar{z}^{p}_{1}\wedge\cdots\wedge d\bar{z}^{p}_{n})_{p}$ $\displaystyle=$
$\displaystyle(-1)^{\frac{n(n-1)}{2}}\frac{\sqrt{-1}^{n}n!c}{4^{n}}\cdot(dz^{p}_{1}\wedge\cdots\wedge
dz^{p}_{n}\wedge d\bar{z}^{p}_{1}\wedge\cdots\wedge d\bar{z}^{p}_{n})_{p}.$

Also, $\Omega_{p}$ is expressed as

$\Omega_{p}=(dz^{p}_{1}\wedge\cdots\wedge dz^{p}_{n})_{p},$

Hence we obtain

$(\omega_{\rho^{h}})_{p}^{n}=(-1)^{\frac{n(n-1)}{2}}\frac{\sqrt{-1}^{n}n!c}{4^{n}}\cdot\Omega_{p}\wedge\overline{\Omega}_{p}.$
$None$

Since this relation $(3.28)$ holds at any point $p$ of ${\rm
Exp}_{o}(W\cdot\mathcal{C})$, it follows from the $G$-invariance of
$\omega_{\rho^{h}}$ and $\Omega$ that $(3.28)$ holds for any point $p$ of
$G\cdot{\rm Exp}_{o}(W\cdot\mathcal{C})(=(G^{\mathbb{C}}/K^{\mathbb{C}})_{\rm
reg})$. Furthermore, since $\omega_{\rho^{h}}^{n}$ and
$\Omega\wedge\overline{\Omega}$ are continuous on
$G^{\mathbb{C}}/K^{\mathbb{C}}$, $(3.28)$ holds on the whole of
$G^{\mathbb{C}}/K^{\mathbb{C}}$. If $c=2^{n}$, then we have

$(\omega_{\rho^{h}})_{p}^{n}=(-1)^{\frac{n(n-1)}{2}}\left(\frac{\sqrt{-1}}{2}\right)^{n}\cdot
n!\cdot\Omega_{p}\wedge\overline{\Omega}_{p}\quad\,\,(p\in
G^{\mathbb{C}}/K^{\mathbb{C}}).$ $None$

Thus the quadruple $({\bf J},\beta_{\rho^{h}},\omega_{\rho^{h}},\Omega)$ gives
an a.e. $C^{\infty}$-Calabi-Yau structure on $G^{\mathbb{C}}/K^{\mathbb{C}}$.
∎

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Department of Mathematics, Faculty of Science

Tokyo University of Science, 1-3 Kagurazaka

Shinjuku-ku, Tokyo 162-8601 Japan

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