# Two variational problems in Kähler geometry

László Lempert  Department of Mathematics, Purdue University, 150N University
Street, West Lafayette, IN 47907-2067, USA

###### Abstract.

On a Kähler manifold $(M,\omega)$ we consider the problems of
maximizing/minimizing Monge–Ampère energy over certain subsets of the space of
Kähler potentials. Under suitable assumptions we prove that solutions to these
variational problems exist, are unique, and have a simple characterization. We
then use the extremals to construct hermitian metrics on holomorphic vector
bundles, and investigate their curvature.

###### 2020 Mathematics Subject Classification:

32Q15, 32U05, 32W20, 52A40, 58E40

##  1\. Introduction

The pair of variational problems studied in this paper goes back to two
sources. One has to do with constructing hermitian metrics on holomorphic
vector bundles; we will discuss this aspect in section 8. The other source is
certain problems in convex geometry: For example, given a bounded set
$S\subset\mathbb{R}^{m}$, among ellipsoids centered at $0$ and containing $S$
characterize the one of least volume. Alternatively, volume minimization can
be considered without prescribing the center. This latter problem was posed
and solved by John in the influential paper [J48], see also [B97]. The result
has also been associated with Löwner’s name. There is a dual problem as well,
maximizing the volume of ellipsoids inscribed in a convex set; either problem
can be easily reduced to the other.

The problems have natural complex variants. For example, consider a convex
$S\subset\mathbb{C}^{m}$, and among all hermitian ellipsoids—images of the
unit ball under complex linear transformations—that are contained in $S$
search for the one with maximal volume. This is still a convex programing
problem, and Arocha, Bracho, and Montejano in [A+22] find that John’s method
will provide a solution. Related earlier work includes Gromov’s, who in [G67]
already dealt with the problem when $S$ is symmetric about the origin, and
[R86, M97] by Rabotin and Ma. It is noteworthy, though, that a variant of the
approach of John and Arocha et al. gives that the complex analog of John’s
result holds under a suitable pseudoconvexity assumption on $S$, more natural
in complex analysis than geometric convexity.

A new perspective is gained if we blow up $\mathbb{C}^{m}$ at the origin. The
blow up is the total space of the tautological line bundle $L$ over complex
projective space $\mathbb{P}_{m-1}$. If $S$ was a balanced111This means
$\lambda S\subset S$ for $\lambda\in\mathbb{C}$, $|\lambda|\leq 1$.
neighborhood of $0$, its blow up will be the unit disc bundle of a hermitian
metric $h$ on $L$. There are several ways to characterize $h$ that arise from
hermitian ellipsoids $S$. These are metrics that can be obtained as the dual
of the Fubini–Study metric on $L^{*}$, composed with arbitrary
$g\in\text{GL}(m,\mathbb{C})$ acting on $L$. Another characterization takes
$-i$ times the curvature (the negative of the Chern form) of $h$, a Kähler
form $\omega$ on $\mathbb{P}_{m-1}$; the requirement is that the scalar
curvature of the metric determined by $\omega$ should be constant
(equivalently, $\omega$ should be Kähler–Einstein).

This suggests a first generalization, a variational problem involving
hermitian metrics on a holomorphic line bundle $L\to M$ over a projective
manifold. We will not spell out the problem at this point, because a further
generalization simplifies matters by abandoning line bundles and hermitian
metrics altogether. Instead, consider a connected compact Kähler manifold
$(M,\omega)$; the role of hermitian metrics will be played by (relative)
Kähler potentials, that form the space

(1.1) ${\mathcal{H}}={\mathcal{H}}(\omega)=\\{u\in
C^{\infty}(M):\omega_{u}=\omega+i\partial\bar{\partial}u>0\\}.$

Here $C^{\infty}(M)$ stands for the Fréchet space of smooth functions
$M\to\mathbb{R}$.

Suppose a connected compact Lie group $G$ acts on the left on $(M,\omega)$ by
holomorphic isometries. The action extends to a holomorphic action of the
complexified group $G^{\mathbb{C}}$

$G^{\mathbb{C}}\times M\ni(g,x)\mapsto gx\in M.$

The induced action of $g$ on differential forms will be denoted $g^{*}$.

######  Definition 1.1.

A Kähler potential $u\in{\mathcal{H}}$ is admissible if there is a $g\in
G^{\mathbb{C}}$ such that $\omega_{u}=g^{*}\omega$.

Since $\omega$ and $g^{*}\omega$ are in the same de Rham cohomology class,
Hodge theory gives that for every $g\in G^{\mathbb{C}}$ there is a $u$ as in
the definition; moreover, $g$ determines $u$ up to an additive constant. If
$g\in G$, $u$ is constant.

Given a bounded $v_{0}:M\to\mathbb{R}$, inscribed/circumscribed ellipsoids of
the convex world correspond in our setting to admissible $v\leq v_{0}$,
respectively $v\geq v_{0}$. It is not hard to check that the logarithm of the
volume of an ellipsoid corresponds to Monge–Ampère energy222The notion, along
with other background material, will be reviewed in section 2. $E(v)$. (A
related result is Lemma 8.2.) Accordingly, the variational problems to be
studied are

(1.2) $\displaystyle\max\\{E(v):v\in{\mathcal{H}}\text{ is admissible, }v\leq
v_{0}\\},$ (1.3) $\displaystyle\min\\{E(v):v\in{\mathcal{H}}\text{ is
admissible, }v\geq v_{0}\\}.$

Under suitable assumptions we will show that solutions exist, are unique, and
can be characterized by a condition involving the moment map of the
$G$-action, analogous to John’s condition. Let $\mathfrak{g}$ denote the Lie
algebra of $G$, $\mathfrak{g}^{*}$ its dual, and $\langle\,,\rangle$ the
pairing between $\mathfrak{g}^{*}$ and $\mathfrak{g}$. The (equivariant)
moment map is a smooth map $\mu:M\to\mathfrak{g}^{*}$ such that for every
$X\in\mathfrak{g}$ the function

$\langle\mu,X\rangle\in C^{\infty}(M)$

is a Hamiltonian of the vector field $X_{M}$ on $M$ that corresponds to $X$
under the action of $G$. This latter condition determines
$\langle\mu,X\rangle$ up to an additive constant, that we choose so that
$\int_{M}\langle\mu,X\rangle\omega^{n}=0$. Not every action has a moment map;
we assume ours has.

The following property of $\mu$ will be relevant for uniqueness. Let
$N_{X}\subset M$ consist of points where $\langle\mu,X\rangle$ vanishes to
second order (at least), and consider the condition:

(1.4) If $X\in\mathfrak{g}$ is nonzero, the convex hull of $\\{\mu(x):x\in
N_{X}\\}\subset\mathfrak{g}^{*}$ avoids $0\in\mathfrak{g}^{*}$.

Again, this property is not guaranteed. In section 5 we will see examples
where it holds and where it does not hold.

To formulate our theorems, for $u\in{\mathcal{H}}$ define the contact set
$C_{u}\subset M$ by

(1.5) $C_{u}=\\{x\in M:u(x)=v_{0}(x)\\}.$

######  Theorem 1.2.

Given a continuous $v_{0}:M\to\mathbb{R}$, the maximum in (1.2) is attained.
An admissible $u\leq v_{0}$, with $\omega_{u}=g^{*}\omega$, $g\in
G^{\mathbb{C}}$, maximizes $E(v)$ in (1.2) if and only if

(1.6) the convex hull of $\\{\mu(gx):x\in C_{u}\\}\subset\mathfrak{g}^{*}$
contains 0.

Furthermore, if $\mu$ satisfies (1.4), the maximizer in (1.2) is unique.

Theorem 1.2 puts no restriction, reminiscent of convexity, on $v_{0}$. This is
natural, since problem (1.2) does not change if $v_{0}$ is replaced by its
“$\omega$–plurisubharmonic envelope”. By contrast, in problem (1.3)
$\omega$–plurisubharmonicity will be important.

######  Theorem 1.3.

Given a continuous $v_{0}:M\to\mathbb{R}$, the minimum in (1.3) is attained.
Assuming $\omega+i\partial\bar{\partial}v_{0}\geq 0$ as a current, an
admissible $u\geq v_{0}$, with $\omega_{u}=g^{*}\omega$, $g\in
G^{\mathbb{C}}$, minimizes $E(v)$ in (1.3) if and only if (1.6) holds.
Finally, if $\mu$ satisfies (1.4), $v_{0}\in C^{2}(M)$, and
$\omega+i\partial\bar{\partial}v_{0}>0$, then the minimizer in (1.3) is
unique.

In section 5 we will see that without assuming (1.4), in both problems
uniqueness may fail. Also, it is not hard to construct examples where (1.4)
holds but $\omega+i\partial\bar{\partial}v_{0}$ fails to be positive, and the
solution of problem (1.3) is not unique.

Above we mentioned in passing that hermitian ellipsoids in $\mathbb{C}^{m}$
correspond to Kähler metrics on $\mathbb{P}_{m-1}$ of constant scalar
curvature. This suggests different variational problems on general Kähler
manifolds $(M,\omega)$: Given $v_{0}:M\to\mathbb{R}$, maximize/minimize $E(v)$
among constant scalar curvature potentials $v\leq v_{0}$, respectively $v\geq
v_{0}$ (i.e., the metric of $\omega_{v}$ should have constant scalar
curvature). Of course, we have to assume ${\mathcal{H}}(\omega)$ contains
constant scalar curvature potentials. If so, no generality is lost by assuming
that the metric of $\omega$ itself has constant scalar curvature. This pair of
variational problems fits with (1.2), (1.3). Indeed, if $v$ is a potential of
constant scalar curvature, then by a result of Berman and Berndtsson it is
admissible for the action of the “reduced” isometry group $G$ of $(M,\omega)$,
see Theorem 4.8 and the comment following it in [BB17]. The converse is also
true, rather obviously. (The reduced isometry group is the connected subgroup
of the group of holomorphic isometries with Lie algebra consisting of Killing
vector fields that vanish somewhere.) By results of Lichnerowicz and
Kobayashi, [K72, L67], to which we will come shortly, such actions always
admit a moment map. In conclusion, the variational problems for constant
scalar curvature potentials are special cases of (1.2), (1.3).

If ellipsoids and Kähler metrics of constant scalar curvature are analogous,
what corresponds in the convex world to the general variational problems
(1.2), (1.3), when $\omega$ is arbitrary? A first guess would be for two
convex bodies $S,T\subset\mathbb{C}^{m}$ that are, say, symmetric about the
origin, finding the linear image of $S$ that is contained in $T$ and has
maximal volume among such images. This problem was solved by V. Milman, see
[T89, Theorem 14.5] and the comments on p. 137 there; the more general problem
without symmetry was treated in [G+01]. The analogy, however, is imperfect.
While in [T89, Theorem 14.5] all linear images are allowed, the spirit of our
theorems would only allow images under $g$ in the complexification of a
subgroup $G\subset\text{GL}(\mathbb{C},m)$ that leaves $S$ invariant. This is
the reason why Theorems 1.2, 1.3 can claim uniqueness, whereas in the convex
geometry problems uniqueness fails sometimes. Nonetheless, the criteria for
the extremals in [T89, Theorem 14.5] and in [G+01] on the one hand, and in
Theorems 1.2, 1.3 above on the other, are quite similar.

Problems (1.2), (1.3) suggest certain ways to construct hermitian metrics on
holomorphic vector bundles. In later sections we investigate one of these
constructions, and curvature properties of the metrics obtained through it.

Contents. In section 2 we review matters concerning plurisubharmonicity in
Kähler manifolds, geodesics in Mabuchi’s space ${\mathcal{H}}$ and beyond,
Monge–Ampère energy, the moment map, and complexified group actions. Sections
3,4, and 5 prove various parts of Theorems 1.2, 1.3: section 3 the existence
of extremals, section 4 their characterization, and section 5 their
uniqueness, occasionally under less restrictive assumptions than what is
formulated in the theorems above. Section 6 proves stability of the extremals
and extrema under perturbation of the barrier $v_{0}$. In section 7 we do
spell out the variational problems for hermitian metrics on line bundles
corresponding to (1.2), (1.3)—something we abandoned earlier—and translate, in
part, Theorems 1.2, 1.3 to this setting. Section 8 then introduces a
construction of hermitian metrics on vector bundles, motivated by problem
(1.3), and sections 9, 10 investigate the curvature of the hermitian metrics
constructed.

During the preparation of this paper I have profited from discussions with
Kuang–Ru Wu, and from his comments on earlier versions.

##  2\. Background

This section reviews notions that will be needed further down.

Spaces of plurisubharmonic functions. Suppose $N$ is a complex manifold and
$\Omega$ a smooth, real $(1,1)$ form, $d\Omega=0$. A function
$f:N\to[-\infty,\infty)$ is called $\Omega$-plurisubharmonic if on every open
$U\subset N$ on which $\Omega$ can be written as $i\partial\bar{\partial}w$
with $w:U\to\mathbb{R}$ smooth, the function $w+f$ is plurisubharmonic. The
collection of such functions is denoted Psh$(\Omega)$. For continuous
functions $f$, $\Omega$-plurisubharmonicity is equivalent to
$\Omega+i\partial\bar{\partial}f\geq 0$ as a current.

Suppose $(M,\omega)$ is an $n$ dimensional, connected compact Kähler manifold.
The space $\text{Psh}(\omega)$ and its subspaces (such as ${\mathcal{H}}$ in
(1.1)) have a rich geometry, see [D15, D17, D19]. A central notion is
geodesics. After Berndtsson and Darvas, they are defined as follows [B18,
D19]. Let $a<b$ be real numbers,
$S_{ab}=\\{s\in\mathbb{C}:a<\text{Re}\,s<b\\}$, and $\pi:S_{ab}\times M\to M$
the projection.

######  Definition 2.1.

(a) A path (i.e., a continuous map) $\varphi:(a,b)\to\text{Psh}(\omega)$ is a
subgeodesic if $\Phi:S\times M\to[-\infty,\infty)$ defined by
$\Phi(s,x)=\varphi(\text{Re}\,s)(x)$ is $\pi^{*}\omega$-plurisubharmonic.

(b) The geodesic $\varphi:(a,b)\to\text{Psh}(\omega)$ connecting
$u,v\in\text{Psh}(\omega)$ is

(2.1) $\varphi=\sup\\{\psi\mid\psi:(a,b)\to\text{Psh}(\omega)\text{ is a
subgeodesic, }\lim_{a}\psi\leq u,\lim_{b}\psi\leq v\\}.$

In (2.1) the pointwise limits exist because for subgeodesics $\psi(\cdot)(x)$
are convex functions for all $x\in M$. Often one extends geodesics by setting
$\varphi(a)=u$, $\varphi(b)=v$, and one still calls $\varphi$ thus extended a
geodesic. A path $\varphi:\mathbb{R}\to\text{Psh}(\omega)$ is called geodesic
if its restriction to any $[a,b]\subset\mathbb{R}$ is.

The geodesics of relevance to this paper are all smooth maps into
${\mathcal{H}}\subset\text{Psh}(\omega)$. This class of geodesics had been
introduced by Mabuchi in [M87] rather differently: ${\mathcal{H}}\subset
C^{\infty}(M)$, as an open subset of a Fréchet space, inherits the structure
of a Fréchet manifold, whose tangent bundle has a canonical trivialization
$T{\mathcal{H}}\approx{\mathcal{H}}\times C^{\infty}(M)$. Mabuchi defines a
Riemannian metric $g_{M}$ on ${\mathcal{H}}$ by letting
$\text{Vol}=\int_{M}\omega^{n}$ $(=\int_{M}\omega_{u}^{n}$ for all
$u\in{\mathcal{H}}$) and

$g_{M}(\xi,\eta)=\dfrac{1}{\text{Vol}}\int_{M}\xi\eta\,\omega_{u}^{n},\qquad\xi,\eta\in
T_{u}{\mathcal{H}}\approx C^{\infty}(M).$

The metric determines a unique Levi–Civita connection $\nabla$ on
$T{\mathcal{H}}$. According to Mabuchi, a smooth
$\varphi:[a,b]\to{\mathcal{H}}$ is a geodesic if its velocity
$\dot{\varphi}:[a,b]\to T{\mathcal{H}}$ is parallel along $\varphi$,
$\nabla_{\dot{\varphi}(t)}\dot{\varphi}(t)=0$, $a\leq t\leq b$.

That Mabuchi’s geodesics are geodesics in the sense of Definition 2.1 follows
from Semmes’ work [S92]. Semmes rediscovered Mabuchi’s connection, and proved
that a smooth $\varphi:[a,b]\to{\mathcal{H}}$ is a geodesic for this
connection if and only if the function

(2.2) $\Phi:\bar{S}_{ab}\times
M\ni(s,x)\mapsto\varphi(\text{Re}\,s)(x)\in\mathbb{R}$

satisfies the Monge–Ampère equation

(2.3) $(\pi^{*}\omega+i\partial\bar{\partial}\Phi)^{n+1}=0,\qquad\text{or,
equivalently,}\qquad\text{rk}\,(\pi^{*}\omega+i\partial\bar{\partial}\Phi)\equiv
n.$

(2.3) implies that $\Phi\in\text{Psh}(\pi^{*}\omega)$, i.e., $\varphi$ is a
subgeodesic, and it follows from [BK77] by Bedford and Kalka that
$\bar{S}_{ab}\times M$ is foliated by graphs of smooth functions
$f:\bar{S}_{ab}\to M$, holomorphic on $S_{ab}$, along which
$\pi^{*}\omega+i\partial\bar{\partial}\Phi$ restricts to $0$. Setting
$u=\varphi(a)$, $v=\varphi(b)$, the maximum principle, applied on the leaves,
then yields that $\varphi$ dominates all subgeodesics $\psi$ in (2.1): it is
therefore the geodesic connecting $u,v$.

######  Proposition 2.2.

If $v_{0}:M\to\mathbb{R}$ is any function, the set
$A=\\{v\in\text{Psh}(\omega):v\leq v_{0}\\}$ is convex in the sense that if
the endpoints of a geodesic $\varphi:[a,b]\to\text{Psh}(\omega)$ are in $A$,
then $\varphi(t)\in A$ for all $t$. Similarly, if
$v_{0}\in\text{Psh}(\omega)$, then $B=\\{v\in\text{Psh}(\omega):v\geq
v_{0}\\}$ is convex.

###### Proof.

Suppose a geodesic, or even a subgeodesic $\varphi:[a,b]\to\text{Psh}(\omega)$
has endpoints in $A$. Then the associated function $\Phi:\bar{S}_{ab}\times
M\to[-\infty,\infty)$ as in (2.2) is $\pi^{*}\omega$-plurisubharmonic on
$S_{ab}\times M$. Since it is translation invariant along the imaginary
direction in $S_{ab}$, for each $x\in M$ the function $\varphi(\cdot)(x)$ is
convex, or identically $-\infty$ on $(a,b)$. Hence $\varphi(t)(x)\leq
v_{0}(x)$ for $t=a,b$ implies the same for all $t\in[a,b]$.

Next suppose our geodesic $\varphi$ has endpoints in $B$. Now $\psi(t)=v_{0}$
for $t\in[a,b]$ is a subgeodesic and $\psi(t)\leq\varphi(t)$ for $t=a,b$.
Hence Definition 2.1 gives $\psi(t)\leq\varphi(t)$ for all $t$, i.e.,
$\varphi(t)\in B$.

######  Proposition 2.3.

Riemannian distance between $u,v\in{\mathcal{H}}$ is $\leq\max_{M}|u-v|$.

###### Proof.

Indeed, the length of the path $\phi(t)=u+t(v-u)$, $0\leq t\leq 1$, connecting
$u,v$ is $\leq\max_{M}|u-v|$.

Mabuchi further introduced a closed smooth 1-form, say $\alpha$, on
${\mathcal{H}}$,

$\alpha(\xi)=\dfrac{1}{\text{Vol}}\int_{M}\xi\,\omega_{u}^{n},\qquad\xi\in
T_{u}{\mathcal{H}}\approx C^{\infty}(M).$

${\mathcal{H}}\subset C^{\infty}(M)$ being convex, $\alpha$ must be exact.
Monge–Ampère energy $E:{\mathcal{H}}\to\mathbb{R}$ is defined as a primitive
of $\alpha$,

(2.4) $dE=\alpha,\qquad E(0)=0.$

The zero level set ${\mathcal{K}}=E^{-1}(0)$ is totally geodesic, and
$u\mapsto\omega_{u}$ is a bijection between ${\mathcal{K}}$ and the set of
Kähler forms $d$–cohomologous to $\omega$.

######  Proposition 2.4.

The map

${\mathcal{H}}\ni
u\mapsto\big{(}u-E(u),E(u)\big{)}\in{\mathcal{K}}\times\mathbb{R}$

is a Riemannian isometry if $\mathbb{R}$ is endowed with the Euclidean metric
(see [M87, top of p. 233]). In particular, $E$ is linear along geodesics in
${\mathcal{H}}$.

The only geodesics needed in the paper are of the following type. Let
${\mathcal{L}}$ stand for Lie derivative, and let $J:TM\to TM$ be the almost
complex structure tensor of $M$.

######  Proposition 2.5.

Suppose a real vector field $Y$ on $M$ vanishes somewhere and generates a one
parameter group $g_{t}$ of biholomorphisms. If ${\mathcal{L}}_{JY}\omega=0$,
$h\in C^{\infty}(M)$ satisfies
$i\partial\bar{\partial}h={\mathcal{L}}_{Y}\omega$ and
$\int_{M}h\,\omega^{n}=0$, then with any $c\in\mathbb{R}$

(2.5)
$\varphi(t)=\int_{0}^{t}g_{\tau}^{*}h\,d\tau\qquad\text{and}\qquad\psi(t)=\varphi(t)+ct$

define geodesics $\varphi:\mathbb{R}\to{\mathcal{K}}$,
$\psi:\mathbb{R}\to{\mathcal{H}}$.

###### Proof.

First observe that

(2.6)
$i\partial\bar{\partial}\psi(t)=i\partial\bar{\partial}\varphi(t)=\int_{0}^{t}g_{\tau}^{*}i\partial\bar{\partial}h\,d\tau=\int_{0}^{t}g_{\tau}^{*}{\mathcal{L}}_{Y}\omega\,d\tau=g_{t}^{*}\omega-\omega.$

The differential of Monge–Ampère energy evaluated on the velocity vector
$\dot{\varphi}(t)$ is

$dE\big{(}\dot{\varphi}(t)\big{)}=\dfrac{1}{\text{Vol}}\int_{M}g_{t}^{*}h\,\omega_{\varphi(t)}^{n}=\dfrac{1}{\text{Vol}}\int_{M}g_{t}^{*}(h\omega^{n})=\dfrac{1}{\text{Vol}}\int_{M}h\,\omega^{n}=0.$

Hence $E\big{(}\varphi(t)\big{)}=E\big{(}\varphi(0)\big{)}=0$ for all $t$, and
$\varphi(t)\in{\mathcal{K}}$. Mabuchi and Donaldson already studied paths in
${\mathcal{K}}$ induced by vector fields on $M$ that generate one parameter
subgroups of biholomorphisms. [M87, Theorem 3.5] implies that $\varphi$ is a
geodesic (for a different perspective, see [D99, §6], and for a
generalization, Theorem 9.7 here). Therefore $\psi$ is also geodesic by
Proposition 2.4.

The moment map. Now let $(M,\omega)$ be a connected compact symplectic
manifold of dimension $2n$. Contraction with tangent vectors

(2.7) $TM\ni\theta\mapsto-\iota(\theta)\omega\in
T^{*}M,\quad\text{resp.}\quad\mathbb{C}TM\ni\theta\mapsto-\iota(\theta)\omega\in\mathbb{C}T^{*}M,$

induces isomorphisms between the tangent and cotangent bundles of $M$ on the
one hand, and their complexifications on the other. The symplectic gradient of
a smooth $h:M\to\mathbb{C}$ is the complex vector field
$\text{sgrad}\,h=\text{sgrad}_{\omega}\,h$ that corresponds to $dh$ under
(2.7). Vector fields that arise in this way are called Hamiltonian. If $h$ is
real valued, $\text{sgrad}\,h$ is a real vector field whose flow preserves
$\omega$.

Consider a smooth left action of a Lie group $G$ on $M$

(2.8) $G\times M\ni(g,x)\mapsto gx\in M.$

The differential of the above map at $\\{e\\}\times M$ induces a homomorphism
from $\mathfrak{g}$, the Lie algebra of $G$, to Vect$(M)$, the Lie algebra of
smooth real vector fields on $M$. Denote the vector field that corresponds to
$X\in\mathfrak{g}$ by $X_{M}$. If $G$ acts by symplectomorphisms, the flow of
these vector fields preserves $\omega$,
$0={\mathcal{L}}_{X_{M}}\omega=d\iota(X_{M})\omega$. Assuming that
$\iota(X_{M})\omega$ is not only closed but exact, we can write
$-\iota(X_{M})\omega=dh_{X}$, or $X_{M}=\text{sgrad}\,h_{X}$ with some
$h_{X}\in C^{\infty}(M)$, that is unique under the condition
$\int_{M}h_{X}\,\omega^{n}=0$. If such $h_{X}$ exists for all
$X\in\mathfrak{g}$, the moment map $\mu:M\to\mathfrak{g}^{*}$ is defined by

$\langle\mu(x),X\rangle=h_{X}(x),\qquad x\in M,\quad X\in\mathfrak{g};$

it is equivariant in the sense that $\mu\circ g=\text{Ad}^{*}_{g^{-1}}\,\mu$,
see e.g. [GS84’, pp. 186–187].

Assume now that $(M,\omega)$ is even Kähler, and the action of $G$ preserves
the complex structure in addition to $\omega$. Then in fact each $g\in G$ acts
by a holomorphic isometry, and there is a good criterion, due to Lichnerowicz
and Kobayashi, for the existence of the moment map. Denote by $J$ the complex
structure tensor of $M$. Any complex vector field $Y$ on $M$ can be decomposed
$Y=Y^{\prime}+Y^{\prime\prime}$ into $(1,0)$ and $(0,1)$ vector fields,

(2.9) $Y^{\prime}=\frac{1}{2}(Y-iJY),\quad
Y^{\prime\prime}=\frac{1}{2}(Y+iJY);\quad\text{if }Y\text{ is real,
}Y=2\text{Re}\,Y^{\prime}=2\text{Re}\,Y^{\prime\prime}.$

That the vector fields $X_{M}$ generate one parameter groups of holomorphic
isometries means that $X_{M}$ is a Killing vector field, and $X_{M}^{\prime}$
is a holomorphic section of the $(1,0)$ tangent bundle $T^{10}M\to M$ [K72, p.
77].

######  Proposition 2.6.

A holomorphic $(1,0)$ vector field $Z$ on $M$ can be written
$Z=(\text{sgrad}\,h)^{\prime}$ with some smooth $h:M\to\mathbb{C}$ if and only
if $\sigma(Z)=0$ for all holomorphic $(1,0)$-forms $\sigma$ on $M$.
$\text{Re}\,Z$ is Killing if and only if $h$ can be chosen real; then
$\text{Re}\,Z=\text{sgrad}\,h/2$.

###### Proof.

See [L67], [K72, pp. 94–95]. In Kobayashi’s notation the $(0,1)$-form $\zeta$
corresponding to the vector field $Z$ is an imaginary constant times
$\iota(Z)\omega$.

Since $\sigma(Z)$ above is a holomorphic function on $M$, hence constant, it
follows that $\sigma(Z)=0$ whenever $Z$ (or $\text{Re}\,\,Z$) vanishes
somewhere. Conversely, if $\text{Re}\,Z=\text{sgrad}\,h$ with a real $h$, then
$Z=0$ where $h$ is maximal. (A fuller result of this sort is [LS94, Theorem 1]
by LeBrun and Simanca.) Hence:

######  Corollary 2.7.

Suppose a connected Lie group $G$ acts by holomorphic isometries on a
connected compact Kähler manifold $M$. The action admits a moment map if and
only if the vector fields $X_{M}$ vanish somewhere, for all $X\in\mathfrak{g}$
or, equivalently, for a family of $X\in\mathfrak{g}$ that spans
$\mathfrak{g}$.

For example, $\text{SU}(n+1)$ acts naturally by holomorphic isometries on
$M=\mathbb{P}_{n}$, endowed with the Fubini–Study metric. The vector fields
$X_{M}$ are real parts of holomorphic sections of $T^{10}\mathbb{P}_{n}$, and
have zeros. We conclude that the action has a moment map. By contrast, if $M$
is a torus with translation invariant Kähler form, the only holomorphic
section of $T^{10}M$ that has a zero is the zero section. Indeed, it is easily
seen directly that only trivial actions will have a moment map ($\equiv 0$).

Assume the $G$-action on the connected compact Kähler manifold $(M,\omega)$,
by holomorphic isometries, admits a moment map $\mu$.

######  Proposition 2.8.

If $X\in\mathfrak{g}$ and $h=-2\langle\mu,X\rangle$, then

(2.10) $i\partial\bar{\partial}h={\mathcal{L}}_{JX_{M}}\omega.$

###### Proof.

By definition, $X_{M}=-\text{sgrad}\,h/2$, or
$dh/2=\iota(X_{M})\omega=\iota(X^{\prime}_{M})\omega+\iota(X_{M}^{\prime\prime})\omega$.
Separating $(1,0)$ and $(0,1)$ parts,
$\bar{\partial}h=2\iota(X_{M}^{\prime})\omega$ follows. Hence

$\displaystyle 0=d\big{(}2\iota(X_{M}^{\prime})\omega-\bar{\partial}h\big{)}$
$\displaystyle=d\,\iota(X_{M})\omega-\sqrt{-1}d\,\iota(JX_{M})\omega-\partial\bar{\partial}h$
$\displaystyle={\mathcal{L}}_{X_{M}}\omega-\sqrt{-1}{\mathcal{L}}_{JX_{M}}\omega-\partial\bar{\partial}h.$

Since ${\mathcal{L}}_{X_{M}}\omega=0$, (2.10) follows.

Complexifications. Assume $G$ is compact and connected. Then the action (2.8)
extends to a holomorphic action

$G^{\mathbb{C}}\times M\ni(g,x)\mapsto gx\in M$

of the complexification of $G$, see e.g. [BD85, III.8] for the notion of
complexification and [K72, III., Theorem 1.1], that explains why
$G^{\mathbb{C}}$ acts. The infinitesimal generators of this action, real
vector fields, are still denoted $X_{M}$, but this time for
$X\in\mathfrak{g}^{\mathbb{C}}=\mathfrak{g}\oplus i\mathfrak{g}$, the Lie
algebra of $G^{\mathbb{C}}$. Thus $(iX)_{M}=JX_{M}$. We will need the
following structure theorem:

######  Theorem 2.9.

$G^{\mathbb{C}}$ is a Stein manifold and the map

$G\times\mathfrak{g}\ni(g,X)\mapsto g\exp iX\in G^{\mathbb{C}}$

is a diffeomorphism.

###### Proof.

We can assume that $G$ is a subgroup of a unitary group ${\rm U}(N)$. In the
construction of $G^{\mathbb{C}}$ in [BD85] we can take $r$ the restriction of
the standard representation ${\rm U}(N)\to\text{GL}(N,\mathbb{C})$, and the
discussion after (8.2) there then identifies $G^{\mathbb{C}}$ with an affine
variety, the zero set
$V(I)\subset\text{GL}(N,\mathbb{C})\subset\mathbb{C}^{N\times N}$ of a certain
ideal $I$ of holomorphic polynomials. In particular, $G^{\mathbb{C}}$ is
Stein. Since $G$ is maximally real in $G^{\mathbb{C}}$ and invariant under the
operation of passing to the adjoint of a matrix, $G^{\mathbb{C}}=V(I)$ is also
invariant. Hence the second statement of the theorem is a special case of
[K02, Proposition 7.14].

######  Lemma 2.10.

If the $G$-action on $(M,\omega)$ admits a moment map $\mu$, then $X_{M}$ has
a zero for all $X\in\mathfrak{g}^{\mathbb{C}}$.

###### Proof.

If $X=Y+iZ$ with $Y,Z\in\mathfrak{g}$, then
$X_{M}^{\prime}=\text{sgrad}\,^{\prime}\big{(}\langle\mu,Y\rangle+i\langle\mu,Z\rangle\big{)}$.
By [LS94, Theorem 1], this implies that $X^{\prime}_{M}$ vanishes somewhere,
and so does $X_{M}$.

######  Proposition 2.11.

Assume that the $\mathfrak{g}$-action on $M$ is effective: $X\in\mathfrak{g}$
and $X_{M}=0$ imply $X=0$. Assume also that the $G$-action on $(M,\omega)$ has
a moment map. If $\gamma\in G^{\mathbb{C}}$ satisfies
$\gamma^{*}\omega=\omega$, then $\gamma\in G$.

###### Proof.

Write $\gamma=g\exp iX$ with $g\in G$, $X\in\mathfrak{g}$ (Theorem 2.9). Thus
$(\exp iX)^{*}\omega=\omega$. By Proposition 2.10 $X$ vanishes at some $y\in
M$, hence $y$ is fixed by all $g_{s}=\exp sX$, $s\in\mathbb{C}$. Choose local
coordinates $x_{1},\dots,x_{n}$ centered at $y$ so that
$\omega=i\sum_{j}dx_{j}\wedge d\bar{x}_{j}$ at $y$. Passing to the Jacobian
matrix defines a homomorphism

$\mathbb{R}\ni t\mapsto\partial
g_{t}/\partial(x_{1},\dots,x_{n})|_{0}\in\text{U}(n).$

The induced homomorphism of Lie algebras associates with $X$ an element
$\tilde{X}\in\mathfrak{u}(n)$. We can arrange the coordinates so that
$\tilde{X}$ is diagonal, diag$(i\lambda_{1},\dots,i\lambda_{n})$ with
$\lambda_{j}\in\mathbb{R}$. This implies $\partial g_{s}/\partial
x|_{0}=\text{diag}(e^{i\lambda_{1}s},\dots,e^{i\lambda_{n}s})$, first for
$s\in\mathbb{R}$, and then for $s\in\mathbb{C}$ by analytic contiuation.
Therefore at $y$

$\omega=(\exp iX)^{*}\omega=i\sum e^{-2\lambda_{j}}dx_{j}\wedge d\bar{x}_{j},$

whence each $\lambda_{j}=0$. But then $\tilde{X},X_{M}$, and so $X$ vanish,
i.e., $\gamma=g\exp iX=g\in G$.

##  3\. Existence of extremals

From now on $(M,\omega)$ is an $n$ dimensional connected compact Kähler
manifold, on which a connected compact Lie group $G$ acts by holomorphic
isometries. The action admits a moment map $\mu:M\to\mathfrak{g}^{*}$, and
extends to a holomorphic action of the complexification $G^{\mathbb{C}}$ on
$M$. Without loss of generality we assume that the induced action of
$\mathfrak{g}$ is effective, if $X_{M}=0$ for some $X\in\mathfrak{g}$, then
$X=0$. The main result of this section is the following.

######  Proposition 3.1.

Given a bounded $v_{0}:M\to\mathbb{R}$, among admissible $v\in{\mathcal{H}}$,
$v\leq v_{0}$, there is at least one which maximizes Monge–Ampère energy
$E(v)$; and among admissible $v\in{\mathcal{H}}$, $v\geq v_{0}$, there is at
least one that minimizes $E(v)$.

We need two auxiliary results.

######  Proposition 3.2.

Suppose $v\in{\mathcal{H}}$, $X\in\mathfrak{g}$, and $\omega_{v}=(\exp
iX)^{*}\omega$. Let $g_{t}=\exp itX$ and $h=-2\langle\mu,X\rangle$. Then

$\varphi(t)=\int_{0}^{t}g_{\tau}^{*}h\,d\tau\qquad\text{and}\qquad\psi(t)=\varphi(t)+E(v)t$

define geodesics $\varphi:\mathbb{R}\to{\mathcal{K}}$,
$\psi:\mathbb{R}\to{\mathcal{H}}$. For each $t$, $\varphi(t),\psi(t)$ are
admissible and $\psi(1)=v$.

###### Proof.

All claims but the last follow by combining Propositions 2.5, 2.8, and (2.6).
As to the last, (2.6) gives $\omega_{\psi(1)}=(\exp iX)^{*}\omega=\omega_{v}$,
whence $\psi(1)=v+\text{const}$. The constant has to be $0$ because
$E\big{(}\psi(1)\big{)}=E(v)$.

######  Proposition 3.3.

If $a,b\in\mathbb{R}$,

(3.1) $\displaystyle K=$ $\displaystyle\\{g\in
G^{\mathbb{C}}:g^{*}\omega=\omega_{v}\text{ with }v\in{\mathcal{H}}\text{ such
that }a\leq E(v),\,\max_{M}v\leq b\\},$ (3.2) $\displaystyle L=$
$\displaystyle\\{g\in G^{\mathbb{C}}:g^{*}\omega=\omega_{v}\text{ with
}v\in{\mathcal{H}}\text{ such that }a\leq\min_{M}v,\,E(v)\leq b\\}$

are compact subsets of $G^{\mathbb{C}}$.

###### Proof.

We start by defining a norm on $\mathfrak{g}$. Let
$\Delta=2\,\text{tr}_{\omega}i\partial\bar{\partial}$ denote the Laplacian on
$C^{\infty}(M,\omega)$. If $X\in\mathfrak{g}$ and
$h_{X}=-2\langle\mu,X\rangle$, by Proposition 2.8
$i\partial\bar{\partial}h_{X}={\mathcal{L}}_{JX_{M}}\omega$. Taking trace with
respect to $\omega$, we obtain $\Delta
h_{X}=2\,\text{tr}_{\omega}{\mathcal{L}}_{JX_{M}}\omega$, hence with the Green
operator $\Gamma:C^{\infty}(M,\omega)\to C^{\infty}(M,\omega)$

(3.3) $h_{X}=\Gamma(2\,\text{tr}_{\omega}{\mathcal{L}}_{JX_{M}}\omega).$

Let

$||X||=\int_{M}|h_{X}|\omega^{n}.$

This is indeed a norm, for if $||X||=0$, then $h_{X}$ and
$X_{M}=-\text{sgrad}\,h_{X}/2$ vanish, whence $X=0$.

Now to the compactness of $K,L$. Write an arbitrary $g\in G^{\mathbb{C}}$ as
$g=\gamma\exp iX$, $\gamma\in G$, $X\in\mathfrak{g}$ (Theorem 2.9). Thus
$g^{*}\omega=(\exp iX)^{*}\omega$. If $\omega_{v}=g^{*}\omega$ then $X$, i.e.
$h_{X}$, can be estimated in terms of $\min_{M}v$, $\max_{M}v$, and $E(v)$ as
follows. Let $g_{t}=\exp itX$, and with $h_{X}$ defined in (3.3),
$\psi(t)=\int_{0}^{t}g_{\tau}^{*}h_{X}\,d\tau+E(v)t$. By Proposition 3.2 this
is a geodesic. In particular, $\psi(\cdot)(x)$ is convex for $x\in M$, whence

$h_{X}+E(v)=\dot{\psi}(0)\leq\psi(1)-\psi(0)\leq\dot{\psi}(1)=g_{1}^{*}h_{X}+E(v).$

Since $\psi(0)=0$ and $\psi(1)=v$ by Proposition 3.2, we obtain

(3.4) $h_{X}\leq v-E(v)\leq g_{1}^{*}h_{X}.$

Suppose $g\in K$. Then (3.4) implies $h_{X}\leq b-a$; with
$h^{+}=\max(h_{X},0)$

$||X||=\int_{M}|h_{X}|\,\omega^{n}=\int_{M}(2h^{+}-h_{X})\,\omega^{n}=\int_{M}2h^{+}\omega^{n}\leq
2(b-a)\int_{M}\omega^{n}.$

Similarly, if $g\in L$, and $h^{-}=\max(-h_{X},0)$, then (3.4) gives
$\min_{M}h_{X}=\min_{M}g_{1}^{*}h_{X}\geq a-b$, and

$||X||=\int_{M}|h_{X}|\,\omega^{n}=\int_{M}(2h^{-}+h_{X})\,\omega^{n}=\int_{M}2h^{-}\omega^{n}\leq
2(b-a)\int_{M}\omega^{n}.$

These estimates show that the closed sets $K,L\subset G^{\mathbb{C}}$ are
contained in the compactum

$\Big{\\{}\gamma\exp iX:\gamma\in G,\,X\in\mathfrak{g},\,||X||\leq
2(b-a)\int_{M}\omega^{n}\Big{\\}},$

therefore themselves are compact.

###### Proof of Proposition 3.1.

Since $G^{\mathbb{C}}$ leaves the de Rham class of $\omega$ invariant, for
each $g\in G^{\mathbb{C}}$ there is a unique $u=U(g)$ that solves

(3.5)
$g^{*}\omega-\omega=i\partial\bar{\partial}u,\qquad\int_{M}u\,\omega^{n}=0.$

As in the previous proof, the map $U:G^{\mathbb{C}}\to C^{\infty}(M)$ can be
expressed through the Green operator $\Gamma$ as

$U(g)=\Gamma\big{(}2\,\text{tr}_{\omega}(g^{*}\omega-\omega)\big{)},$

which shows that $U$ is continuous. Let
$m(g)=\min_{M}\big{(}v_{0}-U(g)\big{)}$. Suppose $g^{*}\omega=\omega_{v}$.
Then $v\leq v_{0}$ if and only if $v=U(g)+m(g)+c$ with a constant $c\leq 0$.
As, in general, $E(w+c)=E(w)+c$, maximizing $E(v)$ among admissible $v\leq
v_{0}$ reduces to maximizing

(3.6) $E\big{(}U(g)+m(g)\big{)},$

a continuous function, over $G^{\mathbb{C}}$. Let $a=\inf_{M}v_{0}$ and
$b=\sup_{M}v_{0}$. Since the constant function $a\leq v_{0}$ is admissible,
maximizing (3.6) over $G^{\mathbb{C}}$ amounts to the same as maximizing over
$K$, see (3.1); and of course, the maximum over this compact set is attained.
The existence of the minimizer in the second statement follows similarly.

##  4\. Characterization of the extremals

We continue with the setup of section 3. To investigate extremals of problems
(1.2), (1.3), it will be convenient to be able to change the Kähler form
$\omega$ to $\omega^{\prime}=\omega_{w}$ with $w\in{\mathcal{H}}$. We start
the section by discussing how notions involved in Theorems 1.2, 1.3 transform
as we pass from $\omega$ to $\omega^{\prime}$.

Let ${\mathcal{H}}^{\prime}={\mathcal{H}}(\omega^{\prime})=\\{u\in
C^{\infty}(M):\omega^{\prime}_{u}=\omega^{\prime}+i\partial\bar{\partial}u>0\\}$.
Consider the map

(4.1) ${\mathcal{H}}\ni v\mapsto v^{\prime}=v-w\in{\mathcal{H}}^{\prime}.$

One checks that $\omega^{\prime}_{v^{\prime}}=\omega_{v}$, and that (4.1) is
an isometry in Mabuchi’s metric. Denote Monge–Ampère energy on
${\mathcal{H}}^{\prime}$ by $E^{\prime}:{\mathcal{H}}^{\prime}\to\mathbb{R}$.
Given $v\in{\mathcal{H}}$, let $\psi:[0,1]\to{\mathcal{H}}$ be a smooth path
connecting $w$ with $v$, and let $\psi^{\prime}(t)=\psi(t)-w$. Continuing to
denote $t$-derivatives by a dot,

(4.2) $\displaystyle E^{\prime}(v^{\prime})$
$\displaystyle=\int_{0}^{1}dE^{\prime}\big{(}\dot{\psi}^{\prime}(t)\big{)}\,dt=\dfrac{1}{\text{Vol}}\int_{0}^{1}\int_{M}\dot{\psi}^{\prime}(t)\big{(}\omega^{\prime}_{\psi^{\prime}(t)}\big{)}^{n}\,dt$
$\displaystyle=\dfrac{1}{\text{Vol}}\int_{0}^{1}\int_{M}\dot{\psi}(t)\omega_{\psi(t)}^{n}\,dt=\int_{0}^{1}dE\big{(}\dot{\psi}(t)\big{)}\,dt=E(v)-E(w).$

Hence finding extremals of $E$ and $E^{\prime}$ is one and the same thing.

Next suppose $w\in{\mathcal{H}}$ is admissible,
$\omega^{\prime}=\omega_{w}=\gamma^{*}\omega$ with some $\gamma\in
G^{\mathbb{C}}$. The action

(4.3) $G^{\mathbb{C}}\times M\ni(g,x)\mapsto\gamma^{-1}g\gamma\,x\in M,$

and its restriction to $G$, will be referred to as the modified action. If
$v\in{\mathcal{H}}$ is admissible, i.e., $\omega_{v}=g^{*}\omega$ with $g\in
G^{\mathbb{C}}$, then
$\omega^{\prime}_{v^{\prime}}=\omega_{v}=(\gamma^{-1}g)^{*}\omega^{\prime}$
shows that $v^{\prime}\in{\mathcal{H}}^{\prime}$ is also admissible; and the
converse holds as well.

The map $\gamma:(M,\omega^{\prime})\to(M,\omega)$ is an isomorphism of
Kählerian $G^{\mathbb{C}}$-spaces, if $(M,\omega^{\prime})$ is endowed with
the modified action (4.3) (and $(M,\omega)$ with the original action). In
particular, the modified action of $G$ is by isometries of
$(M,\omega^{\prime})$. It follows that the moment map of the modified action
is $\mu^{\prime}=\mu\circ\gamma$. If $\gamma\in G$, then
$\mu^{\prime}=\text{Ad}_{g^{-1}}^{*}\,\mu$ by equivariance. To summarize:

######  Proposition 4.1.

Suppose $w\in{\mathcal{H}}$ is admissible, $\omega_{w}=\gamma^{*}\omega$,
$\gamma\in G^{\mathbb{C}}$. The map ${\mathcal{H}}\ni v\mapsto
v-w\in{\mathcal{H}}^{\prime}$ is an isometry that sends Monge–Ampère energy
$E$ on ${\mathcal{H}}$ to Monge–Ampère energy $E^{\prime}$ on
${\mathcal{H}}^{\prime}$, plus a constant. Let $v_{0}:M\to\mathbb{R}$ and
$v^{\prime}_{0}=v_{0}-w$. An admissible $u\in{\mathcal{H}}$ maximizes
Monge–Ampère energy $E(v)$ among admissible $v\in{\mathcal{H}}$, $v\leq
v_{0}$, if and only if $u^{\prime}=u-w$ maximizes $E^{\prime}(v^{\prime})$
among admissible $v^{\prime}\in{\mathcal{H}}^{\prime}$, $v^{\prime}\leq
v_{0}^{\prime}$. For minimizers of $E$ over $\\{v\in{\mathcal{H}}\text{
admissible }:v\geq v_{0}\\}$ a corresponding statement holds.

Furthermore, the moment maps $\mu:(M,\omega)\to\mathfrak{g}^{*}$,
$\mu^{\prime}:(M,\omega^{\prime})\to\mathfrak{g}^{*}$ for the original and
modified $G$-actions (see (4.3)) are related by $\mu^{\prime}=\mu\circ\gamma$.
If $\gamma^{*}\omega=\omega$, then $\gamma\in G$ (by Proposition 2.11), and
$\mu^{\prime}=\text{Ad}_{\gamma^{-1}}^{*}\,\mu$.

Now suppose $v_{0}\in C(M)$. If $u\in{\mathcal{H}}$, let $C_{u}=\\{x\in
M:u(x)=v_{0}(x)\\}$.

######  Proposition 4.2.

Suppose $u\in{\mathcal{H}}$ is admissible, $\omega_{u}=\gamma^{*}\omega$ with
$\gamma\in G^{\mathbb{C}}$. If $u=v$ maximizes $E(v)$ among admissible $v\leq
v_{0}$, or if $u=v$ minimizes $E(v)$ among admissible $v\geq v_{0}$, then the
convex hull of $\mu(\gamma C_{u})\subset\mathfrak{g}^{*}$ contains $0$.

###### Proof.

Consider first the case when $u=0$. When $u$ minimizes, $v_{0}\leq 0$, when
$u$ maximizes, $v_{0}\geq 0$. Since $\omega=\omega_{u}=\gamma^{*}\omega$, by
Proposition 4.1 $\mu\circ\gamma=\text{Ad}_{\gamma^{-1}}^{*}\,\mu$. Hence the
convex hull of $\mu(\gamma C_{u})\subset\mathfrak{g}^{*}$ contains $0$ if and
only if the convex hull of $\mu(C_{u})\subset\mathfrak{g}^{*}$ contains $0$.
Suppose this latter does not contain $0$. Then there is a linear form
$l:\mathfrak{g}^{*}\to\mathbb{R}$ such that $l\big{(}\mu(x)\big{)}<0$ for all
$x\in C_{0}$. Such a form $l$ is evaluation at some $X\in\mathfrak{g}$. Thus

(4.4) $h=-2\langle\mu,X\rangle>0\qquad\text{on }C_{0},$

and in fact $U=\\{x\in M:h(x)>\varepsilon\\}$ is a neighborhood of $C_{0}$ for
some $\varepsilon>0$. Let $g_{t}=\exp itX$ and
$\varphi(t)=\int_{0}^{t}g_{\tau}^{*}h\,d\tau$. By Proposition 3.2
$E\big{(}\varphi(t)\big{)}=0$ and $\varphi(t)$ is admissible for all $t$. Fix
a compact neighborhood $K\subset U$ of $C_{0}$. Since $h|U>\varepsilon$, for
small $t\in\mathbb{R}$ and $y\in K$

$\varphi(t)(y)\geq\varepsilon t\text{ if
}t>0,\qquad\varphi(t)(y)\leq\varepsilon t\text{ if }t<0.$

If $u$ minimizes (maximizes), then $\varphi(0)(x)=0>v_{0}(x)$, respectively
$\varphi(0)(x)=0<v_{0}(x)$, when $x\in M\setminus\text{int}\,K$. It follows
that for small $t$ and on all of $M$

$\displaystyle\varphi(t)>v_{0},\quad\text{ if }t>0\text{ and }u=0\text{
minimizes},$ $\displaystyle\varphi(t)<v_{0},\quad\text{ if }t<0\text{ and
}u=0\text{ maximizes.}$

The inequalities continue to hold if a small constant $c$ is added to
$\varphi(t)$. But this contradicts the extremality of $u=0$, since
$\varphi(t)+c$ is also admissible, and $E\big{(}\varphi(t)+c\big{)}=c=E(u)+c$.

Second, consider a general extremal $u$. Let $\omega^{\prime}=\omega_{u}$,
$v_{0}^{\prime}=v_{0}-u$, and $E^{\prime}:{\mathcal{H}}^{\prime}\to\mathbb{R}$
the Monge–Ampère energy as in Proposition 4.1 (but with $w=u$). Since the
contact sets $\\{u=v_{0}\\}$ and $\\{0=v_{0}^{\prime}\\}$ agree, Proposition
4.1 and the special case we just proved imply the necessity of the condition
in Proposition 4.2 in general.

Next we prove sufficiency.

######  Proposition 4.3.

Suppose $u\in{\mathcal{H}}$ is admissible, $\omega_{u}=\gamma^{*}\omega$ with
$\gamma\in G^{\mathbb{C}}$; and the convex hull of $\mu(\gamma
C_{u})\subset\mathfrak{g}^{*}$ contains $0$. If $u\leq v_{0}$, then $u=v$
maximizes $E(v)$ among all admissible $v\leq v_{0}$. If $u\geq v_{0}$ and
$v_{0}\in\text{Psh}(\omega)$, then $u=v$ minimizes $E(v)$ among all admissible
$v\geq v_{0}$.

The convex hull condition is equivalent to the existence of a Borel
probability measure $m$ (even one with finite support) on the contact set
$C_{u}$ such that

(4.5) $\int_{C_{u}}(\mu\circ\gamma)\,dm=0.$

###### Proof.

Again it suffices to prove under the assumption $u=0$ and $\gamma=e$; the
general case will follow in light of Proposition 4.1. Suppose
$v\in{\mathcal{H}}$ is another admissible potential, $\omega_{v}=g^{*}\omega$.
By Theorem 2.9 $g$ can be taken of form $g=\exp iX$, $X\in\mathfrak{g}$. With
$h=-2\langle\mu,X\rangle$, by Proposition 3.2

$\psi(t)=\int_{0}^{t}(\exp i\tau X)^{*}h\,d\tau+E(v)t,\qquad 0\leq t\leq 1,$

defines a geodesic connecting $0$ with $v$. By convexity (Proposition 2.2), if
$0,v\leq v_{0}$ then $\psi(t)\leq v_{0}$; if $0,v\geq v_{0}$ then $\psi(t)\geq
v_{0}$, for $0\leq t\leq 1$. Also $\psi(0)(x)=0$ when $x\in C_{0}$.

In the first case of the proposition it follows that
$0\geq\dot{\psi}(0)(x)=h(x)+E(v)$ for $x\in C_{0}$. But (4.5) implies
$\int_{C_{0}}h\,dm=0$, and so

$E(u)=0\geq\int_{C_{0}}\big{(}h+E(v)\big{)}\,dm=E(v),$

as claimed. In the second case we have $0\leq\dot{\psi}(0)(x)=h(x)+E(v)$ for
$x\in C_{0}$. Therefore

$E(u)=0\leq\int_{C_{0}}\big{(}h+E(v)\big{)}\,dm=E(v),$

which completes the proof.

##  5\. Uniqueness of extremals

Let $(M,\omega)$, the $G$\- and $G^{\mathbb{C}}$-actions, and the moment map
$\mu:M\to\mathfrak{g}^{*}$ be as in sections 3, 4. With $X\in\mathfrak{g}$ put

$N_{X}=\\{x\in M:\langle\mu,X\rangle\text{ vanishes to second order at }x\\},$

and introduce the assumption

(5.1) $\text{If }X\in\mathfrak{g}\setminus\\{0\\},\text{ the convex hull of
}\\{\mu(x):x\in N_{X}\\}\subset\mathfrak{g}^{*}\text{ avoids }0.$

This latter is the same as requiring that under some linear map
$\mathfrak{g}^{*}\to\mathbb{R}$ the image of $\mu(N_{X})$ be on one side of
$0\in\mathbb{R}$. Thus (5.1) is equivalent to

(5.2) $\text{If }X\in\mathfrak{g}\setminus\\{0\\},\text{ there is a
}Y\in\mathfrak{g}\text{ such that }\langle\mu,Y\rangle>0\text{ on }N_{X}.$

Here are two examples. We use dagger † to indicate adjoint of a matrix. The
standard action of $G=\text{SU}(n+1)$ on $\mathbb{C}^{n+1}$ descends to an
action on projective space
$M=\mathbb{P}_{n}=\big{(}\mathbb{C}^{n+1}\setminus\\{0\\}\big{)}/\sim$,
preserving the Fubini–Study form $\omega$. The Lie algebra $\mathfrak{g}$
consists of traceless skew adjoint matrices $X$, $X^{\dagger}=-X$. By Kirwan
[K84, Lemma 2.5], for example, the pull back $\tilde{\mu}$ of the moment map
$\mu:\mathbb{P}_{n}\to\mathfrak{g}^{*}$ by the projection
$\mathbb{C}^{n+1}\setminus\\{0\\}\to\mathbb{P}_{n}$ is

(5.3)
$\langle\tilde{\mu}(z),X\rangle=-i\dfrac{z^{\dagger}Xz}{z^{\dagger}z},\qquad
z\in\mathbb{C}^{n+1}\setminus\\{0\\}\text{ a column
vector},\,X\in\mathfrak{g},$

up to a positive constant factor that is determined by the normalization of
the Fubini–Study metric.

Assumption (5.2) is satisfied. It suffices to verify this when $X$ is diagonal
with diagonal entries $ia_{0},\dots,ia_{n}\in i\mathbb{R}$, not all $0$. The
simultaneous vanishing of $\langle\tilde{\mu}(z),X\rangle=\sum
a_{j}|z_{j}|^{2}/\sum|z_{j}|^{2}$ and
$d\langle\tilde{\mu}(z),X\rangle(JX_{M})=$

$\partial_{t}|_{t=0}\langle\tilde{\mu}\big{(}(\exp
itX)z\big{)},X\rangle=2\,\frac{\sum a_{j}^{2}|z_{j}|^{2}\sum|z_{j}|^{2}-(\sum
a_{j}|z_{j}|^{2})^{2}}{(\sum|z_{j}|^{2})^{2}}$

implies $a_{j}z_{j}=0$ for all $j$. Say, $a_{0}\neq 0$. Then
$x=(x_{0}:\ldots:x_{n})\in N_{X}$ implies $x_{0}=0$, and so the diagonal
matrix $Y\in\mathfrak{g}$ with entries $-in,i,\dots,i$ satisfies
$\langle\mu,Y\rangle>0$ on $N_{X}$.

As a second example, consider the unit sphere $S^{2}\subset\mathbb{R}^{3}$,
with the inherited conformal, hence complex, structure and rotation invariant
Kähler form (=Fubini–Study form, if $S^{2}$ is identified with
$\mathbb{P}_{1}$). Let $M=(S^{2})^{n}$ and $\omega$ the product Kähler form on
$M$. The group $G=\text{SO}(3)$ acts on $M$ diagonally by rotations. If
$\mathfrak{g}^{*}$ is suitably identified with $\mathbb{R}^{3}$, the moment
map is

$\mu(x_{1},\dots,x_{n})=x_{1}+\dots+x_{n}\in\mathbb{R}^{3}\approx\mathfrak{g}^{*},\qquad(x_{j})\in
M,$

see [K84, 2.8], and $\langle\mu,X\rangle$ is the composition of $\mu$ with a
linear $\pi:\mathbb{R}^{3}\to\mathbb{R}$. Its critical points occur where each
$x_{j}\in S^{2}$ is an extremum point of $\pi|S^{2}$. This means each $x_{j}$
must agree with one of a fixed pair of antipodal points in $S^{2}$. If $n$ is
odd then $\langle\mu,X\rangle\neq 0$ on the critical set, $N_{X}=\emptyset$,
and (5.1) is satisfied. If $n$ is even then $N_{X}$ is nonempty, and along
with any $x$ it contains $-x$ as well. Since $\mu(x)+\mu(-x)=0$, (5.1) fails.

Now to the uniqueness in problems (1.2), (1.3). Let $v_{0}:M\to\mathbb{R}$ be
continuous.

######  Proposition 5.1.

Assume (5.1) (or equivalently (5.2)). There is a unique $v$ that maximizes
$E(v)$ among admissible $v\leq v_{0}$. If $v_{0}\in C^{2}(M)$ and
$\omega+i\partial\bar{\partial}v_{0}>0$, then there is a unique $v$ that
minimizes $E(v)$ among admissible $v\geq v_{0}$.

Later we will need that in the minimization problem a more liberal if slightly
technical condition also implies uniqueness:

######  Proposition 5.2.

Assume (5.1). Consider a $v_{0}\in\text{Psh}(\omega)$ and an admissible $u\geq
v_{0}$, $\omega_{u}=\gamma^{*}\omega$ with some $\gamma\in G^{\mathbb{C}}$. If
there is a Borel probability measure $m$ on the contact set
$C_{u}=\\{u=v_{0}\\}$ such that $\int_{C_{u}}(\mu\circ\gamma)\,dm=0$, $v_{0}$
is $C^{2}$ in a neighborhood of $\text{supp\,}m$, and
$\omega+i\partial\bar{\partial}v_{0}>0$ there, then $u$ is the unique
minimizer of $E(v)$ among admissible $v\geq v_{0}$.

###### Proof of Propositions 5.1, 5.2.

In Proposition 5.1 the existence of extremals is the content of Proposition
3.1. As to uniqueness, the statement of Proposition 5.1 about minimizers
follows from Proposition 5.2. Indeed, Proposition 4.2 implies that if in the
scenario of Proposition 5.1 an admissible $u\geq v_{0}$ minimizes energy, then
there is a Borel probability measure $m$ on $C_{u}$ such that
$\int_{C_{u}}(\mu\circ\gamma)\,dm=0$. Since $v_{0}\in C^{2}(M)$ and
$\omega+i\partial\bar{\partial}v_{0}>0$ everywhere, Proposition 5.2 applies
and gives uniqueness.

This leaves us with proving uniqueness of maximizers in Proposition 5.1, and
Proposition 5.2. We will refer to the former as Case A, to the latter as Case
B. In what follows, let $u,\bar{u}\leq v_{0}$ be two maximizers in Case A,
$E(u)=E(\bar{u})$; or $u$ as in Case B and $\bar{u}\geq v_{0}$ admissible,
$E(u)\geq E(\bar{u})$. In both cases we need to show $u=\bar{u}$.

In light of Proposition 4.1 we can assume $u=0$ and, in case B, $\gamma=e$.
Let $\omega_{\bar{u}}=g^{*}\omega$, $g\in G^{\mathbb{C}}$. By Theorem 2.9 we
can also assume $g=\exp iX$, $X\in\mathfrak{g}$. With $g_{t}=\exp itX$ and
$h=-2\langle\mu,X\rangle$,

(5.4) $\psi(t)=\int_{0}^{t}g_{\tau}^{*}h\,d\tau+tE(\bar{u}),\qquad 0\leq t\leq
1,$

defines a geodesic in ${\mathcal{H}}$, $\psi(0)=0$ and $\psi(1)=\bar{u}$
(Proposition 3.2). By convexity, Proposition 2.2, $\psi(t)\leq v_{0}$ in Case
A and $\psi(t)\geq v_{0}$ in Case B. In Case A $u=0$ is known to be a
maximizer. Therefore by Proposition 4.2 we can find a Borel probability
measure $m$ on $C_{0}$, that we fix, such that $\int_{C_{0}}\mu\,dm=0$. We
claim that in both cases $\psi(\cdot)(y)$ is constant for $m$–almost every
$y\in C_{0}$.

Let us start with Case A. With any $y\in C_{0}$, $\psi(\cdot)(y)$ attains its
maximum at $0$, whence $0\geq\dot{\psi}(0)(y)=h(y)$. Since
$\int_{C_{0}}h\,dm=-2\int_{C_{0}}\langle\mu,X\rangle\,dm=0$, for $m$–almost
every $y$ we have $\dot{\psi}(0)(y)=0$. But $\psi(\cdot)(y)$ is convex, hence
it must be constant, as claimed.

In case B, if $y\in C_{0}$, $\psi(\cdot)(y)$ has a minimum at $0$, so
$0\leq\dot{\psi}(0)(y)=h(y)+E(\bar{u})\leq h(y)$. Again $\int_{C_{0}}h\,dm=0$
implies $\dot{\psi}(0)(y)=0$ for $m$–almost every $y$. Pick a smooth function
$w$ on a neighborhood of $y\in M$ such that $i\partial\bar{\partial}w=\omega$.
The function $\Psi(s,x)=\psi(\text{Re}\,s)(x)$, $s\in\bar{S}_{01}$, $x\in M$,
solves

$\text{rk}\,(\pi^{*}\omega+i\partial\bar{\partial}\Psi)\equiv n\qquad\text{on
}\bar{S}_{01}\times M,$

see section 2. Hence $\text{rk}\,\partial\bar{\partial}(\pi^{*}w+\Psi)\equiv
n$ in a neighborhood of $(0,y)\in\bar{S}_{01}\times M$. The leaf of the
associated Monge–Ampère foliation [BK77] through $(0,y)$ contains the graph of
a smooth function $f:U\to M$, where $U\subset\bar{S}_{01}$ is a connected
(relative) neighborhood of $0\in\bar{S}_{01}$, and $f$ is holomorphic on
$U\cap S_{01}$. Thus $f(0)=y$ and $(\pi^{*}w+\Psi)\big{(}s,f(s)\big{)}$ is a
smooth function of $s\in U$, harmonic when $\text{Re}\,s>0$. As $w+v_{0}$ is
plurisubharmonic, $(w+v_{0})\circ f$ is subharmonic in $U\cap S_{01}$. Also

(5.5) $(w+v_{0})\big{(}f(s)\big{)}\leq
w\big{(}f(s)\big{)}+\Psi\big{(}s,f(s)\big{)}=(\pi^{*}w+\Psi)\big{(}s,f(s)\big{)},$

with equality when $s=0$. But in fact, if $t=\text{Re}\,s$, the
$\partial_{t}$–derivative of the two sides of (5.5) also agree when $s=0$,
because $v_{0}\leq\Psi(0,\cdot)$ with equality at $y$, hence
$dv_{0}=d\Psi(0,\cdot)$ at $y$; and
$\partial_{t}|_{t=0}\Psi(t)(y)=\dot{\psi}(0)(y)=0$ for $m$–almost every $y$.
To sum up, for $m$–almost every $y$ the left hand side of (5.5) minus the
right hand side, a subharmonic function of $s$, attains its maximum $=0$ at
$0$, and there its normal derivative is also $0$. The Hopf–Oleinik lemma,
which in the case needed apparently goes back to Zaremba [H52, O52, Z10], then
implies that the difference function is $\equiv 0$, and equality holds in
(5.5) for all $s\in U$. In particular, $(w+v_{0})\circ f$ is harmonic; since
$i\partial\bar{\partial}(w+v_{0})>0$ near $y$, it follows that $f$ is
constant, $f\equiv y$. But then (5.5), with equality instead of inequality,
gives $v_{0}(y)=\psi(t)(y)$ for $t\in U\cap\mathbb{R}$, and by analytic
continuation, for all $t\in[0,1]$.

With the claim proved, we compute $0=\dot{\psi}(t)(y)=h(g_{t}y)$,

$0=\ddot{\psi}(t)(y)=\partial_{\tau}|_{\tau=0}h\big{(}g_{\tau}(g_{t}y)\big{)}=\iota\big{(}JX_{M}(g_{t}y)\big{)}dh=2\iota\big{(}JX_{M}(g_{t}y)\big{)}\iota\big{(}X_{M}(g_{t}y)\big{)}\omega,$

as $dh=2\iota(X_{M})\omega$. Since $\omega>0$, this implies $X_{M}(g_{t}y)=0$.
In particular, $\langle\mu,X\rangle=-h/2$ vanishes to second order at
$g_{0}y=y$, i.e., $m(C_{0}\setminus N_{X})=0$. On the one hand, this means
that $0=\int_{C_{0}}\mu\,dm=\int_{C_{0}\cap N_{X}}\mu\,dm$, and so $0$ is in
the convex hull of $N_{X}$; on the other, by (5.1) this cannot happen unless
$X=0$. But then $h=0$ and $\bar{u}=\psi(1)=0$ follow.

Putting together Propositions 3.1, 4.2, 4.3, and 5.1 we obtain Theorems 1.2,
1.3.

If in Theorems 1.2, 1.3, or in Proposition 5.1 condition (5.1) is dropped, the
extremals may no longer be unique. For example:

######  Proposition 5.3.

Suppose that for some nonzero $X\in\mathfrak{g}$ the convex hull of
$\\{\mu(x):x\in N_{X}\\}\subset\mathfrak{g}^{*}$ contains $0$. Then there is a
$v_{0}\in{\mathcal{H}}$ such that the minimum of $E(v)$ among admissible
$v\geq v_{0}$ is attained at more than one $v$.

###### Proof.

Write dist for distance on $M$ measured in, say, the Kähler metric of
$\omega$. Let $g_{t}=\exp itX$, $h=-2\langle\mu,X\rangle$, and define a
geodesic $\varphi:\mathbb{R}\to{\mathcal{H}}$ by
$\varphi(t)=\int_{0}^{t}g_{\tau}^{*}h\,d\tau$ (Proposition 3.2). Thus
$g_{t}|N_{X}=\text{id}_{N_{X}}$ and $\varphi(t)|N_{X}=0$. First observe that
for $t\in[-1,1]$ the functions $g_{t}^{*}h$ vanish to second order on $N_{X}$,
and have uniformly bounded Hessians on $M$. Hence

$|\dot{\varphi}(t)(z)|=|(g_{t}^{*}h)(z)|=O\big{(}\text{dist}(z,N_{X})^{2}\big{)},\qquad
t\in[-1,1],\quad z\in M.$

Since $\varphi(0)=0$, integration gives

(5.6) $|\varphi(t)(z)|=O\big{(}|t|\text{dist}(z,N_{X})^{2}\big{)},\qquad
t\in[-1,1],\quad z\in M.$

According to Atiyah, Guillemin, and Sternberg [A82, Lemma 2.2], [GS84, Theorem
5.3], $N_{X}$, the union of certain components of the critical set of $h$, is
a smooth submanifold of $M$. (The weaker result [K72, Theorem 5.1] would also
do.) Choose a smooth $v_{0}:M\to(-\infty,0]$ whose zero set is $N_{X}$, and
whose Hessian in directions normal to $N_{X}$ is negative definite. If needed,
we can replace $v_{0}$ by $\varepsilon v_{0}$ to arrange that
$v_{0}\in{\mathcal{H}}$. Since $v_{0}$ is negative away from $N_{X}$, the
Hessian condition implies $v_{0}(z)\leq c\,\text{dist}(z,N_{X})^{2}$ for $z\in
M$, with some $c<0$. Hence (5.6) yields $\varphi(t)\geq v_{0}$ for small $t$.
Each $\varphi(t)$ is admissible by Proposition 3.2.

But $0=\varphi(0)$ minimizes energy $E(v)$ among admissible $v\geq v_{0}$,
because by assumption the convex hull of $\\{\mu(x):x\in
N_{X}\\}=\\{\mu(x):0=v_{0}(x)\\}$ contains $0$, cf. Proposition 4.3. As
$E\big{(}\varphi(t)\big{)}\equiv 0$ by Proposition 3.2, for small $t$ the
potentials $\varphi(t)$ also minimize.

##  6\. Stability of extremals, extrema

We continue in the setup of section 5. In view of Proposition 5.1, with any
$v_{0}\in C(M)$ one can associate the unique admissible $u_{-}=U_{-}(v_{0})$
that maximizes $E(v)$ among admissible $v\leq v_{0}$. Similarly, if we denote
by $C_{\text{unique}}\subset C(M)$ the set of $v_{0}$ such that $E$ has a
unique minimizer among admissible $v\geq v_{0}$, with each $v_{0}\in
C_{\text{unique}}$ we can associate this unique minimizer $u=U(v_{0})$. For
example, if the assumptions of Proposition 5.1 are met, ${\mathcal{H}}\subset
C_{\text{unique}}$. First we record a stability property of $u_{-},u$; then of
the minimum value in (1.3).

######  Proposition 6.1.

The maps

$U_{-}:C(M)\to{\mathcal{H}}\qquad\text{and}\qquad
U:C_{\text{unique}}\to{\mathcal{H}}$

are continuous, when their domains are endowed with the topology of uniform
convergence and ${\mathcal{H}}\subset C^{\infty}(M)$ with its natural Fréchet
topology.

###### Proof.

Since the proofs for $U_{-}$ and $U$ are almost identical, we will write out
the proof only for $U$. Suppose $v_{0},v_{1}\in C_{\text{unique}}$, and
$\max_{M}|v_{0}-v_{1}|=\delta$. Thus $U(v_{1})\geq v_{1}\geq v_{0}-\delta$,
whence the admissible potential $U(v_{1})+\delta$ is $\geq v_{0}$. Therefore

$E\big{(}U(v_{1})\big{)}+\delta=E\big{(}U(v_{1})+\delta\big{)}\geq
E\big{(}U(v_{0})\big{)}.$

Exchanging the roles of $v_{0},v_{1}$, we obtain
$\big{|}E\big{(}U(v_{0})\big{)}-E\big{(}U(v_{1})\big{)}\big{|}\leq\delta$, and
so $E\circ U$ is continuous.

Now take a sequence of $v_{j}\in C_{\text{unique}}$, $j=1,2,\dots$, converging
uniformly to $v_{0}\in C_{\text{unique}}$. Thus
$\omega_{U(v_{j})}=g_{j}^{*}\omega$ with some $g_{j}\in G^{\mathbb{C}}$,
$j=0,1,\dots$. To prove $\lim_{j}U(v_{j})=U(v_{0})$ we first assume $g_{j}\to
g_{0}$. By Theorem 2.9 $g_{j}=\gamma_{j}\exp iX_{j}$ with $\gamma_{j}\in G$,
$X_{j}\in\mathfrak{g}$, and $X_{j}\to X_{0}$. Since $(\exp
iX_{j})^{*}\omega=\omega_{U(v_{j})}$, by Proposition 3.2

$U(v_{j})=-2\int_{0}^{1}(\exp i\tau
X_{j})^{*}\langle\mu,X_{j}\rangle\,d\tau+E\big{(}U(v_{j})\big{)}.$

This indeed shows that $U(v_{j})\to U(v_{0})$.

Without assuming $g_{j}\to g_{0}$ we can argue like this. Suppose
$U(v_{j})\not\to U(v_{0})$. Thus $U(v_{0})$ has a neighborhood
${\mathcal{V}}\subset{\mathcal{H}}$ that infinitely many $U(v_{j})$ avoid. By
compactness, Proposition 3.3, at the price of passing to a subsequence, we can
arrange that the corresponding $g_{j}$ converge. By what we have already
proved, this implies that the $U(v_{j})$ converge to some admissible $u\neq
U(v_{0})$. However, $u\geq\lim_{j}v_{j}=v_{0}$ and
$E(u)=\lim_{j}E\big{(}U(v_{j})\big{)}=E\big{(}U(v_{0})\big{)}$; this
contradicts uniqueness of the minimizer.

In section 9 we will need a coarser stability property of the minimum value
itself in (1.3). For any upper bounded $v_{0}:M\to[-\infty,\infty)$ let

$I(v_{0})=\inf\\{E(v):v\in{\mathcal{H}}\text{ is admissible, }v\geq
v_{0}\\}\in[-\infty,\infty).$

######  Proposition 6.2.

If a sequence of upper semicontinuous $v_{k}:M\to[-\infty,\infty)$,
$k\in\mathbb{N}$, decreases to $v_{0}$, then $I(v_{k})$ decrease to
$I(v_{0})$.

###### Proof.

The definition implies that $I(v_{k})\geq I(v_{0})$, $k\in\mathbb{N}$, form a
decreasing sequence; call its limit $L\geq I(v_{0})$. Suppose $u\geq v_{0}$ is
admissible. Then the upper semicontinuous functions $\max(v_{k},u)$ decrease
to $u$, perforce uniformly by Dini’s theorem. Thus, if $\varepsilon>0$ there
is a $k$ such that $v_{k}\leq u+\varepsilon$, whence $L\leq I(v_{k})\leq
E(u+\varepsilon)=E(u)+\varepsilon$. This being true for all $\varepsilon>0$
and for all admissible $u\geq v_{0}$, $L\leq I(v_{0})$ follows, and the proof
is complete.

##  7\. Polarized manifolds

With $(M,\omega)$, $G$, and $G^{\mathbb{C}}$ as before we now consider a
holomorphic line bundle $L\to M$ with a smooth hermitian metric $h^{0}$. The
curvature $\Theta(h^{0})$ is a closed form, and we assume
$\omega=i\Theta(h^{0})$. We also assume that the action of each $g\in
G^{\mathbb{C}}$ on $M$ can be lifted to $\tilde{g}:L\to L$; thus $\tilde{g}$
is holomorphic and maps each $L_{x}$ linearly onto $L_{gx}$. This section is
about a reformulation of the variational problems (1.2), (1.3) in this
setting. The material, when $L$ is the hyperplane section bundle over
projective space, will be needed in sections 8, 10.

Any two lifts $L\to L$ of $g\in G^{\mathbb{C}}$ are multiples of one another
by a holomorphic function on $M$, hence by a nonzero constant. All lifts of
all $g\in G^{\mathbb{C}}$ form a group $H^{\mathbb{C}}$, an extension of
$G^{\mathbb{C}}$ by $\mathbb{C}^{\times}$, the multiplicative group of nonzero
complex numbers. While $H^{\mathbb{C}}$ is the complexification of an
extension $H$ of $G$ by $S^{1}$, we will not need this fact. We can let
$H^{\mathbb{C}}$ act on $M$ as well, via the projection $H^{\mathbb{C}}\to
G^{\mathbb{C}}$.

For each $g\in H^{\mathbb{C}}$ there is a $W(g)\in C^{\infty}(M)$ such that

(7.1) $g^{*}h^{0}=e^{-W(g)}h^{0}.$

(In this formula, as usual, $W(g)$ is viewed as a function on $L$, constant on
the fibers.) Thus

$0<i\Theta(g^{*}h^{0})=i\Theta(h^{0})+i\partial\bar{\partial}W(g)=\omega_{W(g)},$

$W(g)\in{\mathcal{H}}$, and $W$ is a smooth map
$H^{\mathbb{C}}\to{\mathcal{H}}$. Since
$i\Theta(g^{*}h^{0})=ig^{*}\Theta(h^{0})=g^{*}\omega$, it follows that $W(g)$
is admissible in the sense of Definition 1.1. The converse is also true:

######  Proposition 7.1.

$v\in{\mathcal{H}}$ is admissible if and only if it is of form $W(g)$, where
$g\in H^{\mathbb{C}}$.

###### Proof.

What remains to prove is the “only if” statement. Note that if
$\lambda\in\mathbb{C}^{\times}\subset H^{\mathbb{C}}$ then

$e^{-W(\lambda g)}h^{0}=(\lambda
g)^{*}h^{0}=g^{*}\lambda^{*}h^{0}=|\lambda|^{2}g^{*}h^{0}=|\lambda|^{2}e^{-W(g)}h^{0}.$

Hence $W(\lambda g)=W(g)-\log|\lambda|^{2}$. Suppose that $v$ is admissible,
$\omega_{v}=g_{0}^{*}\omega$ with some $g_{0}\in G^{\mathbb{C}}$. Choose
$g_{1}\in H^{\mathbb{C}}$ that projects to $g_{0}$. By (7.1)

$\omega+i\partial\bar{\partial}v=g_{1}^{*}\omega=i\Theta(g_{1}^{*}h^{0})=i\Theta(e^{-W(g_{1})}h^{0})=\omega+i\partial\bar{\partial}W(g_{1}),$

whence $W(g_{1})-v=c$ is constant. It follows that
$W(e^{c/2}g_{1})=W(g_{1})-c=v$.

######  Definition 7.2.

We say that hermitian metrics $h,h^{0}$ on $L$ are similar, or are in the same
similarity class, if $h=g^{*}h^{0}$ with some $g\in H^{\mathbb{C}}$. In this
case we put
${\mathcal{E}}(h^{0},h)=E\big{(}W(g)\big{)}=E\big{(}\log(h_{0}/h)\big{)}$.

An arbitrary metric $\bar{h}$ on $L$ is of form $e^{-\bar{v}}h^{0}$, with
$\bar{v}$ a function on $M$. By our discussion:

######  Proposition 7.3.

The extrema in (1.2), (1.3) can be represented as

(7.2) $\displaystyle\max\\{E(v):v\leq\bar{v}\text{ is admissible}\\}=$
$\displaystyle\max\\{E\big{(}W(g)\big{)}:g\in
H^{\mathbb{C}},W(g)\leq\bar{v}\\}$ $\displaystyle=$
$\displaystyle\max\\{{\mathcal{E}}(h^{0},h):h\geq\bar{h}\text{ is similar to
}h^{0}\\}$

and

(7.3) $\displaystyle\min\\{E(v):v\geq\bar{v}\text{ is admissible}\\}=$
$\displaystyle\min\\{E\big{(}W(g)\big{)}:g\in
H^{\mathbb{C}},W(g)\geq\bar{v}\\}$ $\displaystyle=$
$\displaystyle\min\\{{\mathcal{E}}(h^{0},h):h\leq\bar{h}\text{ is similar to
}h^{0}\\}.$

Further, $v$ is extremal in (7.2), resp. (7.3), if and only if $v=W(g)$ with
an extremal $g$.

If $h^{1}$ is similar to $h^{0}$, then
${\mathcal{E}}(h^{1},\cdot)={\mathcal{E}}(h^{0},\cdot)+\text{const}$, cf.
(4.2). It follows that the extremal metrics $h$ in (7.2), (7.3) stay the same
if $h^{0}$ is changed to $h^{1}$.

######  Definition 7.4.

Extremal hermitian metrics in (7.2), (7.3) will be called minimal,
respectively, maximal metrics in the similarity class of $h^{0}$.

If $\bar{v}$ is bounded, these minimal and maximal hermitian metrics exist
according to Proposition 3.1, and if the assumptions of Proposition 5.1 are
satisfied (with $v_{0}=\bar{v})$, then they are unique. For example, if $L$ is
the hyperplane section bundle ${\mathcal{O}}_{\mathbb{P}_{n}}(1)$, and
$h^{0},\bar{h}$ are smooth and positively curved, then by the discussion in
section 5 the extremals exist and are unique.

######  Lemma 7.5.

$E\circ W:H^{\mathbb{C}}\to\mathbb{R}$ is a homomorphism to the additive group
of real numbers.

###### Proof.

If $g_{1},g_{2}\in H^{\mathbb{C}}$, then
$e^{-W(g_{1}g_{2})}h^{0}=g_{2}^{*}g_{1}^{*}h^{0}=e^{-g_{2}^{*}W(g_{1})}g_{2}^{*}h^{0}=e^{-W(g_{2})-g_{2}^{*}W(g_{1})}h^{0}$,
or

$W(g_{1}g_{2})=W(g_{2})+g_{2}^{*}W(g_{1}).$

Now consider $g\in H^{\mathbb{C}}$ and a smooth path $\gamma:[0,1]\to
H^{\mathbb{C}}$, starting at the neutral element. Let
$\varphi(t)=W\big{(}\gamma(t)\big{)}$ and
$\psi(t)=W\big{(}\gamma(t)g\big{)}=W(g)+g^{*}W\big{(}\gamma(t)\big{)}$. Thus
$\dot{\psi}(t)=g^{*}\dot{\varphi}(t)$ as functions on $M$. We compute

$\dfrac{d}{dt}E\big{(}W(\gamma(t)g)\big{)}=dE\big{(}\dot{\psi}(t)\big{)}=\dfrac{1}{\text{Vol}}\int_{M}\dot{\psi}(t)\omega_{\psi(t)}^{n}\\\
=\dfrac{1}{\text{Vol}}\int_{M}\big{(}g^{*}\dot{\varphi}(t)\big{)}\big{(}\gamma(t)g\big{)}^{*}\omega^{n}=\dfrac{1}{\text{Vol}}\int_{M}\dot{\varphi}(t)\gamma(t)^{*}\omega^{n}=\dfrac{d}{dt}E\big{(}W(\gamma(t))\big{)}.$

Hence $E\big{(}W(\gamma(t)g)\big{)}-E\big{(}W(\gamma(t))\big{)}$ is constant.
Setting $t=0$ we find

$E\big{(}W(\gamma(t)g)\big{)}=E\big{(}W(\gamma(t))\big{)}+E\big{(}W(g)\big{)}.$

Since $H^{\mathbb{C}}$ is connected, this shows that $E\circ W$ is indeed a
homomorphism.

##  8\. Hermitian metrics on vector bundles

In the Introduction we suggested that one of the motivations for the
variational problems (1.2), (1.3) had to do with hermitian metrics on vector
bundles. This comes from a fundamental construction in geometry that
associates with a, say, holomorphic vector bundle $F\to B$ a holomorphic line
bundle $\mathcal{O}_{\mathbb{P}F}(1)\to\mathbb{P}F$. Here we use the
convention that $\mathbb{P}F=\bigcup_{b\in B}\mathbb{P}F_{b}$ is the set of
one codimensional subspaces $\Sigma$ in various fibers of $F$, on which the
local trivializations of $F$ induce the structure of a complex manifold
fibered over $B$; and $\mathcal{O}_{\mathbb{P}F}(1)$ is a holomorphic line
bundle whose fiber over $\Sigma\subset F_{b}$ is $F_{b}/\Sigma$. Thus
$\mathcal{O}_{\mathbb{P}F}(1)$ restricts to $\mathbb{P}F_{b}$ as the
hyperplane section bundle. In this section we will write

$\pi:\mathbb{P}F\to B$

for the fibering.

A hermitian metric on a vector space $V$ induces a hermitian metric on any
quotient of $V$. Applying this construction fiberwise, we can associate to a
hermitian metric $h_{F}$ on the vector bundle $F$ a hermitian metric
$h_{\mathbb{P}}$ on the line bundle $\mathcal{O}_{\mathbb{P}F}(1)$. We will
refer to $h_{\mathbb{P}}$ as induced by $h_{F}$. Note that the restrictions
$h_{\mathbb{P}}|\mathcal{O}_{\mathbb{P}F_{b}}(1)$ for $b\in B$ are of
Fubini–Study type, hence positively curved. The question is whether there is a
natural converse construction, that associates a hermitian metric on $F$ to a
hermitian metric $k$ on $\mathcal{O}_{\mathbb{P}F}(1)$. We will be interested
only in $k$ whose restrictions to $\mathcal{O}_{\mathbb{P}F_{b}}(1)$ are
positively curved.

One such construction is as follows. All hermitian metrics on $F$ are induced
from one another: If $h^{0}_{F},h_{F}$ are such, and $b\in B$, then
$h_{F}|F_{b}=g^{*}h^{0}_{F}|F_{b}$ with some $g\in\text{GL}(F_{b})$. The
corresponding metrics $h^{0}_{\mathbb{P}},h_{\mathbb{P}}$ on
${\mathcal{O}}_{\mathbb{P}F_{b}}(1)$ are related by the induced action of $g$
on ${\mathcal{O}}_{\mathbb{P}F_{b}}(1)$. Indeed, if $\Sigma\subset F_{b}$ is a
hyperplane, the fiber of $\mathcal{O}_{\mathbb{P}F}(1)$ over $\Sigma$, i.e.,
$F_{b}/\Sigma$, consists of translates $\Sigma^{\prime}\subset F_{b}$ of
$\Sigma$. Since

$h_{\mathbb{P}}(\Sigma^{\prime})=\min_{\zeta\in\Sigma^{\prime}}h_{F}(\zeta)=\min_{\zeta\in\Sigma^{\prime}}h^{0}_{F}(g\zeta)=\min_{\zeta\in
g\Sigma^{\prime}}h^{0}_{F}(\zeta)=h^{0}_{\mathbb{P}}(g\Sigma^{\prime}),$

we have

(8.1)
$h_{\mathbb{P}}|\mathcal{O}_{\mathbb{P}F_{b}}(1)=g^{*}h^{0}_{\mathbb{P}}|\mathcal{O}_{\mathbb{P}F_{b}}(1),\qquad
g\in\text{GL}(F_{b}).$

We are in the situation of section 7, with
$L=\mathcal{O}_{\mathbb{P}F_{b}}(1)\to M=\mathbb{P}F_{b}$,
$G=\text{SU}(h^{0}_{F}|F_{b})$ and $G^{\mathbb{C}}=\text{SL}(F_{b})$ acting on
$\mathbb{P}F_{b}$, $H^{\mathbb{C}}=\text{GL}(F_{b})$ acting on
$\mathcal{O}_{\mathbb{P}F_{b}}(1)$. (8.1) shows that
$h_{\mathbb{P}}|\mathcal{O}_{\mathbb{P}F_{b}}(1)$ and
$h^{0}_{\mathbb{P}}|\mathcal{O}_{\mathbb{P}F_{b}}(1)$ are similar in the sense
of Definition 7.2, and clearly the converse is also true: Given a metric $h$
on $\mathcal{O}_{\mathbb{P}F}(1)$, if for all $b\in B$ its restrictions to
$\mathcal{O}_{\mathbb{P}F_{b}}(1)$ are similar to the restrictions of
$h^{0}_{\mathbb{P}}$, then it is induced from a hermitian metric $h_{F}$ on
$F$. Moreover, $h_{F}$ is uniquely determined by $h=h_{\mathbb{P}}$: If
$\zeta\in F_{b}$ and $\Sigma\subset F_{b}$ is a hyperplane, write
$\Sigma^{\zeta}\subset F_{b}$ for the equivalence class of $\zeta$ mod
$\Sigma$. Then $h_{\mathbb{P}}(\Sigma^{\zeta})\leq h_{F}(\zeta)$ for all
$\Sigma\in\mathbb{P}F_{b}$; but the two are equal if $\Sigma$ is perpendicular
(measured in $h_{F}$) to $\zeta$. Therefore

$h_{F}(\zeta)=\max\\{h_{\mathbb{P}}(\Sigma^{\zeta}):\Sigma\in\mathbb{P}F_{b}\\}.$

Hence, as established in section 7, given a smooth hermitian metric $k$ on
$\mathcal{O}_{\mathbb{P}F}(1)$, whose restrictions to
$\mathcal{O}_{\mathbb{P}F_{b}}(1)$ are positively curved, in the similarity
class of induced metrics on each $\mathcal{O}_{\mathbb{P}F_{b}}(1)$ we can
find the unique maximal metric $k_{b}$ among metrics $\leq
k|\mathcal{O}_{\mathbb{P}F_{b}}(1)$, cf. (7.3), Definition 7.4. The collection
of $k_{b}$ defines a metric on $\mathcal{O}_{\mathbb{P}F}(1)$, which is
induced by a hermitian metric on $F$. We associate this hermitian metric on
$F$ with $k$, and denote it $k^{F}$. Theorem 6.1 implies that $k^{F}$ is
continuous. However, there is no reason why it should be smooth. To obtain out
of a smooth $k$ a smooth metric on $F$, one may have to regularize $k^{F}$,
for which there are efficient techniques.

This association $k\mapsto k^{F}$ is based on the variational problems (7.3).
It would be equally possible to construct hermitian metrics on $F$ based on
the variational problems (7.2), but those metrics seem to be of less
interest.333Naumann in [N21] proposes yet another way to associate a hermitian
metric on $F$ with a hermitian metric on $\mathcal{O}_{\mathbb{P}F}(1)$, using
a Kähler–Ricci flow on the fibers $\mathbb{P}F_{b}$. It is not clear if the
two constructs are related in some way.

One situation when a smooth metric on $F$ is desired occurs in connection with
Griffiths’ conjecture on the equivalence between amplitude of a holomorphic
vector bundle $F\to B$ and the existence of a hermitian metric on $F$ whose
curvature is Griffiths positive ($B$ is compact). By [G69], amplitude is
equivalent to the existence of a positively curved smooth metric $k$ on
$\mathcal{O}_{\mathbb{P}F}(1)$. One can then ask whether the metric $k^{F}$
constructed above (perhaps regularized) would be positively curved. In section
10 we will see that this is not always true, but $k^{F}$ does have a
positivity property:

######  Theorem 8.1.

Let $F\to B$ be a holomorphic vector bundle of rank $m$, $k$ a smooth
hermitian metric on the line bundle $\mathcal{O}_{\mathbb{P}F}(1)$, positively
curved on each $\mathcal{O}_{\mathbb{P}F_{b}}(1)$, $b\in B$, and $k^{F}$ the
associated hermitian metric on $F$. If $\kappa$ is a continuous real $(1,1)$
form on $B$ such that $i\Theta(k)\geq\pi^{*}\kappa$, then the metric $\det
k^{F}$ on the line bundle $\det F\to B$ satisfies

(8.2) $i\Theta(\det k^{F})\geq m\kappa.$

In particular, if $k$ is positively curved, then so is $\det k^{F}$.

Again, in general $\det k^{F}$ is only continuous, and (8.2) is a comparison
of currents.

Theorem 8.1 will follow from two results. The first expresses $\det k^{F}$,
and in general $\det h_{F}$, through Monge–Ampère energy; the second, for
general $(M,\omega)$, establishes that—loosely stated—if the barrier $v_{0}$
in (1.3) depends on complex parameters in a plurisubharmonic way, then so does
minimal Monge–Ampère energy.

######  Lemma 8.2.

Let $h^{0}_{F}$ and $h_{F}$ be hermitian metrics on $F\to B$,
$h^{0}_{\mathbb{P}},h_{\mathbb{P}}$ the induced metrics on the line bundle
$\mathcal{O}_{\mathbb{P}F}(1)$. Define $u:\mathbb{P}F\to\mathbb{R}$ by
$h_{\mathbb{P}}=e^{-u}h^{0}_{\mathbb{P}}$. With the Kähler forms
$\omega_{b}=i\Theta(h^{0}_{\mathbb{P}}|\mathbb{P}F_{b})$, $b\in B$, the
Monge–Ampère energy of $u_{b}=u|\mathbb{P}F_{b}\in{\mathcal{H}}(\omega_{b})$
satisfies

(8.3) $\log\dfrac{\det h_{F}}{\det h^{0}_{F}}(b)=-mE(u_{b}).$

###### Proof.

Since this is a fiberwise statement, it suffices to prove when $B=\\{b\\}$ is
a singleton, and so $F$ is a vector space. As there is only one fiber, we will
omit the $b$’s in our formulas. We can take $F=\mathbb{C}^{m}$, $m\geq 2$, and
$h^{0}_{F}(\zeta)=\zeta^{\dagger}\zeta$, $\zeta\in\mathbb{C}^{m}$ (a column
vector). There is a $g\in\text{GL}(m,\mathbb{C})$ such that

(8.4) $h_{F}(\zeta)=(g\zeta)^{\dagger}(g\zeta).$

We will use the same symbol for the corresponding hermitian form
$h_{F}(\zeta,\zeta^{\prime})=(g\zeta^{\prime})^{\dagger}(g\zeta)$. Since
$h_{\mathbb{P}}=g^{*}h^{0}_{\mathbb{P}}$ by (8.1), $u_{b}$ in (8.3) is just
$W(g)$ of (7.1). If $e_{1},\dots,e_{m}\in\mathbb{C}^{m}$ is the standard
basis, by definition

$(\det h_{F})(e_{1}\wedge\dots\wedge
e_{m})=\det\big{(}h_{F}(e_{i},e_{j})\big{)}_{i,j}=\det(e_{j}^{\dagger}g^{\dagger}ge_{i})_{i,j}=\det
g^{\dagger}g.$

This holds when $g=I$ as well, hence

(8.5) $\det h_{F}/\det h^{0}_{F}=\det g^{\dagger}g=|\det g|^{2}.$

To see how this is related to the right hand side of (8.3), we vary
$g\in\text{GL}(m,\mathbb{C})$, and with it, $h_{F}$. By Lemma 7.5 $E\circ
W:\text{GL}(m,\mathbb{C})\to\mathbb{R}$ is a homomorphism of Lie groups. Since
$\text{SL}(m,\mathbb{C})$ is simple, $E\circ W$ vanishes on it. If
$\lambda\in\mathbb{C}^{\times}$ and $g=\lambda I\in\text{GL}(m,\mathbb{C})$,
then $h_{F}=|\lambda|^{2}h^{0}_{F}$ and
$h_{\mathbb{P}}=|\lambda|^{2}h^{0}_{\mathbb{P}}$, so $W(\lambda
I)=-\log|\lambda|^{2}$ and $E\big{(}W(\lambda I)\big{)}=-\log|\lambda|^{2}$.
Writing a general $g\in\text{GL}(m,\mathbb{C})$ as $\lambda Ig_{0}$, where
$\lambda^{m}=\det g$ and $g_{0}\in\text{SL}(m,\mathbb{C})$, we obtain

(8.6) $E(u_{b})=E\big{(}W(g)\big{)}=E\big{(}W(\lambda
I)\big{)}+E\big{(}W(g_{0})\big{)}=-\frac{1}{m}\log|\det g|^{2}.$

Comparison with (8.5) proves (8.3).

###### Proof of Theorem 8.1.

Rather than giving a complete proof, here we will reduce the theorem to
Proposition 8.3 below, that will be derived from a more general result in
section 9.

Since curvature is computed locally, we can assume that $F$ is trivial,
$F=B\times\mathbb{C}^{m}\to B$, and $B$ is an open subset of some
$\mathbb{C}^{d}$. We will denote points of $\mathbb{C}^{d}$ by
$z=(z_{1},\dots,z_{d})$. It will suffice to prove with any $\varepsilon>0$
that $i\Theta(\det k^{F})\geq
m(\kappa-i\varepsilon\partial\bar{\partial}|z|^{2})$ in some neighborhood of
an arbitrary $b\in B$. Suppose
$\kappa=i\sum_{\alpha,\beta}\kappa_{\alpha\beta}dz_{\alpha}\wedge
d\bar{z}_{\beta}$, and let
$q(z)=\sum_{\alpha,\beta}\kappa_{\alpha\beta}(b)z_{\alpha}\bar{z}_{\beta}-\varepsilon|z|^{2}/2$.
Then
$i\partial\bar{\partial}q=\kappa-i\varepsilon\partial\bar{\partial}|z|^{2}/2$
at $b$, and therefore close to $b$

$\kappa-i\varepsilon\partial\bar{\partial}|z|^{2}\leq
i\partial\bar{\partial}q\leq\kappa.$

Hence it will suffice to prove that
$i\Theta(k)\geq\pi^{*}i\partial\bar{\partial}q$ implies $i\Theta(\det
k^{F})\geq i\,m\partial\bar{\partial}q$, the special case of the theorem when
$\kappa=i\partial\bar{\partial}q$ with a smooth function $q$.

In this case, along with $k$ consider the metric $e^{\pi^{*}q}k$ on
$\mathcal{O}_{\mathbb{P}F}(1)$ and the metric $(e^{\pi^{*}q}k)^{F}=e^{q}k^{F}$
on $F$ associated with it. Since

$\Theta(e^{\pi^{*}q}k)=\Theta(k)-\pi^{*}\kappa\qquad\text{and}\qquad\Theta\big{(}\det(e^{q}k^{F})\big{)}=\Theta(\det
k^{F})-m\kappa,$

$\Theta(k)\geq\pi^{*}\kappa$ precisely when $\Theta(e^{\pi^{*}q}k)\geq 0$, and
$\Theta(\det k^{F})\geq m\kappa$ precisely when
$\Theta\big{(}\det(e^{q}k^{F})\big{)}\geq 0$. This shows that it will be
enough to prove when $\kappa=0$.

For the last reduction step let
$p:\mathbb{P}F=B\times\mathbb{P}_{m-1}\to\mathbb{P}_{m-1}$ denote the
projection, so that $\mathcal{O}_{\mathbb{P}F}(1)$ is the pullback of
${\mathcal{O}}_{\mathbb{P}_{m-1}}(1)$ by $p$. Define a reference metric
$h^{0}_{F}$ by $h_{F}^{0}(b,\zeta)=|\zeta|^{2}$, $(b,\zeta)\in
B\times\mathbb{C}^{m}=F$. The induced metric $h^{0}_{\mathbb{P}}$ on
$\mathcal{O}_{\mathbb{P}F}(1)$ is the pullback of the Fubini–Study metric on
${\mathcal{O}}_{\mathbb{P}_{m-1}}(1)$. Denoting the curvature of the latter by
$-i\omega$, therefore $i\Theta(h^{0}_{\mathbb{P}})=p^{*}\omega$.

If $\bar{v}\in C^{\infty}(\mathbb{P}F)$ is defined by
$k=e^{-\bar{v}}h^{0}_{\mathbb{P}}$, we have

$0\leq
i\Theta(k)=i\Theta(h^{0}_{\mathbb{P}})+i\partial\bar{\partial}\bar{v}=p^{*}\omega+i\partial\bar{\partial}\bar{v}.$

Since $k$ restricted to $\mathcal{O}_{\mathbb{P}F_{b}}(1)$ is positively
curved, $\bar{v}(b,\cdot)\in{\mathcal{H}}(\omega)$. Denote Monge–Ampère energy
in ${\mathcal{H}}(\omega)$ by $E$, and let

(8.7) $\chi(b)=\min\\{E(v):v\in{\mathcal{H}}(\omega)\text{ is admissible,
}v\geq\bar{v}(b,\cdot)\\},\qquad b\in B.$

By (7.3) and by Lemma 8.2 $\log(\det k^{F}/\det h^{0}_{F})=-m\chi$. Since
$h^{0}_{F}$ and $\det h^{0}_{F}$ are flat, $i\Theta(\det
k^{F})=i\,m\partial\bar{\partial}\chi$ follows. Hence the next proposition
implies Theorem 8.1:

######  Proposition 8.3.

If $\bar{v}\in C^{\infty}(B\times\mathbb{P}_{m-1})$ satisfies
$p^{*}\omega+i\partial\bar{\partial}\bar{v}\geq 0$, then $\chi:B\to\mathbb{R}$
defined in (8.7) is plurisubharmonic.

The proof will be given in the next section.

##  9\. Plurisubharmonic variation

In this section we prove a generalization of Proposition 8.3. Let us return to
our connected compact Kähler manifold $(M,\omega)$, on which a connected
compact Lie group $G$ acts by holomorphic isometries, with a moment map
$\mu:M\to\mathfrak{g}^{*}$ and complexified action $G^{\mathbb{C}}\times M\to
M$. In addition, let $B$ be a complex manifold, $p:B\times M\to M$ the
projection.

######  Theorem 9.1.

If $\bar{v}:B\times M\to[-\infty,\infty)$ is $p^{*}\omega$–plurisubharmonic,
then

(9.1) $\chi(b)=\inf\\{E(v):v\in{\mathcal{H}}\text{ is
admissible},v\geq\bar{v}(b,\cdot)\\}$

defines a plurisubharmonic function $\chi:B\to[-\infty,\infty)$.

In particular, Proposition 8.3 holds.—The proof will need some preparation. We
start with a variant of Kiselman’s minimum principle [K78].

######  Lemma 9.2.

Suppose $w$ is a plurisubharmonic function on $B\times G^{\mathbb{C}}$, left
invariant under $G$ in the sense that $w(b,g_{0}g)=w(b,g)$ if $b\in B$,
$g_{0}\in G$, $g\in G^{\mathbb{C}}$. Then the function

$\chi:B\ni b\mapsto\inf\\{w(b,g):g\in G^{\mathbb{C}}\\}\in[-\infty,\infty)$

is plurisubharmonic. The same holds if instead of left, we assume right
invariance: $w(b,gg_{0})=w(b,g)$ if $g_{0}\in G$.

###### Proof.

Chafi and Loeb in [C01, L85] already proved this under the extra condition
that $w$ is smooth, strictly plurisubharmonic, and exhaustive. Deng, Zhang,
and Zhou in [D+14] gave a new proof and a substantial generalization; they
also sketched an argument to reduce general functions $w$ to those satisfying
the extra conditions. Since in the setup of our lemma this reduction greatly
simplifies, we explain how to do it. We assume $w$ is left invariant.

Our starting point is [C01, Théorème 2.1] that implies our claim under the
extra assumptions, and we will show how to get rid of the extra assumptions in
two steps. Of course, it suffices to show that $\chi$ is plurisubharmonic in a
neighborhood of an arbitrary $b\in B$. Therefore we can assume $B$ is a
pseudoconvex open subset of some $\mathbb{C}^{k}$.

Suppose $w$ is just smooth. Since $G^{\mathbb{C}}$ is Stein (Theorem 2.9),
$B\times G^{\mathbb{C}}$ admits a smooth strictly plurisubharmonic exhaustion
function $f:B\times G^{\mathbb{C}}\to[0,\infty)$. By averaging, we can arrange
that $f(b,gh)=f(b,h)$ if $g\in G$. Postcomposing with a suitable increasing,
convex, and smooth function $\mathbb{R}\to\mathbb{R}$ we can further arrange
that $w+\varepsilon f$, for every $\varepsilon>0$, is exhaustive and strictly
plurisubharmonic. By [C01], then

$\chi^{\varepsilon}(b)=\inf\\{w(b,g)+\varepsilon f(b,g):g\in
G^{\mathbb{C}}\\}$

is plurisubharmonic. Since $\chi$ is the decreasing limit of
$\chi^{\varepsilon}$ as $\varepsilon\to 0$, our lemma holds under the sole
supplementary assumption that $w$ is smooth.

Now take a general $w$. Consider the complex Lie group $H=\mathbb{C}^{k}\times
G^{\mathbb{C}}$, with Lie algebra $\mathfrak{h}$. Let $|\,\,|$ be a Euclidean
norm and $\lambda$ a translation invariant measure on $\mathfrak{h}$. Choose a
compactly supported smooth function $\rho:\mathfrak{h}\to[0,\infty)$ such that
$\rho(X)$ depends only on $|X|$, and $\int_{\mathfrak{h}}\rho\,d\lambda=1$. If
$\varepsilon>0$ put
$\rho_{\varepsilon}(X)=\rho(X/\varepsilon)/\varepsilon^{\dim_{\mathbb{R}}\mathfrak{h}}$,
and define

(9.2) $w_{\varepsilon}(h)=\int_{\text{supp\,}\rho_{\varepsilon}}w(h\exp
X)\rho_{\varepsilon}(X)\,d\lambda(X).$

The domains of $w_{\varepsilon}$ are open sets of form $B_{\varepsilon}\times
G^{\mathbb{C}}\subset H$ that increase to $B\times G^{\mathbb{C}}$ as
$\varepsilon\to 0$; $w_{\varepsilon}$ is clearly plurisubharmonic and left
invariant under $G$. Choose a neighborhood $D\subset\mathfrak{h}$ of $0$ on
which $\exp$ is diffeomorphic, and let $\log:\exp D\to D$ denote its inverse.
If $h\in H$, pull back $\lambda$ to a measure $\lambda_{h}$ on $h\exp D$ by
the map $t\mapsto\log(h^{-1}t)$. This is absolutely continuous with respect to
a fixed smooth measure $\nu$ on $H$, and the Radon–Nikodým derivative
$(d\lambda_{h}/d\nu)(t)$ is smooth, as a function of $(t,h)$, $h^{-1}t\in\exp
D$. If $\varepsilon$ is so small that $\text{supp\,}\rho_{\varepsilon}\subset
D$, by a change of variables

$w_{\varepsilon}(h)=\int_{h\exp D}w(t)\rho_{\varepsilon}(\log
h^{-1}t)\,d\lambda_{h}(t).$

This shows that $w_{\varepsilon}$ is smooth where defined. Finally, for fixed
$h$ (9.2) can be seen as a Euclidean convolution of two functions on
$\mathfrak{h}$, $w(h\exp\cdot)$ and $\rho_{\varepsilon}$, evaluated at
$0\in\mathfrak{h}$. As $w(h\exp\cdot)$ is plurisubharmonic, in particular
subharmonic, $w_{\varepsilon}(h)$ decreases to $w(h)$ as $\varepsilon$
decreases to $0$, see e.g. [D12, Chapter I (4.18)]. By what we have already
proved,

$\chi_{\varepsilon}(b)=\inf\\{w_{\varepsilon}(b,g):g\in G^{\mathbb{C}}\\}$

is plurisubharmonic, therefore so is the decreasing limit
$\chi=\lim_{\varepsilon\to 0}\chi_{\varepsilon}$.

We will need to compare minimal Monge–Ampère energy in
${\mathcal{H}}={\mathcal{H}}(\omega)$ with that in
${\mathcal{H}}(\lambda\omega)$, $\lambda\in(0,\infty)$. Let $E_{\lambda}$ be
Monge–Ampère energy in ${\mathcal{H}}(\lambda\omega)$ and, for
$v_{0}:M\to[-\infty,\infty)$ bounded above,

$I_{\lambda}(v_{0})=\inf\\{E_{\lambda}(v):v\in{\mathcal{H}}(\lambda\omega)\text{
is admissible, }v\geq v_{0}\\}\in[-\infty,\infty).$

Thus $I$ of section 6 is $I_{1}$. The following is straightforward to check:

######  Proposition 9.3.

The map ${\mathcal{H}}(\omega)\ni v\mapsto\lambda
v\in{\mathcal{H}}(\lambda\omega)$ is a bijection; it is also a bijection
between admissible elements of ${\mathcal{H}}(\omega)$,
${\mathcal{H}}(\lambda\omega)$. Further, $E_{\lambda}(\lambda v)=\lambda E(v)$
and $I_{\lambda}(\lambda v_{0})=\lambda I(v_{0})$.

In section 3 we have already introduced a smooth
$U:G^{\mathbb{C}}\to{\mathcal{H}}$ such that $u=U(g)$ solves
$i\partial\bar{\partial}u=g^{*}\omega-\omega$, or $\omega_{u}=g^{*}\omega$.
The normalization for $U$ was $\int_{M}U(g)\omega^{n}=0$. For our present
purposes it is better to replace $U(g)$ by $V(g)=U(g)-E\big{(}U(g)\big{)}$; it
is still true that $V$ is smooth and $\omega_{V(g)}=g^{*}\omega$, but now
$E\big{(}V(g)\big{)}=0$. A potential $v\in{\mathcal{H}}$ is admissible if and
only if $v=V(g)+c$ with some $g\in G^{\mathbb{C}}$ and $c\in\mathbb{R}$, and
$v\geq v_{0}$ if and only if $c\geq\max_{M}\big{(}v_{0}-V(g)\big{)}$. Thus
$\chi$ of (9.1) is given by

(9.3) $\displaystyle\chi(b)$
$\displaystyle=\inf\\{E(v):v\geq\bar{v}(b,\cdot)\text{ is admissible}\\}$
$\displaystyle=\inf_{g\in
G^{\mathbb{C}}}E\big{(}V(g)+\max_{M}(\bar{v}(b,\cdot)-V(g))\big{)}=\inf_{g\in
G^{\mathbb{C}}}\max_{x\in M}\big{(}\bar{v}(b,x)-V(g)(x)\big{)}.$

######  Lemma 9.4.

$V(h)=V(gh)$ if $g\in G$, $h\in G^{\mathbb{C}}$, and

(9.4) $V(h)-V(gh)+h^{*}V(g)=0\qquad\text{if }g,h\in G^{\mathbb{C}}.$

(9.4) means that $V:G^{\mathbb{C}}\to C^{\infty}(M)$ is a cocycle for the
group cohomology of $G^{\mathbb{C}}$ with values in the right $G^{\mathbb{C}}$
module $C^{\infty}(M)$ (the action of $G^{\mathbb{C}}$ on $C^{\infty}(M)$ is
by pullback).

###### Proof.

When $g\in G$, $V(g)=0$, so the first statement follows from (9.4). In turn,
(9.4) follows from a result of Darvas and Rubinstein. In [DR17, Lemma 5.8]
they show—with different notation—that

$g.u=V(g)+g^{*}u,\qquad g\in G^{\mathbb{C}},\,u\in{\mathcal{H}},\,E(u)=0,$

defines a right action of $G^{\mathbb{C}}$. Writing out $(gh).u=h.(g.u)$ one
obtains (9.4).

######  Corollary 9.5.

$V(g^{-1})=-(g^{-1})^{*}V(g)$ if $g\in G^{\mathbb{C}}$.

This follows by putting $h=g^{-1}$ in (9.4).

$V(g)(x)$ was defined through its $\partial\bar{\partial}$ with respect to
$x\in M$. It turns out that there is a simple formula for its
$\partial\bar{\partial}$ with respect to $g\in G^{\mathbb{C}}$ as well.
Consider for $x\in M$ the orbit map

$o_{x}:G^{\mathbb{C}}\ni g\mapsto gx\in M.$

######  Proposition 9.6.

For fixed $x\in M$, $i\partial\bar{\partial}V(\cdot)(x)=o_{x}^{*}\omega$.

###### Proof.

Below we will write $(\partial\bar{\partial})_{G}$,
$(\partial\bar{\partial})_{M}$ to indicate operators acting on the $G$ or $M$
variables of functions that are defined on product spaces.—We need to show
that if $\xi\in T^{10}G$ then the two forms in question, evaluated on
$\xi,\bar{\xi}$, give the same result; or that with any $g\in G^{\mathbb{C}}$,
$X\in\mathfrak{g}^{\mathbb{C}}$, and $\phi(s)=g\exp sX$ ($s\in\mathbb{C}$),
abbreviating $\partial_{s}=\partial/\partial s$,
$\partial_{\bar{s}}=\partial/\partial\bar{s}$,

$\displaystyle P(s,x)=$
$\displaystyle\big{(}i(\partial\bar{\partial})_{G}V(\cdot)(x)\big{)}\big{(}\phi(s)_{*}\partial_{s}\,,\,\phi(s)_{*}\partial_{\bar{s}}\big{)}=i\partial_{s}\partial_{\bar{s}}V\big{(}\phi(s)\big{)}(x)\qquad\text{and}$
$\displaystyle Q(s,x)=$
$\displaystyle\omega\big{(}o_{x*}\phi(s)_{*}\partial_{s}\,,\,o_{x*}\phi(s)_{*}\partial_{\bar{s}}\big{)}$

are equal.

First we compute $(\partial\bar{\partial})_{M}P$ and
$(\partial\bar{\partial})_{M}Q$. If locally $\omega=i\partial\bar{\partial}w$,
then $Q(s,x)=i\partial_{s}\partial_{\bar{s}}w\big{(}\phi(s)x\big{)}$,

$\displaystyle(\partial\bar{\partial})_{M}Q(s,\cdot)$
$\displaystyle=i(\partial\bar{\partial})_{M}\partial_{s}\partial_{\bar{s}}w\big{(}\phi(s)\cdot\big{)}=i\partial_{s}\partial_{\bar{s}}(\partial\bar{\partial})_{M}\big{(}w(\phi(s)\cdot)\big{)},$
$\displaystyle(\partial\bar{\partial})_{M}P(s,\cdot)$
$\displaystyle=i\partial_{s}\partial_{\bar{s}}(\partial\bar{\partial})_{M}V\big{(}\phi(s)\big{)}=\partial_{s}\partial_{\bar{s}}\big{(}\phi(s)^{*}\omega-\omega\big{)}=i\partial_{s}\partial_{\bar{s}}(\partial\bar{\partial})_{M}\big{(}w(\phi(s)\cdot)\big{)}.$

Thus $\partial\bar{\partial}P(s,\cdot)=\partial\bar{\partial}Q(s,\cdot)$,
whence $P(s,\cdot)-Q(s,\cdot)=c$ is constant. Now $X_{M}$ vanishes at some
$y\in M$ according to Lemma 2.10. For such $y$, $\phi(s)(y)$ is independent of
$s$, and $Q(s,y)=0$. All we need to show is that $P(s,y)=0$ as well: this will
imply $c=0$, and with it, the proposition.

We compute $P(s,y)$ as follows. Since $(\exp sX)y=y$, Lemma 9.4 gives $V(g\exp
sX)(y)=V(\exp sX)(y)+V(g)(y)$. Substituting $g\exp\sigma X$ for $g$,

(9.5) $\displaystyle V\big{(}g\exp(\sigma+s)X\big{)}(y)=V(\exp\sigma
X)(y)+V(\exp sX)(y)+V(g)(y)\quad\text{and}$ (9.6)
$\displaystyle\partial_{s}\partial_{\bar{s}}V\big{(}g\exp(\sigma+s)X\big{)}(y)=\partial_{s}\partial_{\bar{s}}V(\exp
sX)(y)$

follow. Setting $\sigma=-s$ in (9.5), and taking
$\partial_{s}\partial_{\bar{s}}$, we obtain

$0=\partial_{s}\partial_{\bar{s}}V(\exp(-sX))(y)+\partial_{s}\partial_{\bar{s}}V(\exp
sX)(y).$

This implies $\partial_{s}\partial_{\bar{s}}|_{s=0}V(\exp sX)(y)=0$. Letting
$s=0$ in (9.6) then gives
$P(\sigma,y)=i\partial_{\sigma}\partial_{\bar{\sigma}}V(g\exp\sigma X)(y)=0$,
as needed.

###### Proof of Theorem 9.1.

First we assume that $\bar{v}$ is smooth). We start with representation (9.3)
of $\chi$, but replace $g$ by $g^{-1}$. Thus $\chi(b)=\inf_{g\in
G^{\mathbb{C}}}w(b,g)$, where

(9.7) $\displaystyle w(b,g)=$ $\displaystyle\max_{x\in
M}\big{(}\bar{v}(b,x)-V(g^{-1})(x)\big{)}=\max_{x\in
M}\big{(}\bar{v}(b,x)+V(g)(g^{-1}x)\big{)}$ $\displaystyle=$
$\displaystyle\max_{x\in M}\big{(}\bar{v}(b,gx)+V(g)(x)\big{)}$

(here we used Corollary 9.5). Let $\pi_{G}:B\times G\to G$ be the projection.
With $x\in M$ fixed, consider functions $\bar{u},u:B\times G\to\mathbb{R}$,

$\bar{u}(b,g)=\bar{v}(b,gx),\qquad u(b,g)=\bar{u}(b,g)+V(g)(x).$

If $(\text{id}_{B}\times o_{x})(b,g)$ is defined as
$\big{(}b,o_{x}(g)\big{)}$, we can write $\bar{u}=(\text{id}_{B}\times
o_{x})^{*}\bar{v}$. By assumption

$i\partial\bar{\partial}\bar{u}=i(\text{id}_{B}\times
o_{x})^{*}\partial\bar{\partial}\bar{v}\geq-(\text{id}_{B}\times
o_{x})^{*}p^{*}\omega=-\pi_{G}^{*}o_{x}^{*}\omega,$

and
$i\partial\bar{\partial}u\geq-\pi_{G}^{*}o_{x}^{*}\omega+\pi_{G}^{*}i\partial\bar{\partial}V(\cdot)(x)=0$
by Proposition 9.6. In other words, $u$ is plurisubharmonic, and so the
continuous function $w$, the maximum of plurisubharmonic functions, is itself
plurisubharmonic. But the second expression in (9.7), in conjunction with
Lemma 9.4, shows that $w$ is right invariant, $w(b,g)=w(b,gg_{0})$ if
$g_{0}\in G$. Therefore by Lemma 9.2 $\chi(b)=\inf_{g\in
G^{\mathbb{C}}}w(b,g)$ is plurisubharmonic.

Next consider a $p^{*}\omega$–plurisubharmonic $\bar{v}$ that is just bounded.
We can assume $B\subset\mathbb{C}^{k}$ is open and bounded. Denoting points of
$\mathbb{C}^{k}$ by $z$ and letting $q(z,x)=|z|^{2}$, $(z,x)\in B\times M$,
the form $\theta=i\partial\bar{\partial}q+p^{*}\omega$ is positive on $B\times
M$. Regularization techniques of Demailly and Pǎun [D92, DP04], but especially
[BK07, Theorem 2] of Błocki and Kołodziej imply that there is a sequence of
smooth $(\theta/j+p^{*}\omega)$–plurisubharmonic functions $u_{j}$ decreasing
to $\bar{v}$. (To apply the Błocki–Kołodziej result we may need to shrink
$B$.) The functions $v_{j}(z,x)=u_{j}(z,x)+|z|^{2}/j$ are then
$(1+1/j)p^{*}\omega$–plurisubharmonic, and also decrease to $\bar{v}$. At the
price of adding a constant, we can assume that $\bar{v}$ is negative; then
$u_{j},v_{j}$ can be made negative as well, for large $j$. By what we have
already proved

$\chi_{j}(b)=\inf\big{\\{}E_{1+1/j}(v):v\in{\mathcal{H}}\big{(}(1+1/j)\omega\big{)}\text{
is admissible, }v\geq v_{j}(b,\cdot)\big{\\}}$

is plurisubharmonic, and eventually $\leq 0$. Now

$\chi_{j}(b)=I_{1+1/j}\big{(}v_{j}(b,\cdot)\big{)}=\frac{j+1}{j}I\bigg{(}\dfrac{jv_{j}\big{(}b,\cdot)}{j+1}\bigg{)}$

by Proposition 9.3. Since the $v_{j}$ are (eventually) negative,
$jv_{j}/(j+1)$ decrease to $\bar{v}$ as $j\to\infty$. In light of Proposition
6.2 $\chi(b)=I\big{(}\bar{v}(b,\cdot)\big{)}$ is the decreasing limit of
$j\chi_{j}(b)/(j+1)$, therefore plurisubharmonic.

Finally, consider a general $\bar{v}$. With $j\in\mathbb{N}$, set this time
$v_{j}=\max(\bar{v},-j)$. By what we have already proved,
$\chi_{j}(b)=I\big{(}v_{j}(b,\cdot)\big{)}$ are plurisubharmonic functions of
$b$, hence so is their, again by Proposition 6.2 decreasing, limit
$I\big{(}\bar{v}(b,\cdot)\big{)}=\chi(b)$.

To conclude this section we digress to describe how the function
$V:G\to{\mathcal{H}}$ gives rise to solutions of the Monge–Ampère equation.
While this is not needed for the main purpose of the paper, it generalizes
Mabuchi’s [M87, Theorem 3.5], that we used in the proof of Proposition 2.5,
and provides an independent proof. Let $\pi:\mathbb{C}\times M\to M$ be the
projection and, as before, $n=\dim_{\mathbb{C}}M$.

######  Theorem 9.7.

If $g\in G^{\mathbb{C}}$ and $Z\in\mathfrak{g}^{\mathbb{C}}$ then
$\Phi(s,x)=V\big{(}(\exp sZ)g\big{)}(x)$, $s\in\mathbb{C}$, $x\in M$,
satisfies

(9.8) $(\pi^{*}\omega+i\partial\bar{\partial}\Phi)^{n+1}=0.$

In particular, let $g$ be the neutral element, $X\in\mathfrak{g}$, and $Z=iX$.
Writing $s=\sigma+it$,

$V(\exp sZ)=V\big{(}\exp(-tX)\exp(i\sigma X)\big{)}=V(\exp i\sigma X)$

by Lemma 9.4, and so $\Phi(s,x)=\Phi(\text{Re}\,s,x)$. Hence (9.8) implies
that $\sigma\mapsto\Phi(\sigma,\cdot)$ is a geodesic, cf. the discussion
around (2.3); this is equivalent to [M87, Theorem 3.5].

###### Proof.

Any $\mathbb{R}$-linear form on $\mathfrak{g}$ extends to a unique
$\mathbb{C}$-linear form on $\mathfrak{g}^{\mathbb{C}}$. Thus
$\mathfrak{g}^{*}$ embeds into $(\mathfrak{g}^{\mathbb{C}})^{*}$, and
accordingly, the moment map $\mu$ can be viewed as a map
$\mu:M\to(\mathfrak{g}^{\mathbb{C}})^{*}$. Let $g_{s}=\exp sZ$. Let $Z=X+iY$
with $X,Y\in\mathfrak{g}$, and write $\bar{Z}=X-iY$.

To compute $\pi^{*}\omega+i\partial\bar{\partial}\Phi$ we start by recalling

(9.9)
$\omega+i\partial\bar{\partial}V(g_{s}g)=(g_{s}g)^{*}\omega=g^{*}g_{s}^{*}\omega.$

Therefore, with $\sigma\in\mathbb{R}$,

$i\partial\bar{\partial}\,\partial_{\sigma}|_{\sigma=0}V(g_{\sigma}g)=\partial_{\sigma}|_{\sigma=0}\,g^{*}(\exp
i\sigma Y)^{*}(\exp\sigma
X)^{*}\omega=g^{*}{\mathcal{L}}_{JY_{M}}\omega=-2ig^{*}\partial\bar{\partial}\langle\mu,Y\rangle$

by Proposition 2.8. Hence
$\partial_{\sigma}|_{\sigma=0}V(g_{\sigma}g)=-2g^{*}\langle\mu,Y\rangle+c$,
where $c$ is a constant function on $M$. With $s=\sigma+it$ it follows upon
replacing $g$ by $g_{s_{0}}g$

$\partial_{\sigma}V(g_{s}g)=-2g^{*}g_{s}^{*}\langle\mu,Y\rangle+c,\qquad\partial_{t}V(g_{s}g)=-2g^{*}g_{s}^{*}\langle\mu,X\rangle+c^{\prime},$

and with $\partial_{s}=(\partial_{\sigma}-i\partial_{t})/2$,
$\partial_{\bar{s}}=(\partial_{\sigma}+i\partial_{t})/2$,

(9.10)
$\partial_{s}V(g_{s}g)=ig^{*}g_{s}^{*}\langle\mu,Z\rangle+c^{\prime\prime},\qquad\partial_{\bar{s}}V(g_{s}g)=-ig^{*}g_{s}^{*}\langle\mu,\bar{Z}\rangle+c^{\prime\prime\prime}.$

If $Z\in\mathfrak{g}$ then
$d\langle\mu,Z\rangle=-\iota(Z_{M})\omega=-\iota(Z_{M}^{\prime})\omega-\iota(Z_{M}^{\prime\prime})\omega$,
whence $\bar{\partial}\langle\mu,Z\rangle=-\iota(Z_{M}^{\prime})\omega$. By
$\mathbb{C}$-linearity, the same holds for $Z\in\mathfrak{g}^{\mathbb{C}}$,
and passing to conjugates
$\partial\langle\mu,\bar{Z}\rangle=-\iota(Z_{M}^{\prime\prime})\omega$
follows. Therefore (9.10) implies

(9.11) $\displaystyle\partial_{s}\bar{\partial}V(g_{s}g)$
$\displaystyle=\sqrt{-1}g^{*}g_{s}^{*}\bar{\partial}\langle\mu,Z\rangle=-\sqrt{-1}g^{*}g_{s}^{*}\iota(Z_{M}^{\prime})\omega,$
$\displaystyle\partial_{\bar{s}}\partial V(g_{s}g)$
$\displaystyle=-\sqrt{-1}g^{*}g_{s}^{*}\partial\langle\mu,\bar{Z}\rangle=\sqrt{-1}g^{*}g_{s}^{*}\iota(Z_{M}^{\prime\prime})\omega.$

Since we can change $g$ to $g_{s_{0}}g$, it suffices to prove (9.8) at points
$(0,x)\in\mathbb{C}\times M$. Choose a smooth function $w$ on a neighborhood
of a fixed $x_{0}\in M$ such that $\omega=i\partial\bar{\partial}w$ there. By
Proposition 9.6 $i\partial\bar{\partial}V(\cdot)(x)=o_{x}^{*}\omega$, hence
near $x_{0}$, when $s=0$

(9.12)
$\partial_{s}\partial_{\bar{s}}V(g_{s}g)=g^{*}\partial_{s}\partial_{\bar{s}}(\exp
sZ)^{*}w=-\sqrt{-1}g^{*}\iota(Z_{M}^{\prime})\iota(Z_{M}^{\prime\prime})\omega.$

Extend the action of $g$ to $\mathbb{C}\times M$, by letting it act as the
identity on $\mathbb{C}$. (9.9), (9.11), and (9.12) give (still when $s=0$)

$(g^{*})^{-1}(\pi^{*}\omega+\sqrt{-1}\partial\bar{\partial}\Phi)=\omega+ds\wedge\iota(Z_{M}^{\prime})\omega+\iota(Z_{M}^{\prime\prime})\omega\wedge
d\bar{s}+\iota(Z_{M}^{\prime})\iota(Z_{M}^{\prime\prime})\omega\,ds\wedge
d\bar{s}.$

(On the right $ds,d\bar{s}$, and $\omega$ are pulled back to $\mathbb{C}\times
M$.) One checks that $\partial_{s}-Z_{M}^{\prime}$ is in the kernel of the
above $2$-form. This means that
$\text{rk}\,\pi^{*}\omega+i\partial\bar{\partial}\Phi\leq n$, which proves the
theorem.

##  10\. An example

Let $F\to B$ be a holomorphic vector bundle,
$\mathcal{O}_{\mathbb{P}F}(1)\to\mathbb{P}F$ the associated line bundle, $k$ a
smooth hermitian metric on $\mathcal{O}_{\mathbb{P}F}(1)$, and $k^{F}$ the
associated hermitian metric on $F$, as in section 8. The question we entertain
here is how the association $k\mapsto k^{F}$ impacts curvature: If $k$ is
positively curved, must $k^{F}$ also be positively, or at least
semipositively, curved? According to Griffiths [G69, (2.36), (2.37)], a smooth
hermitian metric $h_{F}$ on $F$ is (semi)positively curved if and only if the
induced metric $h_{\mathbb{P}}$ on $\mathcal{O}_{\mathbb{P}F}(1)$ is. For this
reason the question above, in its weaker form, can be rephrased as follows:
Consider a complex manifold $B$ and the product $B\times\mathbb{P}_{n}$,
$n\in\mathbb{N}$, with projection $p:B\times\mathbb{P}_{n}\to\mathbb{P}_{n}$.
Let $\omega$ be the Fubini–Study Kähler form on $\mathbb{P}_{n}$, and endow
$\mathbb{P}_{n}$ with the standard actions of $G=\text{SU}(n+1)$,
$G^{\mathbb{C}}=\text{SL}(n+1,\mathbb{C})$.

mmn Suppose $\bar{v}\in C^{\infty}(B\times\mathbb{P}_{n})$ satisfies
$p^{*}\omega+i\partial\bar{\partial}\bar{v}>0$. Let
$u:B\times\mathbb{P}_{n}\to\mathbb{R}$ be such
(10.1)m that for each $b\in B$, $u(b,\cdot)\in{\mathcal{H}}(\omega)$ minimizes
Monge–Ampère energy $E(v)$ among
mmmm admissible $v\geq\bar{v}(b,\cdot)$. Is $u$ $p^{*}\omega$-plurisubharmonic
on $B\times\mathbb{P}_{n}$?

Indeed, with $F=B\times\mathbb{C}^{n+1}\to B$ the trivial bundle,
$\mathcal{O}_{\mathbb{P}F}(1)$ is the pullback of the hyperplane section
bundle ${\mathcal{O}}_{\mathbb{P}_{n}}(1)$ by $p$, and the trivial metric
$h^{0}_{F}(b,\zeta)=\zeta^{\dagger}\zeta$ on $F$ induces a metric
$h^{0}_{\mathbb{P}}$ on $\mathcal{O}_{\mathbb{P}F}(1)$, which is just the
pullback of the Fubini–Study metric on ${\mathcal{O}}_{\mathbb{P}_{n}}(1)$. If
$\bar{v},u$ are as in (10.1), define metrics
$k=e^{-\bar{v}}h^{0}_{\mathbb{P}}$ and $e^{-u}h^{0}_{\mathbb{P}}$ on
$\mathcal{O}_{\mathbb{P}F}(1)$. On the one hand, that $k$ is positively curved
means $i\Theta(k)=p^{*}\omega+i\partial\bar{\partial}\bar{v}>0$; on the other,
$i\Theta(k^{F})\geq 0$ if and only if the metric induced by $k^{F}$, namely
$e^{-u}h^{0}_{\mathbb{P}}$, satisfies
$i\Theta(e^{-u}h^{0}_{\mathbb{P}})=p^{*}\omega+i\partial\bar{\partial}u\geq
0$. Therefore an affirmative answer to the original question implies the same
for its reformulation (10.1).

However, the answer to (10.1) is “not always”. Hence Griffiths’ conjecture
cannot be proved just by applying the construction $k\mapsto k^{F}$ to a
positively curved metric $k$ on $\mathcal{O}_{\mathbb{P}F}(1)$.

######  Theorem 10.1.

There are an open $B\subset\mathbb{C}$ and $\bar{v}\in
C^{\infty}(B\times\mathbb{P}_{1})$ such that
$p^{*}\omega+i\partial\bar{\partial}\bar{v}>0$, but $u\in
C(B\times\mathbb{P}_{1})$ constructed in (10.1) fails to be
$p^{*}\omega$-plurisubharmonic.

In constructing the example it is easier to start with $u$. Let
$A\subset\mathbb{C}$ be the unit disc. Using homogeneous coordinates
$(x_{0}:x_{1})$ on $\mathbb{P}_{1}$, $u$ will be a perturbation of

(10.2)
$u_{0}(z,x)=\log\dfrac{e^{2|z|^{2}}|x_{0}|^{2}+e^{-|z|^{2}}|x_{1}|^{2}}{|x_{0}|^{2}+|x_{1}|^{2}}=-|z|^{2}+\log\dfrac{e^{3|z|^{2}}|x_{0}|^{2}+|x_{1}|^{2}}{|x_{0}|^{2}+|x_{1}|^{2}},$

$z\in A$, $x=(x_{0}:x_{1})\in\mathbb{P}_{1}$. (The coefficient $2$ in front of
$|z|^{2}$ could be replaced by any number $>1$.) This is certainly not
subharmonic when restricted to $A\times\\{(0:1)\\}$. The immediate question
is, is there a strictly $p^{*}\omega$-plurisubharmonic $\bar{v}\in
C^{\infty}(A\times\mathbb{P}_{1})$ (i.e.,
$i\partial\bar{\partial}\bar{v}+p^{*}\omega>0$) for which the construction in
(10.1) outputs this $u=u_{0}$? Of course, $u_{0}$ must dominate $\bar{v}$, but
not by much, to ensure it minimizes energy. One may try to take $\bar{v}$ the
$p^{*}\omega$-plurisubharmonic envelope of $u_{0}$, the largest
$p^{*}\omega$-plurisubharmonic function $\leq u_{0}$. This will essentially
work, except that such envelopes are rarely smooth or strictly
$p^{*}\omega$-plurisubharmonic. Therefore the true $\bar{v}$ will be a
regularization of the envelope.

In what follows, we will construct this envelope, $v_{0}$—although, in the end
we will not need to know that the $v_{0}$ we have found is the envelope.—It is
determined by three regimes

$\displaystyle
D_{0}=\big{\\{}|x_{1}|^{2}<2e^{3|z|^{2}}|x_{0}|^{2}\big{\\}},\qquad
D_{2}=\big{\\{}2e^{3}|x_{0}|^{2}<|x_{1}|^{2}\big{\\}},$ $\displaystyle
D_{1}=\big{\\{}2e^{3|z|^{2}}|x_{0}|^{2}<|x_{1}|^{2}<2e^{3}|x_{0}|^{2}\big{\\}},$

and three functions, in addition to $u_{0}$,

(10.3) $\displaystyle u_{1}(x)$
$\displaystyle=-\dfrac{1}{3}\log\dfrac{|x_{1}|^{2}}{2|x_{0}|^{2}}+\log\dfrac{3|x_{1}|^{2}/2}{|x_{0}|^{2}+|x_{1}|^{2}},\qquad\text{and}$
$\displaystyle u_{2}(x)$
$\displaystyle=-1+\log\dfrac{e^{3}|x_{0}|^{2}+|x_{1}|^{2}}{|x_{0}|^{2}+|x_{1}|^{2}}.$

In these formulae $x=(x_{0}:x_{1})\in\mathbb{P}_{1}$ and $z\in A$. The value
of $u_{1}$ when $x_{0}$ or $x_{1}=0$ is computed as a limit $=-\infty$.

######  Proposition 10.2.

$u_{1},u_{2}$ are $\omega$-subharmonic on $\mathbb{P}_{1}$ and $u_{0}$ is
strictly $p^{*}\omega$-plurisubharmonic on $D_{0}$.

###### Proof.

With the projection $\pi:\mathbb{C}^{2}\setminus\\{0\\}\to\mathbb{P}_{1}$, the
first question is whether $\pi^{*}u_{1},\pi^{*}u_{2}$ are
$\pi^{*}\omega$-plurisubharmonic on $\mathbb{C}^{2}\setminus\\{0\\}$. Since
$\pi^{*}\omega=i\partial\bar{\partial}\log\big{(}|x_{0}|^{2}+|x_{1}|^{2}\big{)}$,
the question is whether the functions

$\displaystyle-\dfrac{1}{3}\log\dfrac{|x_{1}|^{2}}{2|x_{0}|^{2}}+\log\dfrac{3|x_{1}|^{2}}{2}=\dfrac{1}{3}\log(2|x_{0}|^{2})+\log\dfrac{3|x_{1}|^{4/3}}{2}\quad\text{and}$
$\displaystyle-1+\log(e^{3}|x_{0}|^{2}+|x_{1}|^{2})$

are plurisubharmonic on $\mathbb{C}^{2}\setminus\\{0\\}$: they obviously are.

To discuss $u_{0}$, we start with a formula for smooth functions $r,s$:

$\partial\bar{\partial}\log(e^{r}+e^{s})=\dfrac{e^{r}\partial\bar{\partial}r+e^{s}\partial\bar{\partial}s}{e^{r}+e^{s}}+e^{r+s}\dfrac{\partial(r-s)\wedge\bar{\partial}(r-s)}{(e^{r}+e^{s})^{2}}.$

Since $x_{0}\neq 0$ on $D_{0}$, we can use the local coordinate
$\xi=x_{1}/x_{0}$. At least when $\xi\neq 0$

$u_{0}(z,x)=-|z|^{2}+\log\dfrac{e^{3|z|^{2}}|x_{0}|^{2}+|x_{1}|^{2}}{|x_{0}|^{2}+|x_{1}|^{2}}=-|z|^{2}+\log(e^{r}+e^{s})-\log(1+|\xi|^{2}),$

where $r=3|z|^{2}$, $s=\log|\xi|^{2}$. Hence

$\displaystyle p^{*}\omega+i\partial\bar{\partial}u_{0}$
$\displaystyle=-idz\wedge d\bar{z}+\dfrac{3ie^{3|z|^{2}}\,dz\wedge
d\bar{z}}{e^{3|z|^{2}}+|\xi|^{2}}+ie^{r+s}\dfrac{\partial(r-s)\wedge\bar{\partial}(r-s)}{(e^{r}+e^{s})^{2}}$
$\displaystyle=\dfrac{i(2e^{3|z|^{2}}-|\xi|^{2})}{e^{3|z|^{2}}+|\xi|^{2}}dz\wedge
d\bar{z}+ie^{r}\dfrac{\alpha\wedge\bar{\alpha}}{(e^{r}+e^{s})^{2}},$

where $\alpha=3\bar{z}\bar{\xi}\,dz-d\xi$. By continuity, this formula holds
when $\xi=0$, too. Above, the coefficients of $i\,dz\wedge d\bar{z}$ and
$i\alpha\wedge\bar{\alpha}$ are positive in $D_{0}$, and the forms $dz$ and
$\alpha$ are everywhere linearly independent. It follows that
$p^{*}\omega+i\partial\bar{\partial}u_{0}$ is indeed positive.

######  Proposition 10.3.

The formula

(10.4) $u_{12}(x)=\begin{cases}u_{1}(x)\quad\text{if}\quad
2e^{3}|x_{0}|^{2}\leq|x_{1}|^{2}\\\ u_{2}(x)\quad\text{if}\quad
2e^{3}|x_{0}|^{2}\geq|x_{1}|^{2}\end{cases}$

defines a continuous $\omega$-subharmonic function
$\mathbb{P}_{1}\to[-\infty,\infty)$.

###### Proof.

Proposition 10.2 implies that $u_{12}$ is $\omega$-subharmonic away from the
circle $K=\\{x:2e^{3}|x_{0}|^{2}=|x_{1}|^{2}\\}$. On $K$ itself $u_{1},u_{2}$
agree to second order: writing $t=|x_{1}/x_{0}|^{2}$, near $K$

$u_{2}(x)-u_{1}(x)=-1+\log(t+e^{3})+\dfrac{1}{3}\log(t/2)-\log(3t/2),$

and the function on the right as well as its derivative vanish when
$t=2e^{3}$. In particular, in (10.4) the two definitions for $u_{12}(x)$ when
$x\in K$ are consistent, and $u_{12}$ is continuous. It also follows that near
$K$ $du_{12}$ exists and is Lipschitz continuous locally. Hence near $K$ the
distributional Laplacian of $u_{12}$ is locally $L^{\infty}$. Since
$\omega+i\partial\bar{\partial}u_{12}\geq 0$ as a current on
$\mathbb{P}_{1}\setminus K$, it follows that
$\omega+i\partial\bar{\partial}u_{12}\geq 0$ everywhere, and $u_{12}$ is
indeed $\omega$-subharmonic.

######  Proposition 10.4.

The formula

(10.5) $v_{0}(z,x)=\begin{cases}u_{0}(z,x)&\text{if}\quad|x_{1}|^{2}\leq
2e^{3|z|^{2}}|x_{0}|^{2}\\\ u_{12}(x)&\text{if}\quad|x_{1}|^{2}\geq
2e^{3|z|^{2}}|x_{0}|^{2}\end{cases}$

defines a continuous $p^{*}\omega$-plurisubharmonic function
$v_{0}:A\times\mathbb{P}_{1}\to\mathbb{R}$, and for each $z\in A$ the function
$u_{0}(z,\cdot)\geq v_{0}(z,\cdot)$ is the unique minimizer of $E(v)$ among
admissible $v\in{\mathcal{H}}(\omega)$, $v\geq v_{0}(z,\cdot)$.

###### Proof.

Fix $x\in\mathbb{P}_{1}$. Differentiation shows that as a function of $|z|$,
$0\leq|z|\leq 1$,

$e^{2|z|^{2}}|x_{0}|^{2}+e^{-|z|^{2}}|x_{1}|^{2}$

has a strict minimum. If $|x_{1}|^{2}\leq 2e^{3}|x_{0}|^{2}$, the minimum
occurs when $|x_{1}|^{2}=2e^{3|z|^{2}}|x_{0}|^{2}$ (write $|z|_{\text{min}}$
for this value of $|z|$); if $|x_{1}|^{2}\geq 2e^{3}|x_{0}|^{2}$, the minimum
occurs when $|z|=1$. It follows that if $|x_{1}|^{2}<2e^{3}|x_{0}|^{2}$ then

$u_{0}(z,x)\geq u_{0}(|z|_{\text{min}},x)=u_{1}(x),$

and if $|x_{1}|^{2}\geq 2e^{3}|x_{0}|^{2}$ then $u_{0}(z,x)\geq
u_{0}(1,x)=u_{2}(x)$. From this we first infer that in (10.5) the two
definitions for $v_{0}(z,x)$, when $|x_{1}|^{2}=2e^{3}|x_{0}|^{2}$, are
consistent, and $v_{0}\leq u_{0}$ is continuous. The second inference is that
$u_{0}$ and $u_{12}$ agree to second order at

$\tilde{K}=\\{(z,x)\in
A\times\mathbb{P}_{1}:|x_{1}|^{2}=2e^{3|z|^{2}}|x_{0}|^{2}\\}.$

Since $v_{0}$ is obviously $p^{*}\omega$-plurisubharmonic on
$(A\times\mathbb{P}_{1})\setminus\tilde{K}$, it follows—as in the proof of
Proposition 10.3—that it is $p^{*}\omega$-plurisubharmonic on all of
$A\times\mathbb{P}_{1}$.

Now fix $z\in A$. To show that $u_{0}(z,\cdot)$ uniquely minimizes
Monge–Ampère energy among admissible $v\in{\mathcal{H}}(\omega)$, $v\geq
v_{0}(z,\cdot)$, we will apply Proposition 5.2. The diagonal matrix

$\gamma=\text{diag}(e^{3|z|^{2}/4},e^{-3|z|^{2}/4})\in\text{SL}(2,\mathbb{C})$

acts on $\mathbb{P}_{1}$ as the diagonal matrix
$\text{diag}(e^{|z|^{2}},e^{-|z|^{2}/2})$, and pulls back
$\omega=i\partial\bar{\partial}\log(|x_{0}|^{2}+|x_{1}|^{2})$ to
$\omega+i\partial\bar{\partial}u_{0}(z,\cdot)$. Let $X\in\mathfrak{su}(2)$. It
is a matrix of form

$X=i\begin{pmatrix}P&Q\\\ \bar{Q}&-P\end{pmatrix},$

with $P\in\mathbb{R}$, $Q\in\mathbb{C}$. Using formula (5.3) for the moment
map in $\mathbb{P}_{n}$,

(10.6)
$\big{\langle}\mu(\gamma(x)),X\big{\rangle}=\dfrac{Pe^{3|z|^{2}/2}|x_{0}|^{2}+Q\bar{x}_{0}x_{1}+\bar{Q}x_{0}\bar{x}_{1}-Pe^{-3|z|^{2}/2}|x_{1}|^{2}}{e^{3|z|^{2}/2}|x_{0}|^{2}+e^{-3|z|^{2}/2}|x_{1}|^{2}}$

up to a constant factor that will play no role. If we denote by $m_{r}$ the
rotation invariant probability measure on the circle
$K_{r}=\\{(1:re^{it})\in\mathbb{P}_{1}\mid t\in\mathbb{R}\\}$, $r>0$, (10.6)
gives

$\displaystyle\int_{K_{r}}\langle\mu\circ\gamma,X\rangle\,dm_{r}=$
$\displaystyle\dfrac{1}{2\pi}\int_{0}^{2\pi}\dfrac{Pe^{3|z|^{2}/2}+Qre^{it}+\bar{Q}re^{-it}-Pr^{2}e^{-3|z|^{2}/2}}{e^{3|z|^{2}/2}+r^{2}e^{-3|z|^{2}/2}}\,dt$
$\displaystyle=$
$\displaystyle\dfrac{Pe^{3|z|^{2}/2}-Pr^{2}e^{-3|z|^{2}/2}}{e^{3|z|^{2}/2}+r^{2}e^{-3|z|^{2}/2}}.$

Thus $\int(\mu\circ\gamma)\,dm_{r}=0$ if $r=e^{3|z|^{2}/2}$. By (10.5)
$\bar{v}(z,\cdot)$ in a neighborhood of $K_{r}$ agrees with $u_{0}(z,\cdot)$,
and $u_{0}(z,\cdot)$ is smooth and strictly $\omega$-subharmonic there
(Proposition 10.2). Hence Proposition 5.2 gives that $u_{0}(z,\cdot)$ is
indeed the unique minimizer of $E$.

###### Proof of Theorem 10.1.

In the setup of Proposition 10.4, let $B\subset A$ be a non–empty, relatively
compact open subset. It follows from [BK07, Theorem 2] and from Dini’s theorem
that there is a decreasing sequence of strictly $p^{*}\omega$-plurisubharmonic
functions $v^{j}\in C^{\infty}(B\times\mathbb{P}_{1})$ converging uniformly to
$v_{0}|B\times\mathbb{P}_{1}$. By Propositions 5.1, 10.4
$v^{j}(b,\cdot),v_{0}(b,\cdot)\in C_{\text{unique}}$ for each $b\in B$ (see
section 6 for this notation). Define
$u^{j}:B\times\mathbb{P}_{1}\to\mathbb{R}$ so that
$u^{j}(b,\cdot)\in{\mathcal{H}}(\omega)$ minimizes $E(v)$ among admissible
$v\geq v^{j}(b,\cdot)$, for each $b\in B$. Since $v^{j}$ decrease as $j$
grows, so do $u^{j}$. Hence Proposition 6.1 implies that the $u^{j}$ are
continuous and decrease to $u_{0}$ on $B\times\mathbb{P}_{1}$, locally
uniformly by Dini’s theorem. Not all $u^{j}$ can be
$p^{*}\omega$-plurisubharmonic, since their limit is not. This means that one
of the $v^{j}$ will do as $\bar{v}$ of the theorem.

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