# Asymptotic geometry of toric Kähler instantons

Brian Weber

###### Abstract.

The symplectic reduction of a complete toric Kähler manifold need not be
closed or be a polygon. In the scalar-flat Kähler case we establish natural
geometric criteria for these reductions to be closed, and classify the
asymptotic geometries that may occur.

##  1\. Introduction

A Kähler manifold $(M^{4},J,g)$ with a 2-torus action that preserves the
symplectic structure and metric is said to be a toric Kähler 4-manifold. If
the manifold is compact, the quotient is a compact Delzant polygon (see [8]
[18]), but if the manifold is only complete, its quotient need not be closed
and need not be a polygon; see the example in Section 6.1. We give necessary
and sufficient conditions for the quotient to be closed, and when $M^{4}$ is
scalar-flat we classify its possible metrics.

Throughout, $M^{4}=(M^{4},J,g,\mathcal{X}^{1},\mathcal{X}^{2})$ will be a
complete Kähler 4-manifold where $\mathcal{X}^{1}$, $\mathcal{X}^{2}$ are two
symplectomorphic Killing fields that commute. This means
$\mathcal{L}_{\mathcal{X}^{i}}\omega=\mathcal{L}_{\mathcal{X}^{i}}g=\mathcal{L}_{\mathcal{X}^{i}}J=0$,
and $[\mathcal{X}^{1},\mathcal{X}^{2}]=0$. The fields $\mathcal{X}^{1}$,
$\mathcal{X}^{2}$ may be generators of a torus action, but dealing with
Killing fields rather than action tori allows additional flexibility, not
least because we can take arbitrary linear combination of the fields without
worrying about whether it may come from a torus automorphism. It is convenient
to assume $M^{4}$ is simply connected; we may pass to the universal cover if
not. Because $\mathcal{L}_{\mathcal{X}^{i}}\omega=0$ we have
$di_{\mathcal{X}^{i}}\omega=0$, and therefore functions $\varphi^{1}$,
$\varphi^{2}$ exist with $\omega(\mathcal{X}^{i},\cdot)=-d\varphi^{i}$. These
are called the momentum or action coordinates on $M^{4}$. Then
$\nabla\varphi^{1}=-J\mathcal{X}^{1}$, $\nabla\varphi^{2}=-J\mathcal{X}^{2}$,
and the usual Nijenhuis relation implies that also
$[\nabla\varphi^{1},\nabla\varphi^{2}]=0$. To complete
$(\varphi^{1},\varphi^{2})$ into a coordinate system we first choose a single
leaf of the $\nabla\varphi^{1}$-$\nabla\varphi^{2}$ distribution, assign it
coordinates $(\theta_{1},\theta_{2})=(0,0)$, and then push $\theta_{1}$,
$\theta_{2}$ along the action created by the
$\mathcal{X}^{1}$-$\mathcal{X}^{2}$ fields; these are the cyclic or angle
coordinates. The coordinates $(\varphi^{1},\theta_{1},\varphi^{2},\theta_{2})$
are known as action-angle coordinates; see [3]. The Arnold-Liouville reduction
is

(1)
$\Phi:M^{4}\rightarrow\mathbb{R}^{2},\quad\quad\Phi(p)=\left(\varphi^{1}(p),\varphi^{2}(p)\right).$

The set $\Sigma^{2}=\Phi(M^{4})$ in the $\varphi^{1}$-$\varphi^{2}$ plane is
called the reduction of $M^{4}$. Because $g$ is invariant the reduction is
generically a Riemannian submersion and $Int(\Sigma^{2})$ inherits a
Riemannian metric, $g_{\Sigma}$. We call $(\Sigma^{2},g_{\Sigma})$ the metric
reduction of $M^{4}$.

Although $\Sigma^{2}$ need not be closed nor a polygon, we recover convexity.

######  Proposition 1.1 (cf. Proposition 3.3).

Assume $(M^{4},J,g,\mathcal{X}^{1},\mathcal{X}^{2})$ is geodesically complete.
Then its reduction $\Sigma^{2}$ is convex in the $\varphi^{1}$-$\varphi^{2}$
plane.

We use the term “boundary point” of $\Sigma^{2}$ to mean the extrinsic
boundary of $\Sigma^{2}$ in the topology of the coordinate plane (as opposed
to the intrinsic metric topology on $(\Sigma^{2},g_{\Sigma})$), and we use
$\partial\Sigma^{2}$ to indicate the set of all such points in the
$\varphi^{1}$-$\varphi^{2}$ plane. We define the “included boundary points” to
be $(\partial\Sigma^{2}){}_{I}=\partial\Sigma^{2}\cap\Sigma^{2}$ and the “non-
included boundary points” to be
$(\partial\Sigma^{2}){}_{N}=\partial\Sigma^{2}\setminus\Sigma^{2}$. If $p$ is
a point in the $\varphi^{1}$-$\varphi^{2}$ coordinate plane and $\epsilon>0$,
we use the notation $D_{p}(\epsilon)$ to mean the coordinate disk of radius
$\epsilon$ around $p$; then $p\in\partial\Sigma^{2}$ if and only if every
$D_{p}(\epsilon)$ contains both points of $\Sigma^{2}$ and
$\mathbb{R}^{2}\setminus\Sigma^{2}$. See Figure 1.

|  |
---|---|---
Figure 1.  (a) A typical reduction showing included and non-included boundary
points with (b) the included boundary points and (c) the non-included boundary
points.

######  Proposition 1.2 (Structure of $(\partial\Sigma^{2})_{I}$).

The included boundary points $p\in\Sigma^{2}\cap\partial\Sigma^{2}$ are
precisely those for which $\Phi^{-1}(p)\subset{}M^{4}$ consists of points
where the distribution $\\{\mathcal{X}^{1},\mathcal{X}^{2}\\}$ has rank 1 or
0. In this case, a coordinate disk $D_{p}(\epsilon)$, $\epsilon>0$ exists so
that, if the rank is $1$ at $\Phi^{-1}(p)$, then a linear function
$m(\varphi^{1},\varphi^{2})$ exists so that
$\Sigma^{2}\cap{}D_{p}(\epsilon)=\\{m\geq 0\\}\cap{}D_{p}(\epsilon)$, and the
rank is 0 then two linear functions $m_{1}$, $m_{2}$ exist so that
$\Sigma^{2}\cap{}D_{p}(\epsilon)=\\{m_{1}\geq 0\\}\cap\\{m_{2}\geq
0\\}\cap{}D_{p}(\epsilon)$.

######  Proposition 1.3 (Structure of $(\partial\Sigma^{2})_{N}$).

If $q\in\partial\Sigma^{2}\setminus\Sigma^{2}$ is a non-included boundary
point, then it is infinitely far away in the sense that if
$\\{p_{i}\\}\subset{}M^{4}$ is any sequence of points with
$\Phi(p_{i})\rightarrow{}q$ in the coordinate topology, then $p_{i}$ diverges
in $M^{4}$.

By Proposition 1.2, the included boundary $(\partial\Sigma^{2})_{I}$ consists
of segments, rays, and lines in the $\varphi^{1}$-$\varphi^{2}$ plane, joined
at vertex points. Proposition 1.4 says that such a segment, ray, or line
$l\in(\partial\Sigma^{2})_{I}$, is the image of the zero-set of a Killing
field. When $l$ is maximally extended, we call its preimage $L^{2}=\Phi(l)$ a
polar submanifold—this is an analogy with the fact that the zero-set of a
rotational Killing field on $\mathbb{S}^{2}$ consists of its north and south
pole. See Figure 2 for a depiction.

######  Proposition 1.4 (Polar submanifolds).

Assume $l\subset(\partial\Sigma^{2})_{I}$ is a maximally extended boundary
segment, ray, or line that has one or both of its termini at included vertex
points or non-included points. Then $L^{2}=\Phi^{-1}(l)$ is a holomorphically
embedded, geodesically complete codimension-2 submanifold with a Killing field
$\mathcal{X}$.

Any terminus of $l$ at a non-included point corresponds to a manifold end of
$L^{2}$. A terminus of $l$ at an included vertex point corresponds to a zero
of the Killing field on $L^{2}$.

Figure 2.  A polar submanifold $L^{2}\subset{}M^{2}$ over a segment
$l\subset\Sigma^{2}$, with distance function $r$ and momentum function
$\varphi^{1}$. Depicted is the possibility that $l$ has a terminus on a non-
included point.

We give geometric criteria on $M^{4}$ that forces the reduction $\Sigma^{2}$
to be closed. First we remark that the Killing field $\mathcal{X}$ on a polar
submanifold $L^{2}$ gives rise to a momentum function
$\varphi:L^{2}\rightarrow\mathbb{R}$ in the usual way:
$d\varphi=-i_{\mathcal{X}}\omega$. Then using $\varphi$ we have a standard
construction for a distance function $r:L^{2}\rightarrow\mathbb{R}^{2}$, this
being $dr=|d\varphi|^{-1}d\varphi$. The trajectories of $\nabla{}r$ are
perpendicular to the Killing field $\mathcal{X}$, and we call such a distance
function a radial distance function.

######  Theorem 1.5 (Criteria for closedness of $\Sigma^{2}$).

Whenever $L^{2}\subset{}M^{4}$ is a polar submanifold that has an unbounded
radial distance function $r$, assume one of the following holds:

  * •

The Killing field $\mathcal{X}$ on $L^{2}$ decays slowly (or not at all): a
constant $C_{1}>0$ exist so when $|r|$ is sufficiently large, then
$|\mathcal{X}|\geq{}C_{1}|r|^{-1}$,

  * •

The negative part of the Gaussian curvature $K$ of $L^{2}$ decays quickly:
when $|r|$ is sufficiently large, then $K\geq-2/r^{2}$.

Then the image $l\subset\Sigma^{2}$ of any polar submanifold $L^{2}$ is
closed, if $L^{2}$ has an unbounded distance function then $l$ has at most one
terminal point (which is an included point). If every component of
$\partial\Sigma^{2}$ has an included point, then $\Sigma^{2}$ is closed.

We address the case that $(\partial\Sigma^{2})_{I}$ is empty.

######  Theorem 1.6 (c.f. Theorem 4.4).

Assume $(M^{4},J,g,\mathcal{X}^{1},\mathcal{X}^{2})$ has non-negative scalar
curvature $s\geq 0$ and that the distribution
$\\{\mathcal{X}^{1},\mathcal{X}^{2}\\}$ is always rank 2 (this is the same as
$M^{4}\rightarrow\Sigma^{2}$ being a Riemannian submersion, or
$(\partial\Sigma^{2})_{I}=\varnothing$).

Then $M^{4}$ is flat $\mathbb{C}^{2}$, its Killing fields are translations,
and $(\Sigma^{2},g_{\Sigma})$ is flat $\mathbb{R}^{2}$.

Finally we consider the scalar-flat case, and classify these metrics based on
their natural asymptotic characteristics. We require a broader range of
asymptotic models that the usual ALE-ALF-ALG-ALH schema of [7] [11]. The ALF
asymptotic model in particular is much too rigid, due to the fact that it
requires not only quadratic curvature decay and cubic volume growth, but a
very particular collapsing condition: the Hopf circles on the $\mathbb{S}^{3}$
level-sets of the distance function must asymptotically be the fibers of a
collapsing F-structure (in the sense of Cheeger-Gromov [5]). This condition is
too inflexible to model the needed range of phenomena, as the examples of the
generalized Taub-NUT metrics demonstrates [23].

To create the needed range of asymptotic models, we use the following
definition.

######  Definition 1.

Assume $(M^{n},g)$ and $(\widetilde{M}{}^{n},\tilde{g})$ are Riemannian
manifolds and that a compact set $\widetilde{K}\subset\widetilde{M}{}^{n}$ and
covering map $\pi:\widetilde{M}{}^{4}\setminus\widetilde{K}\rightarrow{}M^{4}$
exist so that

(2)
$\left\|\pi^{*}g\,-\,\tilde{g}\right\|_{\tilde{g}}\;=\;O(\tilde{\rho}{}^{-l})\quad\text{for
$l>0$, or $o(1)$ for $l=0$}$

where $\tilde{\rho}$ is a distance function on $\widetilde{M}^{4}$. Then we
say $(M^{n},g)$ is _asymptotically modeled on
$(\widetilde{M}{}^{n},\tilde{g})$_ to degree $l$. If, in addition, the $M^{n}$
has Killing fields $\\{\mathcal{X}^{i}\\}$ and the model $\widetilde{M}{}^{4}$
has Killing fields $\\{\widetilde{\mathcal{X}}{}^{i}\\}$ so that
$\pi_{*}\widetilde{\mathcal{X}}{}^{i}=\mathcal{X}^{i}$, we say $M^{4}$ is
_equivariantly asymptotically modeled_ on $\widetilde{M}^{4}$ to degree $l$.

This is similar to typical definitions in the literature except we do not
require that the manifold’s curvature be similar to the model curvature.

---
Figure 3.  The five closed polygon types: the general case, parallel-ray case,
half-plane case, parallel-line case, and the plane.

Depicted in Figure 3 are the five kinds of non-compact closed polygons.

From [22], we know any manifold $M^{4}$ whose reduction is a half-plane has
only has a single nontrivial scalar-flat metric up to homothety: the
“exceptional half-plane” instanton metric. The fourth case, the strip, is not
well studied. The fifth case, the plane, is always flat by Theorem 1.6. The
general case and parallel-ray case give larger varieties of metrics, all of
which were described explicitly in the classification theorem of [22].

######  Theorem 1.7.

Assume $(M^{4},J,g,\mathcal{X}^{1},\mathcal{X}^{2})$ is a toric Kähler
4-manifold that is scalar-flat, geodesically complete, and has finite
topological type.

Then its reduction $\Sigma^{2}$ is closed if and only if $M^{4}$ is one of the
following:

  1. (1)

$M^{4}$ is asymptotically “general,” meaning it is one of the following:

     * •

_“Asymptotically ALE”_ , modeled on Euclidean space to order $1$

     * •

_“Asymptotically ALF”_ , modeled on the Taub-NUT to order $1$

     * •

_“Asymptotically ALF-like”_ , modeled on a generalized Taub-NUT with chirality
$k\in(-1,0)\cup(0,1)$, to order $1$

     * •

_“Asymptotically Exceptional”_ , modeled on a maximally chiral Taub-NUT (with
chirality $k=-1$ or $k=1$), to order $1$

  2. (2)

$M^{4}$ is “asymptotically equivariantly
$\mathbb{R}^{2}\times\mathbb{S}^{2}$,” meaning its asymptotic model is any of
the model metrics in (53) of Section 5.2, to order $0$,

  3. (3)

$M^{4}$ is an exceptional half-plane instanton of [22]

  4. (4)

$\Sigma^{2}$ is a closed strip

  5. (5)

$M^{4}$ is flat

To define the terms in Theorem 1.7, a metric is “asymptotically ALE” if it is
modeled on Euclidean space in the sense of Definition 1 (this differs from the
standard definition of “ALE” in that we do not require a curvature decay
condition). The “asymptotically ALF” metrics are modeled on the Taub-NUT
metric of the classic literature [15]. The “asymptotically ALF-like” metrics
are modeled on the family of generalized Taub-NUT metrics on $\mathbb{C}^{2}$
discovered by Donaldson [10]; in the language of [23] they have “chirality”
$k$ in $(-1,0)\cup(0,1)$. They have quadratic curvature decay $|\mathop{\rm
Rm}|=O(r^{-2})$, bounded $L^{2}$ curvature energy, cubic volume growth, and a
Killing field that is asymptotically bounded. They are not ALF, except in the
chirality 0 case, which is the classic Taub-NUT. The “asympotically
exceptional” metrics are modeled on the exceptional Taub-NUTs of [23], which
have maximum chirality $k=-1$ or $k=1$ and are complete scalar-flat Kähler
metrics on $\mathbb{C}^{2}$. However curvature does not decay, volume growth
is quartic, and $L^{2}(|\text{Ric}|)=\infty$. The aymptotic models of case (1)
metrics were studied carefully in [23], and the metrics themselves were
explicitly written down in [22].

The case (2) metrics, which all have polygon reductions with parallel rays,
were all written down and classified in [22]. This being the case, to date the
geometries of their asymptotic models not been studied (although in Section 5
we characterize some aspects of their geometries). The models themselves are
given explicitly in (53) of Section 5.2. At present we only mention that the
family of models all have half-strip reductions with three edge and parallel
rays, and are the simplest possible of the case (2) metrics; see Figure 5.

Case (3) consists of just one non-trivial metric; see [22]. Case (4), the
closed strip, is the only case where the metrics have not been classified. In
case (5), the flat case, is any metric whose reduction has no edges, and any
metric with compact reduction (which in the scalar-flat toric case are always
flat). Also in case (5) is flat $\mathbb{C}^{2}$ with one rotational and one
translational field, which has a half-plane reduction, and flat
$\mathbb{C}^{2}$ with two rotational fields, which has a quarter-plane
reduction.

In Section 6 we give some examples.

Remark. This paper is a companion to [21] and [22]. Using analytic results of
[21], the paper [22] classified the possible metrics on scalar-flat toric
Kähler 4-manifolds whose reductions are closed, with the exception of the
strip and plane (cases (4) and (5)). That paper begins with hypotheses on
$\Sigma^{2}$ rather than $M^{4}$, the main extrinsic assumption being
“Hypothesis A,” that the reduction $\Sigma^{2}$ is closed. In this paper we
have laid out what intrinsic conditions on the parent manifold $M^{4}$ are
equivalent to the extrinsic “Hypothesis A” of that paper.

##  2\. Preliminaries and Notation

This section recalls the theory of scalar-flat toric Kähler manifolds, and
sets notation for the paper. See [13] [1] [9] [10] [2] [4] and references
therein. The companion paper [22] has similar notation. Section 2.3 has a new
result, but otherwise this section’s material is well-known so we are brief.

###  2.1. Properties of the reduction

From $J\nabla\varphi^{i}=\frac{\partial}{\partial\theta_{i}}$, in action-angle
coordinates $(\varphi^{1},\varphi^{2},\theta_{1},\theta_{2})$ we can write

(3) $g\;=\;\left(\begin{array}[]{c|c}G_{ij}&0\\\ \hline\cr
0&G^{ij}\end{array}\right),\;J\;=\;\left(\begin{array}[]{c|c}0&-G^{ij}\\\
\hline\cr
G_{ij}&0\end{array}\right),\;\omega\;=\;\left(\begin{array}[]{c|c}0&-Id\\\
\hline\cr Id&0\end{array}\right).$

where $G_{ij}$ and its inverse $G^{ij}$ are the $2\times 2$ matrices
$G_{ij}=\big{<}\frac{\partial}{\partial\varphi^{i}},\frac{\partial}{\partial\varphi^{j}}\big{>}\;=\;\left<\nabla\theta_{i},\nabla\theta_{j}\right>$,
$G^{ij}=\left<\nabla\varphi^{i},\nabla\varphi^{j}\right>=\big{<}\frac{\partial}{\partial\theta_{i}},\frac{\partial}{\partial\theta_{j}}\big{>}$.
We have $\mathcal{X}^{1}=\frac{\partial}{\partial\theta_{1}}$,
$\mathcal{X}^{2}=\frac{\partial}{\partial\theta_{2}}$, and the generating
functions $\theta_{1}$, $\theta_{2}$ of the
$\mathcal{X}^{1}$-$\mathcal{X}^{2}$ action are pluriharmonic, meaning
$d(Jd\theta_{i})=0$; see Lemma 2.1 of [22]. Therefore conjugate pluriharmonic
functions exist, that we denote $\xi_{1}$, $\xi_{2}$, given by
$d\xi_{i}=Jd\theta_{i}$. This gives a holomorphic chart
$z_{1}=\xi_{1}+\sqrt{-1}\theta_{1}$, $z_{2}=\xi_{2}+\sqrt{-1}\theta_{2}$, with
a coordinate frame and coframe given by

(4) $\displaystyle\begin{aligned}
\frac{\partial}{\partial{z}_{1}}&\;=\;\frac{1}{2}\left(\nabla\varphi^{1}-\sqrt{-1}\,\mathcal{X}^{1}\right),\quad\frac{\partial}{\partial{z}_{2}}\;=\;\frac{1}{2}\left(\nabla\varphi^{2}-\sqrt{-1}\,\mathcal{X}^{2}\right)\\\
dz_{1}&\;=\;d\xi_{1}+\sqrt{-1}\,d\theta_{1},\quad\quad\quad\;\;dz_{2}\;=\;d\xi_{2}+\sqrt{-1}\,d\theta_{2}.\end{aligned}$

In this holomorphic frame the Hermitian metric is
$h^{i\bar{\jmath}}=\big{<}\frac{\partial}{\partial{}z_{i}},\frac{\partial}{\partial{}\bar{z}_{j}}\big{>}=\frac{1}{2}G^{ij}$.
Because the $\xi_{i}$ are $\mathcal{X}^{i}$-invariant, we have
$\xi_{i}=\xi_{i}(\varphi^{1},\varphi^{2})$. Consequently the distributions
$\\{\nabla\varphi^{1},\nabla\varphi^{2}\\}$ and
$\\{\nabla\xi_{1},\nabla\xi_{2}\\}$ coincide, and both are integrable as
$[\nabla\varphi^{i},\nabla\varphi^{j}]=0$.

Because the reduction
$\Phi(\varphi^{1},\theta_{1},\varphi^{2},\theta_{2})=(\varphi^{1},\varphi^{2})$
is $\mathcal{X}^{1}$-$\mathcal{X}^{2}$ invariant, this is generically a
Riemannian submersion, and we have a metric and complex structure at interior
points of $\Sigma^{2}$. Abbreviating
$\mathcal{V}=det(G^{ij})=|\nabla\varphi^{1}|^{2}|\nabla\varphi^{2}|^{2}-\big{<}\nabla\varphi^{1},\nabla\varphi^{2}\big{>}{}^{2}$,

(5) $g_{\Sigma}=(G_{ij})\quad\text{and}\quad
J_{\Sigma}\;=\;\mathcal{V}^{-\frac{1}{2}}\left(\begin{array}[]{cc}\left<\mathcal{X}{}^{1},\,\mathcal{X}{}^{2}\right>&-|\mathcal{X}{}^{1}|^{2}\\\
|\mathcal{X}{}^{2}|^{2}&-\left<\mathcal{X}{}^{1},\,\mathcal{X}{}^{2}\right>\end{array}\right).$

The metric $g_{\Sigma}$ is inherited from the parent manifold $(M^{4},g)$, but
the complex structure $J_{\Sigma}$ is not. Rather, it is the (dualization of)
the Hodge-* on $(\Sigma^{2},g_{\Sigma})$, and is obtained from
$J_{\Sigma}{}^{i}_{j}=(det\,g_{ij})g_{ik}\epsilon^{jk}$ where $\epsilon$ is
the Levi-Civita symbol. Citing Proposition 2.5 of [22], we have degenerate-
elliptic equations for $\varphi^{1}$, $\varphi^{2}$

(6) $d(\mathcal{V}^{-\frac{1}{2}}J_{\Sigma}{d}\varphi^{k})=0.$

From Proposition 2.2 of [22], the Ricci and scalar curvatures on $M^{4}$ are

(7) $\displaystyle\begin{aligned}
\rho\;=\;-\sqrt{-1}\partial\bar{\partial}\log\,\mathcal{V},\quad\quad
s\;=\;-\triangle\log\,\mathcal{V}.\end{aligned}$

The scalar curvature $s$ on $M^{4}$ can be computed on $\Sigma^{2}$, where

(8)
$\triangle_{\Sigma}\mathcal{V}^{\frac{1}{2}}+\frac{1}{2}s\mathcal{V}^{\frac{1}{2}}\;=\;0.$

This is equivalent to the 4th-order Abreu equation for scalar curvature
$s=-\frac{1}{2}\frac{\partial^{2}G^{ij}}{\partial\varphi^{i}\varphi^{j}}$ (see
Eq. (10) of [1]), after a Trudinger-Wang style reduction [20]. See for example
Corollary 2.4 of [22]. Therefore when the $M^{4}$ scalar curvature is zero
there exists a natural analytic coordinate $z=x+\sqrt{-1}y$ on $\Sigma^{2}$
given by $y=\sqrt{\mathcal{V}}$, and $x$ is an harmonic conjugate of $y$.
Because $y=\sqrt{\mathcal{V}}$ is a volume, we call the isothermal system
$(x,y)$ the volumetric normal coordinates and the analytic coordinate
$z=x+\sqrt{-1}y$ the volumetric normal function. From Proposition 1.1 of [22],
when $\Sigma^{2}$ is closed and has connected boundary, the map to the upper
half-space

(9) $z:\Sigma^{2}\rightarrow\\{y\geq 0\\}$

is a biholomorphism with $\partial\Sigma^{2}$ mapping bijectively onto the
half-space boundary $\\{y=0\\}$. In [23], the Ricci form $\rho$ and the Weyl
tensor on $M^{4}$ were both computed from data on its reduction $\Sigma^{2}$.
For example letting $K_{\Sigma}$ be the Gaussian curvature on
$(\Sigma^{2},g_{\Sigma})$, Proposition A.2 of [23] states
$|W^{-}|^{2}=24(K_{\Sigma})^{2}$.

###  2.2. The Kähler and symplectic potential

We describe the so-called Kähler and symplectic potentials on $M^{4}$. Let
$P^{2}\subset{}M^{4}$ be the union of the polar submanifolds, so in particular
$\Phi:P^{2}\rightarrow\Sigma^{2}$ maps $P$ onto $(\partial\Sigma^{2})_{I}$ and
$\Phi:M^{4}\setminus{}P^{2}\rightarrow\Sigma^{2}$ is a Riemannian submersion
onto the interior of $\Sigma^{2}$. Any leaf of the
$\\{\nabla\varphi^{1},\nabla\varphi^{2}\\}$-distribution maps bijectively onto
$\Sigma^{2}$, and $M^{4}\setminus{}P^{2}$ is diffeomorphic to the product of
$Int(\Sigma^{2})$ with any of the $\mathcal{X}^{1}$-$\mathcal{X}^{2}$ leafs.
Because $Int(\Sigma^{2})$ is contractable, we can write
$\omega=\sqrt{-1}\partial\bar{\partial}{F}$ for some smooth pseudoconvex
function ${F}={F}(\xi^{1},\xi^{2})$ called the Kähler potential of $g$.
Because ${F}$ is pseudoconvex and the distribution
$\\{\nabla\xi^{1},\nabla\xi^{2}\\}$ is Legendrian—meaning
$\omega(\nabla\xi^{i},\nabla\xi^{j})=0$ for $i,j=1,2$—we have that ${F}$ is a
_convex_ function of $\xi^{1},\xi^{2}$. From
$d\varphi^{i}=\omega\big{(}\frac{\partial}{\partial\theta_{i}},\,\big{)}$ and
$d\xi_{i}=Jd\theta_{i}$,

(10)
$\varphi^{1}\;=\;\frac{\partial{F}}{\partial\xi_{1}},\quad\varphi^{2}\;=\;\frac{\partial{F}}{\partial\xi_{2}},$

see for example Eq. (4.4) of [13]. These are precisely the hodographs usually
seen with the Legendre transform; indeed because $F=F(\xi_{1},\xi_{2})$ is
convex it has a Legendre transform $G=G(\varphi^{1},\varphi^{2})$, called the
manifold’s symplectic potential. Referring to the $2\times 2$ matrix $G_{ij}$
above, it turns out that
$G_{ij}=\frac{\partial^{2}{G}}{\partial\varphi^{i}\partial\varphi^{j}}$. See
[13] for a justification of this fact. We remark that
$G(\varphi^{1},\varphi^{2})$ is a smooth convex function on
$M^{4}\setminus{}P^{2}$, and passes to a smooth convex function on
$Int(\Sigma^{2})$.

###  2.3. A conformal change of the metric

We compute the Gaussian curvature $K_{\Sigma}$ of $\Sigma^{2}$ and create a
conformal-change formula (17) that will be used in Section 4. We use
“$s_{\Sigma}$” to indicate the scalar curvature of $(\Sigma^{2},g_{\Sigma})$,
and “$s$” to indicate the scalar curvature on the parent 4-manifold
$(M^{4},g)$, which is $\mathcal{X}^{1}$-$\mathcal{X}^{2}$ invariant so passes
to a function on $\Sigma^{2}$. Of course $K_{\Sigma}=\frac{1}{2}s_{\Sigma}$.
Due to the fact that $G_{ij}$ is a matrix of second partial derivatives, the
metric on $g_{\Sigma}=(G_{ij})$ on $\Sigma^{2}$ obeys

(11)
$\displaystyle\frac{\partial{G}_{ij}}{\partial\varphi^{k}}\;=\;\frac{\partial{G}_{ik}}{\partial\varphi^{j}}.$

Using
$\mathcal{V}=\text{det}(G^{ij})=|\nabla\varphi^{1}|^{2}|\nabla\varphi^{2}|^{2}-\big{<}\nabla\varphi^{1},\nabla\varphi^{2}\big{>}{}^{2}$,
the Christoffel symbols are

(12) $\displaystyle\begin{aligned}
\Gamma_{ij}^{k}\;=\;\frac{1}{2}\frac{\partial{G}_{ij}}{\partial\varphi^{s}}G^{sk},\quad\quad\Gamma^{k}\;\stackrel{{\scriptstyle\textit{def}}}{{=}}\;G^{ij}\Gamma_{ij}^{k}\;=\;-G^{ks}\frac{\partial}{\partial\varphi^{s}}\log\mathcal{V}^{\frac{1}{2}}.\end{aligned}$

The usual formula for scalar curvature in terms of Christoffel symbols is

(13) $\displaystyle
s_{\Sigma}\;=\;G^{ij}\frac{\partial\Gamma_{ij}^{s}}{\partial\varphi^{s}}-G^{ij}\frac{\partial\Gamma_{sj}^{s}}{\partial\varphi^{i}}+G_{st}\Gamma^{s}\Gamma^{t}\,-\,G^{is}G^{jt}G_{kl}\Gamma_{ij}^{k}\Gamma_{st}^{l}.$

From (11) and (12) we find that
$\frac{\partial}{\partial\varphi^{s}}\left(G^{ij}\Gamma_{ij}^{s}\right)-\frac{\partial}{\partial\varphi^{i}}\left(G^{ij}\Gamma_{sj}^{s}\right)=0$,
so

(14) $\displaystyle\begin{aligned}
s_{\Sigma}&\;=\;\frac{\partial}{\partial\varphi^{s}}\left(G^{ij}\Gamma_{ij}^{s}\right)-\frac{\partial}{\partial\varphi^{i}}\left(G^{ij}\Gamma_{sj}^{s}\right)+G^{is}G^{jt}G_{kl}\Gamma_{ij}^{k}\Gamma_{st}^{l}-G_{st}\Gamma^{s}\Gamma^{t}\\\
&\;=\;G^{is}G^{jt}G_{kl}\Gamma_{ij}^{k}\Gamma_{st}^{l}-G_{st}\Gamma^{s}\Gamma^{t}\\\
&\;=\;\left|\Gamma_{ij}^{k}\right|^{2}\,-\,\left|\Gamma^{k}\right|^{2}\;=\;\left|\Gamma_{ij}^{k}\right|^{2}\,-\,\big{|}\nabla\log\mathcal{V}^{\frac{1}{2}}\big{|}^{2}.\end{aligned}$

Now modify the metric by
$\widetilde{g}_{\Sigma}=\mathcal{V}^{\frac{1}{2}}{g}_{\Sigma}$. The usual
conformal-change formula gives

(15) $\displaystyle\begin{aligned}
\widetilde{s}_{\Sigma}&\;=\;\mathcal{V}^{-\frac{1}{2}}\left(s_{\Sigma}\,-\,\triangle_{\Sigma}\log\mathcal{V}^{\frac{1}{2}}\right)\\\
&\;=\;\mathcal{V}^{-\frac{1}{2}}\left(\left|\Gamma_{ij}^{k}\right|^{2}\,-\,\big{|}\nabla\log\mathcal{V}^{\frac{1}{2}}\big{|}^{2}\,-\,\triangle_{\Sigma}\log\mathcal{V}^{\frac{1}{2}}\right).\end{aligned}$

But
$\triangle_{\Sigma}\log\mathcal{V}^{\frac{1}{2}}+\big{|}\nabla\log\mathcal{V}^{\frac{1}{2}}\big{|}{}^{2}=\mathcal{V}^{-\frac{1}{2}}\triangle_{\Sigma}\mathcal{V}^{\frac{1}{2}}$,
so along with (7) we see the conformally related scalar curvature is simply

(17)
$\displaystyle\begin{array}[]{ll}\widetilde{s}_{\Sigma}&\;=\;\mathcal{V}^{-\frac{1}{2}}\left(\left|\Gamma_{ij}^{k}\right|^{2}\,-\,\mathcal{V}^{-\frac{1}{2}}\triangle_{\Sigma}\mathcal{V}^{\frac{1}{2}}\right)\;=\;\mathcal{V}^{-\frac{1}{2}}\left(\left|\Gamma_{ij}^{k}\right|^{2}\,+\,\frac{1}{2}s\right).\end{array}$

In particular $s\geq 0$ on $M^{4}$ implies $\widetilde{s}_{\Sigma}\geq 0$.
This is the central observation behind the proof of Theorem 1.6.

##  3\. Characteristics of the reduction

We characterize points of $\partial\Sigma^{2}$ and prove Propositions 1.1,
1.2, 1.3, and 1.4.

###  3.1. Structure included and non-included boundary points

In the $\varphi^{1}$-$\varphi^{2}$ plane it is convenient to refer to a
coordinate disk around a point $p=(p^{1},p^{2})$ by

(18)
$D_{p}(r)\;=\;\left\\{\,(\varphi^{1},\varphi^{2})\in\mathbb{R}^{2}\;\big{|}\;\sqrt{(\varphi^{1}-p^{1})^{2}+(\varphi^{2}-p^{2})^{2}}<r\,\right\\}.$

We remark that $\Phi^{-1}(D_{p}(r))\subseteq{}M^{4}$ is an equivariant
neighborhood of any $\tilde{p}\in\Phi^{-1}(p)\subset{}M^{4}$. Proposition 3.1
is well-known in the case of compact symplectic manifolds [8], [18]; we give a
geometrically-flavored proof in the Kähler setting.

######  Proposition 3.1 (Structure of $\Sigma^{2}$ near included boundary
points).

Assume $p=(p^{1},p^{2})$ is a point of $(\partial\Sigma^{2})_{I}$. Then on
$\Phi^{-1}(p)$ the distribution $\\{\mathcal{X}^{1},\mathcal{X}^{2}\\}$ has
rank $0$ or $1$, and a coordinate disk $D_{p}(\epsilon)$ exists so that one of
the following holds:

  1. (1)

the distribution $\\{\mathcal{X}^{1},\mathcal{X}^{2}\\}$ has rank 1, and a
linear function
$m(\varphi^{1},\varphi^{2})=\alpha(\varphi^{1}-p^{1})+\beta(\varphi^{2}-p^{2})$
exists so that $D_{p}(\epsilon)\cap\Sigma^{2}=D_{p}(\epsilon)\cap\\{m\geq
0\\}$.

  2. (2)

the distribution $\\{\mathcal{X}^{1},\mathcal{X}^{2}\\}$ has rank 0, then two
linear functions
$m_{0}(\varphi^{1},\varphi^{2})=\alpha_{0}(\varphi^{1}-p^{1})+\beta_{0}(\varphi^{2}-p^{2})$,
$m_{1}(\varphi^{1},\varphi^{2})=\alpha_{1}(\varphi^{1}-p^{1})+\beta_{1}(\varphi^{2}-p^{2})$
exist so that $D_{p}(\epsilon)\cap\Sigma^{2}=D_{p}(\epsilon)\cap\\{m_{0}\geq
0\\}\cap\\{m_{1}\geq 0\\}$.

In particular, $\Sigma^{2}\cap{}D_{p}(\epsilon)$ is convex.

###### Proof.

Let $p\in{}M^{4}$ be any point in $M^{4}$ where
$\\{\mathcal{X}^{1},\mathcal{X}^{2}\\}$ has rank 2. Then the distribution
$\\{\nabla\varphi^{1},\nabla\varphi^{2}\\}$ also has rank 2, so therefore near
$p$ the projection $\Phi:M^{4}\rightarrow\Sigma^{2}$ is a Riemannian
submersion, and maps neighborhoods of $p$ to neighborhoods $\Phi(p)$. Thus if
the rank of the distribution $\\{\nabla\varphi^{1},\nabla\varphi^{2}\\}$ is
$2$, then $\Phi(p)\notin{}(\partial\Sigma^{2})_{I}$. Therefore
$\Phi(p)\in{}(\partial\Sigma^{2})_{I}$ implies the rank is 0 or 1.

Now assume the rank of $\\{\mathcal{X}^{1},\mathcal{X}^{2}\\}$ is 1; then
numbers $\alpha,\beta$ exist so the Killing field
$\mathcal{X}=\alpha\mathcal{X}^{1}+\beta\mathcal{X}^{2}$ vanishes at $p$. The
zero-set $\\{\mathcal{X}=0\\}$ of any Killing field is a totally geodesic
submanifold of even dimension. Because the rank of the distribution at $p$ is
$1$, and because the Killing fields commute, a second non-zero Killing field
exists that preserves the zero-set of $\mathcal{X}$, so the zero-set
$\\{\mathcal{X}=0\\}$ through $p$ has dimension at least $1$. Recalling that
the dimension of any zero-locus of any Killing field on any manifold has even
dimension, the zero-set $L^{2}=\\{\mathcal{X}=0\\}$ is 2-dimensional. Because
$\mathcal{X}$ is a holomorphic vector field also $L^{2}$ is a complex
submanifold.

Setting $\varphi=\alpha\varphi^{1}+\beta\varphi^{2}$, we have that
$\nabla\varphi=-J\mathcal{X}=0$ on $L^{2}$; thus $L^{2}$ consists of critical
points of the function $\varphi$. To see that $L^{2}$ is either a local max or
a min of $\varphi$ (not, for example, a saddle), consider any short segment
perpendicular to $L^{2}$, and consider its orbit via the field $\mathcal{X}$.
Since $\mathcal{X}$ is zero on $L^{2}$, one endpoint of the segment will
remain fixed on $L^{2}$ while the other endpoint creates a circle; the orbit
of the segment therefore forms a disk. Because $\varphi$ is also
$\mathcal{X}$-invariant, when restricted to this disk the level-sets of
$\varphi$ are the integral curves of $\mathcal{X}$, which are therefore
circles. Because its level-sets are compact, $\varphi$ either has a max or a
min at the disk’s center. Along $L^{2}$ the function $\varphi$ remains
constant, so therefore the entire locus $L^{2}$ is a local max or min of
$\varphi$. Replacing our choice of $\alpha$, $\beta$ with $-\alpha$, $-\beta$
if necessary, we may assume $L^{2}$ is a local minimum of $\varphi$. Finally,
from $\varphi^{1}(p)=p^{1}$ and $\varphi^{2}(p)=p^{2}$, we have
$\alpha(\varphi^{1}-p^{1})+\beta(\varphi^{2}-p^{2})\geq 0$ in a neighborhood
of $L^{2}$ with equality if and only if $(\varphi^{1},\varphi^{2})\in{}L^{2}$.

In the case that $\\{\mathcal{X}^{1},\mathcal{X}^{2}\\}$ has rank 0, then it
is the intersection of two totally-geodesic submanifolds on which the
distribution has rank $1$. Following the above argument twice, once for each
submanifold, leads to two linear relations. ∎

In the case of complete Kähler manifolds, the reduction $\Sigma^{2}$ might not
contain all of its boundary points. We show that when the parent $M^{4}$ is
complete, these non-included points are “points at infinity” in a precise
sense.

######  Proposition 3.2 (Structure of $\Sigma^{2}$ near non-included boundary
points).

Assume $(M^{4},J,g,\mathcal{X}^{1},\mathcal{X}^{2})$ is geodesically complete,
and let $q\in(\partial\Sigma^{2})_{N}$ be a non-included boundary point. Then
if $\\{p_{i}\\}\subset{}M^{4}$ is any sequence of points so that
$\Phi(p_{i})\rightarrow{}q$ in the coordinate topology on $\mathbb{R}^{2}$,
then $p_{i}$ diverges in the metric topology on $M^{4}$.

###### Proof.

First, if $K\subset{}M^{4}$ is any compact set, then its image $\Phi(K)$ is
compact in the coordinate topology on $\mathbb{R}^{2}$ by continuity of
$\Phi$. Given such a compact set $K\subset{}M^{4}$ we show that at most
finitely many $p_{i}$ lie in $K$. If not, there is a subsequence (still
denoted $p_{i}$) that remains within $K$ but so that $\Phi(p_{i})$ converges
to $p$ in the $\mathbb{R}^{2}$-topology. But then $\Phi(p_{i})$ remains within
the compact set $\Phi(K)\in\Sigma^{2}$ so a subsequence converges to some
point in $\Phi(K)$. But this is impossible, because $\Phi(p_{i})$ converges to
$q\notin\Sigma^{2}$. ∎

###  3.2. Convexity of the reduction

In the purely symplectic setting, if $M^{4}$ is non-compact it is impossible
to recover the classic result of [8] that reductions are polygons when $M^{4}$
is compact. But if $M^{4}$ has a complete Kähler metric we recover convexity.
The argument is based on the following very basic computation: letting
$\textbf{a}=(a^{1},a^{2})$ and $\textbf{b}=(b^{1},b^{2})$ be points in the
$\varphi^{1}$-$\varphi^{2}$ plane and $\gamma(t)=(1-t)\textbf{a}+t\textbf{b}$
the segment between them, then
$\frac{d}{dt}=(b^{1}-a^{1})\frac{\partial}{\partial\varphi^{1}}+(b^{2}-a^{2})\frac{\partial}{\partial\varphi^{2}}$
and because the coefficients are constant we have

(19)
$\frac{d^{2}}{dt^{2}}\;=\;(b^{1}-a^{1})^{2}\frac{\partial^{2}}{(\partial\varphi^{1})^{2}}+2(b^{1}-a^{1})(b^{2}-a^{2})\frac{\partial^{2}}{\partial\varphi^{1}\partial\varphi^{2}}+(b^{2}-a^{2})^{2}\frac{\partial^{2}}{(\partial\varphi^{2})^{2}}.$

Because $g_{ij}=\frac{\partial^{2}G}{\partial\varphi^{i}\varphi^{j}}$, along
this segment we have

(20) $\displaystyle\frac{d^{2}G}{dt^{2}}$
$\displaystyle=(b^{1}-a^{1})^{2}\frac{\partial^{2}G}{(\partial\varphi^{1})^{2}}+2(b^{1}-a^{1})(b^{2}-a^{2})\frac{\partial^{2}G}{\partial\varphi^{1}\partial\varphi^{2}}+(b^{2}-a^{2})^{2}\frac{\partial^{2}G}{(\partial\varphi^{2})^{2}}$
$\displaystyle=\;g\left(\frac{d}{dt},\,\frac{d}{dt}\right)\;>\;0.$

######  Proposition 3.3 (cf. Proposition 1.1).

Assume the metric on $(M^{4},J,g,\mathcal{X}^{1},\mathcal{X}^{2})$ is
complete. Then its reduction $\Sigma^{2}$ is convex.

###### Proof.

Let ${\boldsymbol{\varphi}}_{0},{\boldsymbol{\varphi}}_{1}\in\mathbb{R}^{2}$
be points within $Int(\Sigma^{2})$ under the condition that the line segment
$\gamma(t)=(1-t){\boldsymbol{\varphi}}_{0}+t{\boldsymbol{\varphi}}_{1}$,
$t\in[0,1]$ between them remains in $\Sigma^{2}$. From (20),

(21)
$g\left(\frac{d}{dt},\,\frac{d}{dt}\right)\;=\;\frac{d^{2}G}{dt^{2}},\quad\text{and}\quad\left|\frac{d}{dt}\right|\;=\;\sqrt{\frac{d^{2}G}{dt^{2}}}.$

If the segment lies within $Int(\Sigma^{2})$, its length is bounded: from
Hölder’s inequality

(22) $\displaystyle Len(\gamma)$
$\displaystyle\;=\;\int_{0}^{1}\left|\frac{d}{dt}\right|\,dt\;=\;\int_{0}^{1}\left(\frac{d^{2}G}{dt^{2}}\right)^{\frac{1}{2}}\,dt$
$\displaystyle\;\leq\;\left(\int_{0}^{1}\frac{d^{2}G}{dt^{2}}\,dt\right)^{\frac{1}{2}}\;=\;\left(\frac{dG}{dt}({\boldsymbol{\varphi}}_{1})\;-\;\frac{dG}{dt}({\boldsymbol{\varphi}}_{0})\right)^{\frac{1}{2}}$

so the length of the line segment is bounded in terms of its endpoints.
Because $Int(\Sigma^{2})$ is connected we can vary
${\boldsymbol{\varphi}}_{0}$, ${\boldsymbol{\varphi}}_{1}$ throughout
$Int(\Sigma^{2})$. If $\Sigma^{2}$ is non-convex then there exists some point
$p\in(\partial\Sigma^{2})_{N}$ that has a supporting segment that is in the
closure of $\Sigma^{2}$ and has endpoints in $\Sigma^{2}$. By Proposition 3.2
the segment $\gamma$ must have infinite length. However (22) gives a uniform
upper bound on the length of $l$. From this contradiction we conclude that
$\Sigma^{2}$ is convex. ∎

###  3.3. Structure of interior points

######  Lemma 3.4.

If $p\in\Sigma^{2}$ then its preimage $\Phi^{-1}(p)\subset{}M^{4}$ is a
connected submanifold with tangent bundle
$\text{span}\\{\mathcal{X}^{1},\mathcal{X}^{2}\\}\subset{}TM^{4}$ restricted
to $\Phi^{-1}(p)$. Finally, $p\in{}Int(\Sigma^{2})$ if and only if
$\\{\mathcal{X}^{1},\mathcal{X}^{2}\\}$ has rank $2$ on $\Phi^{-1}(p)$, if and
only if a coordinate disk $D_{p}(\epsilon)$ exists so
$\Phi:\Phi^{-1}(D_{p}(\epsilon))\rightarrow{}D_{p}(\epsilon)$ is a Riemannian
submersion.

###### Proof.

Because $\mathcal{X}^{1}$, $\mathcal{X}^{2}$ are Killing, the orbit of any
point $\tilde{p}\in{}M^{4}$ under $\mathcal{X}^{1}$, $\mathcal{X}^{2}$ is a
manifold with tangent bundle being the distribution
$\\{\mathcal{X}^{1},\mathcal{X}^{2}\\}$ restricted to the orbit. Because
$\Sigma^{2}$ is $M^{4}$ modulo the action of these fields, the image under
$\Phi$ of the orbit of any point $\tilde{p}\in{}M^{4}$ is a single point in
$\Sigma^{2}$. The distribution $span\\{\nabla\varphi^{1},\nabla\varphi^{2}\\}$
has the same rank as $span\\{\mathcal{X}^{2},\mathcal{X}^{2}\\}$ and is
perpendicular to it, so the pushforward $\Phi_{*}$ is an isometry on
$span\\{\nabla\varphi^{1},\nabla\varphi^{2}\\}$. Therefore, because
$span\\{\nabla\varphi^{1},\nabla\varphi^{2}\\}$ has rank 2, $\Phi$ is a
Riemannian submersion on a neighborhood of $p$.

It remains to show that a point of $\Sigma^{2}$ has connected pre-image.
Assume a point $p\in\Sigma^{2}$ has two components
$T_{1},T_{2}\subset\Phi^{-1}(\Sigma^{2})$. Let $\gamma(s)$, $s\in[0,1]$ be a
geodesic from $T_{1}$ to $T_{2}$ that represents a shortest path between these
submanifolds. The vector $\dot{\gamma}$ is pependicular to $T_{1}$ and $T_{2}$
at the two points of contact:
$\left<\dot{\gamma}(0),\mathcal{X}^{i}\right>=\left<\dot{\gamma}(1),\mathcal{X}^{i}\right>=0$
for both fields $\mathcal{X}^{1}$, $\mathcal{X}^{2}$. To show
$\left<\dot{\gamma}(t),\mathcal{X}^{i}\right>=0$ for all $t$, we compute

(23) $\displaystyle\frac{d^{2}}{ds^{2}}\left<\mathcal{X},\dot{\gamma}\right>$
$\displaystyle=\dot{\gamma}\dot{\gamma}\left<\mathcal{X},\,\dot{\gamma}\right>=\left<\nabla_{\dot{\gamma}}\nabla_{\dot{\gamma}}\mathcal{X},\,\dot{\gamma}\right>=\left<\mathop{\rm
Rm}(\dot{\gamma},\mathcal{X})\dot{\gamma},\,\dot{\gamma}\right>=0.$

In particular $\dot{\gamma}\in{}span\\{\nabla\varphi^{1},\nabla\varphi^{2}\\}$
so the path $s\mapsto\Psi(\gamma(s))$ in $\Sigma^{2}$ has the same length as
the path $s\mapsto\gamma$ in $M^{4}$. Given any $s\in[0,1]$, draw the line
segment in $\Sigma^{2}$

(24) $\eta_{s}(t)=(1-t)\,\Phi(\gamma(0))\,+\,t\,\Phi(\gamma(s))$

from $\Phi(\gamma(0))$ to $\Phi(\gamma(s))$. Then $(s,t)\mapsto\eta_{s}(t)$ in
$\Sigma^{2}$ is a spanning surface for the closed loop $\Phi(\gamma(s))$.

We can lift each segment $t\mapsto\eta_{s}(t)$ to a path in $M^{4}$ in the
following way. In action-angle coordinates, $\gamma(s)$ has the expression
$\gamma(s)=(\varphi^{1}(s),\varphi^{2}(s),c_{1},c_{2})$ due to the fact that,
as we have shown, the angle coordinates
$(\theta_{1},\theta_{2})=(c_{1},c_{2})$ remain constant along $\gamma$. Then
for each $s$ the path (24) lifts directly to $M^{4}$ by using the
$(\varphi^{1},\varphi^{2})$-coordinates from its expression in $\Sigma^{2}$
and assigning it the $(\theta_{1},\theta_{2})$ coordinates $(c_{1},c_{2})$.
This produces a continuous lift of $(s,t)\rightarrow\eta_{s}(t)$ in
$\Sigma^{2}$ to a 2-disk $(s,t)\mapsto\tilde{\eta}_{s}(t)$ in $M^{4}$.

Because the path $t\mapsto\eta_{t}(1)$ is a trivial path in $\Sigma^{2}$,
therefore its lift $t\mapsto\tilde{\eta}_{t}(1)$ remains on a single leaf of
the $\\{\mathcal{X}{}^{1},\mathcal{X}{}^{2}\\}$ distribution. This contradicts
the assumption that $\gamma(0)$ and $\gamma(1)$ are on different leaves of
this distribution. ∎

##  4\. Reductions without an included boundary

We prove Theorem 1.6, that if $s\geq 0$ and
$(\partial\Sigma^{2})_{I}=\varnothing$ then $M^{4}$ is flat. We do so in the
following stages. Because $\Phi:M^{4}\rightarrow\Sigma^{2}$ is now a global
Riemannian submersion by Lemma 3.4, in particular $(\Sigma^{2},g_{\Sigma})$ is
complete. Recalling the quantity $\mathcal{V}$ from Section 2.1, we first show
that if $\mathcal{V}$ is constant, the metric is flat. By (8) the scalar
curvature condition $s\geq 0$ is equivalent to
$\triangle_{\Sigma}\sqrt{\mathcal{V}}\leq 0$; we use a classical Liouville
theorem to show this implies $\mathcal{V}$ is constant.

But to use the classical Liouville theorem, me must determine that the
complete manifold $(\Sigma^{2},g_{\Sigma})$ is actually biholomorphic to
$\mathbb{C}$, or, equivalently, that it is simply connected and parabolic in
the sense of potential theory [6]. To do this, we show that if
$(\Sigma^{2},g_{\Sigma})$ is complete then the conformal change from §2.3
gives a metric $\widetilde{g}_{\Sigma}$ that actually remains complete, and,
crucially, gives a manifold $(\Sigma^{2},\widetilde{g}_{\Sigma})$ with non-
negative Gaussian curvature. Thus $\Sigma^{2}$ is biholomorphic with
$\mathbb{C}$, by the Cheng-Yau criterion for parabolicity. The classical
Liouville theorem then forces $\mathcal{V}$ to be constant, and we conclude
that $g_{\Sigma}$ is a flat metric.

######  Lemma 4.1.

Assume $(\Sigma^{2},g_{\Sigma})$ is geodesically complete. If $\mathcal{V}$ is
constant, then $(\Sigma^{2},g_{\Sigma})$ is biholomorphic to $\mathbb{C}$, and
$g_{\Sigma}$ is a flat metric with constant coefficients when expressed in
$\varphi^{1}$, $\varphi^{2}$ coordinates.

###### Proof.

With $\mathcal{V}$ constant, then
$d(\mathcal{V}^{-\frac{1}{2}}J_{\Sigma}{d}\varphi^{k})=0$ from (6) gives
$d(J_{\Sigma}{}d\varphi^{k})=0$, so $\varphi^{1}$, $\varphi^{2}$ are harmonic
on $\Sigma^{2}$. Therefore each determines an analytic function which we can
write $z=\varphi^{1}+\sqrt{-1}\eta^{1}$, $w=\varphi^{2}+\sqrt{-1}\eta^{2}$
(where $d\eta^{k}=J_{\Sigma}{}d\varphi^{k}$). Away from possible critical
points these are each a holomorphic coordinate on $\Sigma^{2}$, and

(25) $\displaystyle\begin{aligned}
\frac{d}{dz}&\;=\;\frac{1}{2|\nabla\varphi^{1}|^{2}}\left(\nabla\varphi^{1}\,+\,\sqrt{-1}J_{\Sigma}\nabla\varphi^{1}\right)\\\
\frac{d}{dw}&\;=\;\frac{1}{2|\nabla\varphi^{2}|^{2}}\left(\nabla\varphi^{2}\,+\,\sqrt{-1}J_{\Sigma}\nabla\varphi^{2}\right)\end{aligned}$

and we easily compute the transition function:

(26)
$\displaystyle\frac{dw}{dz}\;=\;g\left(\frac{d}{dz},\,\overline{\nabla}{w}\right)\;=\;\frac{\left<\nabla\varphi^{1},\,\nabla\varphi^{2}\right>}{|\nabla\varphi^{1}|^{2}}\,-\,\sqrt{-1}\frac{\sqrt{\mathcal{V}}}{|\nabla\varphi^{1}|^{2}}.$

But $\frac{dw}{dz}$ is holomorphic, so its imaginary part is harmonic, so
$|\nabla\varphi^{1}|^{-2}$ is harmonic. In the $z$-coordinate the Hermitian
metric is
$h_{\Sigma}=\left|\frac{d}{dz}\right|^{2}=\frac{1}{2}|\nabla\varphi^{1}|^{-2}$,
so $h_{\Sigma}$ is an harmonic function. Using
$\triangle_{\Sigma}{}h_{\Sigma}=0$, the Gaussian curvature is

(27) $\displaystyle
K_{\Sigma}\;=\;-\frac{1}{2}h_{\Sigma}^{-1}\triangle_{\Sigma}\log{h}_{\Sigma}\;=\;4|\nabla|\nabla\varphi^{1}||^{2}$

which is non-negative, forcing the complete manifold $(\Sigma^{2},g_{\Sigma})$
to be parabolic—this is due to the Cheng-Yau condition for parabolicity; see
[6] or the remark below. The region $\Sigma^{2}\subset\mathbb{R}^{2}$ is
simply connected due to the fact that it is convex by Proposition 3.3.
Consequently $(\Sigma^{2},J_{\Sigma})$ is biholomorphic to $\mathbb{C}$. But
then the classic Liouville theorem implies $|\nabla\varphi^{1}|^{-2}$ is
actually constant, because it is an harmonic function on $\mathbb{C}$ bounded
from below. In particular, (27) now gives $K_{\Sigma}=0$.

Similarly $|\nabla\varphi^{2}|^{-2}$ is constant. The fact that $\mathcal{V}$
is constant and
$\mathcal{V}=|\nabla\varphi^{1}|^{2}|\nabla\varphi^{2}|^{2}-\left<\nabla\varphi^{1},\nabla\varphi^{2}\right>{}^{2}$
means $\left<\nabla\varphi^{1},\nabla\varphi^{2}\right>$ is constant. Because
$g_{\Sigma}{}^{ij}=\left<\nabla\varphi^{i},\nabla\varphi^{j}\right>$, all
components of the metric are constants when measured in
$(\varphi^{1},\varphi^{2})$ coordinates. ∎

######  Lemma 4.2.

Assume $(\Sigma^{2},g_{\Sigma})$ is geodesically complete,
$(\Sigma^{2},J_{\Sigma})$ is biholomorphic to $\mathbb{C}$, and
$\triangle_{\Sigma}\sqrt{\mathcal{V}}\leq 0$. Then $(\Sigma^{2},g_{\Sigma})$
is flat and the metric $g_{\Sigma}$ has constant coefficients when expressed
in $\varphi^{1}$, $\varphi^{2}$ coordinates.

###### Proof.

Since the function $\mathcal{V}^{\frac{1}{2}}$ is superharmonic and bounded
from below on $\mathbb{C}$, it is constant. Lemma 4.1 now provides the result.
∎

Lemmas 4.1 and 4.2 together show $\mathcal{V}=const$ if and only if
$(\Sigma^{2},J_{\Sigma})$ is biholomorphic to $\mathbb{C}$. Before this
equivalence can be made useful, we require a fact about the conformal change
discussed in §2.3.

######  Lemma 4.3.

Assume $(\Sigma^{2},g_{\Sigma})$ is geodesically complete and
$\triangle_{\Sigma}\sqrt{\mathcal{V}}\leq 0$. Setting
$\widetilde{g}_{\Sigma}=\mathcal{V}^{\frac{1}{2}}g_{\Sigma}$, then
$(\Sigma^{2},\widetilde{g}_{\Sigma})$ is also geodesically complete.

###### Proof.

For a proof by contradiction suppose $(\Sigma^{2},\tilde{g}_{\Sigma})$ is not
geodesically complete. Let $\gamma:[0,R]\rightarrow\Sigma^{2}$ be a shortest
inextendible unit-speed geodesic in the $\widetilde{g}_{\Sigma}$ metric. Then
$\gamma$ must have _infinite_ length in the $g_{\Sigma}$ metric; the reason is
that, because the conformal factor $\sqrt{\mathcal{V}}$ is smooth and non-
negative, if $\gamma$ has finite total length in $g_{\Sigma}$ it will remain
within a compact subset $K$ of $\Sigma^{2}$. But on any compact subset the
conformal factor $\mathcal{V}$ is smooth and nonzero, so the defining ODE for
$\gamma$ remains smooth in $\tilde{g}_{\Sigma}$, so it is extendible.

For a contradiction, we prove that $\gamma$ has finite length in the
$g_{\Sigma}$ metric. Set $p=\gamma(0)$ and let $r$ be the distance function
$r=dist(p,\cdot)$ for the $\tilde{g}_{\Sigma}$ metric. We create a lower
barrier for the superharmonic function $\mathcal{V}$ by using a Bochner-style
Laplacian comparison (a technique appearing in [19]). This is available
because the curvature of $(\Sigma^{2},\tilde{g}_{\Sigma})$ is positive:
$\widetilde{K}_{\Sigma}=\frac{1}{2}\tilde{s}_{\Sigma}\geq 0$ by (17).
Therefore comparing the $\widetilde{g}_{\Sigma}$ Laplacian
$\widetilde{\triangle}_{\Sigma}$ to the Laplacian on flat Euclidean space
$\triangle_{Eucl}$, the usual Laplacian comparison for the distance function
provides $\widetilde{\triangle}_{\Sigma}{r}\leq\triangle_{Eucl}r={r}^{-1}$.
Therefore $\widetilde{\triangle}_{\Sigma}{r}^{-1}\geq{r}^{-3}$.

Now we create a barrier for $\mathcal{V}$. We consider the annulus
$B_{p}(R)\setminus{}B_{p}(R/2)$. On the outer boundary $\partial{}B_{p}(R)$
the function $\left(r^{-1}-R^{-1}\right)$ equals 0. Because the inner boundary
$\partial{}B_{p}(R/2)$ is compact in $\Sigma^{2}$, there exists some
$\epsilon>0$ so
$\mathcal{V}^{\frac{1}{2}}\geq\epsilon\left(r^{-1}-R^{-1}\right)$. Using
$\widetilde{\triangle}{}_{\Sigma}\mathcal{V}^{\frac{1}{2}}=0$, we now have

(28) $\displaystyle\begin{aligned}
&\mathcal{V}^{\frac{1}{2}}\,-\,\epsilon\left(r^{-1}\,-\,R^{-1}\right)\;\geq\;0\quad\text{on
$\partial\mathcal{A}$}\quad\text{and}\\\
&\widetilde{\triangle}_{\Sigma}\left(\mathcal{V}^{\frac{1}{2}}\,-\,\epsilon\left(r^{-1}\,-\,R^{-1}\right)\right)\;\leq\;-\epsilon{r}^{-3}\;<\;0.\end{aligned}$

Thus $\mathcal{V}^{\frac{1}{2}}-\epsilon(r^{-1}-R^{-1})$ is superharmonic on
the annulus and non-negative on its boundary. So
$\mathcal{V}^{\frac{1}{2}}\geq\epsilon(r^{-1}-R^{-1})$ on the annulus by the
maximum principle. Because the path $\gamma$ lies within the annulus for
$t\in[R/2,R]$ we immediately have

(29) $\displaystyle\begin{aligned}
\big{(}\left(\mathcal{V}\circ\gamma\right)(t)\big{)}^{\frac{1}{2}}&\;>\;\epsilon\left(t^{-1}-R^{-1}\right),\quad\text{which
is}\\\
\big{(}\left(\mathcal{V}\circ\gamma\right)(t)\big{)}^{-\frac{1}{4}}&\;<\;\epsilon^{-\frac{1}{2}}\left(t^{-1}-R^{-1}\right)^{-\frac{1}{2}}\;=\;\epsilon^{-\frac{1}{2}}(tR)^{\frac{1}{2}}\left(R-t\right)^{-\frac{1}{2}}.\end{aligned}$

This allows us to estimate the $g_{\Sigma}$-length of $\gamma$. From
$|\dot{\gamma}|_{g_{\Sigma}}=\mathcal{V}^{-\frac{1}{4}}|\dot{\gamma}|_{\tilde{g}_{\Sigma}}$,
we have

(30) $\displaystyle\begin{aligned}
|\dot{\gamma}|_{g_{\Sigma}}&\;=\;\big{(}\mathcal{V}\circ\gamma\big{)}^{-\frac{1}{4}}\cdot|\dot{\gamma}|_{\tilde{g}_{\Sigma}}\;<\;\epsilon^{-\frac{1}{2}}(tR)^{\frac{1}{2}}\left(R-t\right)^{-\frac{1}{2}}\cdot|\dot{\gamma}|_{\tilde{g}_{\Sigma}}\end{aligned}$

within the annulus. Because also $|\dot{\gamma}|_{\tilde{g}_{\Sigma}}=1$, we
have

(31) $\displaystyle\begin{aligned}
Len_{g_{\Sigma}}(\gamma[R/2,R])&\;=\;\int_{R/2}^{R}|\dot{\gamma}|_{g_{\Sigma}}dt\;=\;\int_{R/2}^{R}(\mathcal{V}{}^{-\frac{1}{4}}\circ\gamma)(t)\,dt\\\
&\quad\;\leq\;\epsilon^{-\frac{1}{2}}R^{\frac{1}{2}}\int_{R/2}^{R}t^{\frac{1}{2}}(R-t)^{-\frac{1}{2}}dt\;=\;\epsilon^{\frac{1}{2}}\frac{2+\pi}{4}R^{\frac{3}{2}}.\end{aligned}$

The total length of $\gamma$ in the $g_{\Sigma}$ metric is
$\int_{0}^{R/2}|\dot{\gamma}|_{g_{\Sigma}}dt+\int_{R/2}^{R}|\dot{\gamma}|_{g_{\Sigma}}dt$.
The first integral is finite because $\gamma(t)$ remains within a compact
region of $\Sigma^{2}$ for $t\in[0,R/2]$. Thus $g_{\Sigma}$-length of $\gamma$
is finite. This contradiction completes the proof. ∎

######  Theorem 4.4 (cf. Theorem 1.6).

Assume $(\Sigma^{2},g_{\Sigma})$ is geodesically complete and
$\triangle_{\Sigma}\sqrt{\mathcal{V}}\leq 0$. Then $(\Sigma^{2},g_{\Sigma})$
and $(\Sigma^{2},\tilde{g}_{\Sigma})$ are flat.

###### Proof.

By Lemma 4.3, $(\Sigma^{2},\widetilde{g}_{\Sigma})$ is complete, and by (17)
$\widetilde{K}_{\Sigma}=\frac{1}{2}\tilde{s}_{\Sigma}\geq 0$. By the Cheng-Yau
criterion for parabolicity, $(\Sigma^{2},\widetilde{g}_{\Sigma})$ is parabolic
so is biholomorphic to $\mathbb{C}$. Then $\mathcal{V}^{\frac{1}{2}}$ is a
positive superharmonic function on $\mathbb{C}$, so is constant. By Lemma 4.1,
$g_{\Sigma}$ is flat and therefore so is $\tilde{g}_{\Sigma}$. ∎

######  Corollary 4.5 (cf. Theorem 1.6).

Assume $(M^{4},J,g,\mathcal{X}^{1},\mathcal{X}^{2})$ has $s\geq 0$, and assume
the distribution $\\{\mathcal{X}^{1},\mathcal{X}^{2}\\}$ is always rank 2.
Then $M^{4}$ is flat $\mathbb{C}^{2}$ and $\mathcal{X}^{1}$, $\mathcal{X}^{2}$
are translational Killing fields.

###### Proof.

Since $\mathcal{V}=0$ if and only if $\\{\mathcal{X}^{1},\mathcal{X}^{2}\\}$
is a linearly dependent set, the reduction $\Sigma^{2}$ has no edges. The
moment map is therefore a Riemannian submersion, so the metric polytope
$(\Sigma^{2},g_{\Sigma})$ is complete. By equation (17) we have
$\triangle_{\Sigma}\sqrt{\mathcal{V}}\leq 0$. Theorem 4.4 now implies
$g_{\Sigma}$ is flat, and $g_{\Sigma}$ is a constant matrix when expressed in
$\varphi^{1}$-$\varphi^{2}$ coordinates. Referring to (3), we have that $G$ is
a constant matrix, so $g$ and $J$ are constant. Thus $M^{4}$ is also flat.

The fact that $M^{4}\approx\mathbb{C}^{2}$ comes from our earlier assumption
that $M^{4}$ is simply connected. The fact that the Killing fields are
translational is due to the fact that they are nowhere zero. ∎

Remark. Crucial to the proofs of Lemma 4.1 and Theorem 4.4 is the fact that a
complete, simply connected $\Sigma^{2}$ with $K_{\Sigma}\geq 0$ is
biholomorphic to $\mathbb{C}$. This is a simple consequence of volume
comparison and the Cheng-Yau criterion for parabolicity, namely that
$\int_{1}^{\infty}\frac{t}{Vol\,B_{t}}dt=\infty$. The assertion that a simply
connected, boundaryless, complete Riemann surface is parabolic if and only if
it is actually $\mathbb{C}$ is a consequence of the uniformization theorem.
The subject of parabolicity has received a great deal of attention; for a tiny
sampling of this large subject see [16] [17] [14] [12] [22] and references
therein.

##  5\. The Asymptotic Model Geometries

Here we prove Theorem 1.7. The earlier paper [22] showed that if the reduction
$\Sigma^{2}$ of $M^{4}$ is closed and has at least one edge, then we can write
down all possible metrics on $M^{4}$. Theorem 1.6 removed the need for the
polygon to have an edge. We remark that cases (3) and (5) consist of one
metric each, and (4) assumes that the polygon is closed. Thus to complete the
“$\Rightarrow$” part of Theorem 1.7, we must show, in cases (1) and (2), that
the metrics written down in [22] indeed meet the criteria of being
“asymptotically general” and “asymptotically equivariantly
$\mathbb{R}^{2}\times\mathbb{S}^{2}$” in the sense discussed in the
introduction. We check this in Lemmas 5.2 and 5.3 below.

We must also prove the converse, that if the metric on $M^{4}$ has the
asymptotic structure of any of the cases (1)-(5), then its reduction
$\Sigma^{2}$ is closed. Again we must only check this in the cases (1) and
(2), as the closure of $\Sigma^{2}$ is built into cases (3) and (4) already.
In 5.3 we prove the following lemma.

######  Lemma 5.1.

Assume $M^{4}$ is either “equivariantly asymptotically general,” or
“equivariantly asymptotically $\mathbb{R}^{2}\times\mathbb{S}^{2}$.” Then the
parent manifold $M^{4}$ has two unbounded polar submanifolds, and each has a
Killing field which remains bounded away from zero. By Theorem 1.5
consequently any such manifold $M^{4}$ has closed reduction.

In cases (1), (2), and (3) we have two different bijections between
$\Sigma^{2}$ and the closed upper half-plane. These are

(32)
$z:\Sigma^{2}\rightarrow\overline{H}{}^{2}\quad\text{and}\quad\Phi:\overline{H}{}^{2}\rightarrow\Sigma^{2}.$

The first is the “volumetric normal function” $z=x+\sqrt{-1}y$ discussed in
§2.1, which is a biholomorphism by Proposition 1.1 of [22]. The second is
given by $\Phi(x,y)=(\varphi^{1}(x,y),\varphi^{2}(x,y))$, which is smooth
everywhere except at corner points, where it is Lipschitz. As discussed in
Section 2 the geometry of $M^{4}$ is completely determined by
$(\Sigma^{2},g_{\Sigma})$, which itself is completely determined by the
relationship between the $(\varphi^{1},\varphi^{2})$ and $(x,y)$ coordinates.
From (2.13) of [22] the metric is

(33) $\displaystyle
g_{\Sigma}\;=\;\frac{1}{y}\text{det}(A)\left(dx\otimes{}dx\,+\,dy\otimes{}dy\right)$

where $A$ is the coordinate transition matrix
$A=\left(\frac{\partial\\{\varphi^{1},\varphi^{2}\\}}{\partial\\{x,y\\}}\right)$.

As discussed in the introduction, we say the manifold $(M^{4},g)$ is
equivariantly asymptotically modeled on $(\widetilde{M}{}^{4},\widetilde{g})$
to order $l$ if, outside some compact set
$\widetilde{K}\subset\widetilde{M}{}^{4}$ there exists an equivariant covering
map $\pi:\widetilde{M}{}^{4}\setminus{}\widetilde{K}\rightarrow{}M^{4}$ so
that

(34)
$\left\|\pi^{*}g\;-\;\widetilde{g}\right\|_{\widetilde{g}}\;=\;O(\tilde{\rho}{}^{-l}),\text{
or $o(1)$ if $l=0$}.$

In what follows, the models $\widetilde{M}{}^{4}$ will always be a toric
Kahler manifold with momentum functions
$\tilde{\varphi}^{1},\tilde{\varphi}^{2}:\overline{H}{}^{2}\rightarrow\widetilde{\Sigma}{}^{2}$.
We can use the same $(x,y)$ coordinates to express either metric, and take the
$\pi$ to simply be the identity. Then the comparison is

(35) $\displaystyle\left\|\pi^{*}g\;-\;\tilde{g}\right\|_{\widetilde{g}}$
$\displaystyle\;=\;\frac{1}{y}\left|\text{det}(A)-\text{det}(\widetilde{A})\right|\left\|dx\otimes{}dx+dy\otimes{}dy\right\|_{\widetilde{g}}$
$\displaystyle\;=\;\frac{1}{y}\left|\text{det}(A)-\text{det}(\widetilde{A})\right|\sqrt{2}(\|dx\|_{\widetilde{g}})^{2}$
$\displaystyle\;=\;\frac{1}{y}\left|\text{det}(A)-\text{det}(\widetilde{A})\right|\sqrt{2}\frac{y}{\text{det}(\widetilde{A})}\;=\;\sqrt{2}\left|1-\frac{\text{det}(A)}{\text{det}(\widetilde{A})}\right|.$

Therefore proving that $\widetilde{M}{}^{4}$ is an equivariant asymptotic
model for $M^{4}$ comes down to comparing coordinate transition Jacobians for
the two manifolds.

###  5.1. The “general” case.

We show that any metric on a polygon with non-parallel rays is asymptotically
modeled on a generalized Taub-NUT metric. From [22], after possible affine
recombination, the momentum functions on such a polygon always have the form

(36) $\displaystyle\varphi^{1}$
$\displaystyle\;=\;\frac{\alpha}{2}(y)^{2}+\frac{s_{d}}{2}\left((x-x_{d})+\sqrt{(x-x_{d})^{2}+(y)^{2}}\right)$
$\displaystyle+\sum_{i=1}^{d-1}\frac{m_{i+1}-m_{i}}{2(x_{i+1}-x_{i})}\left(x_{i+1}-\sqrt{(x-x_{i+1})^{2}+(y)^{2}}-x_{i}+\sqrt{(x-x_{i})^{2}+(y)^{2}}\right)$
$\displaystyle\varphi^{2}$
$\displaystyle\;=\;\frac{\beta}{2}(y)^{2}+n_{1}+\frac{s_{0}}{2}\left(-(x-x_{1})+\sqrt{(x-x_{1})^{2}+(y)^{2}}\right)$
$\displaystyle+\sum_{i=1}^{d-1}\frac{n_{i+1}-n_{i}}{2(x_{i+1}-x_{i})}\left(x_{i+1}-\sqrt{(x-x_{i+1})^{2}+(y)^{2}}-x_{i}+\sqrt{(x-x_{i})^{2}+(y)^{2}}\right)$

where the values $s_{0}$, $s_{d}$, $m_{i}$, $n_{i}$, and the differences
$x_{i+1}-x_{i}$ are determined uniquely from the labeled polygon. The values
$\alpha,\beta\geq 0$ are “free” parameters, meaning in this case that changing
$\alpha$ or $\beta$ changes the metric but does not change the polygon or its
labels. Specifically, by the recipe of [22], the values
$\\{x_{1},\dots,x_{d}\\}$ are the Lipschitz points on the $x$-axis in
$\mathbb{C}$ and $\\{p_{1},\dots,p_{d}\\}$ are the vertex point on the
boundary of $\Sigma^{2}$ where $p_{i}=(m_{i},n_{i})$. The polygon boundary
labels are $s_{0}$ and $s_{d}$ on the rays, and
$s_{i}=\frac{|p_{x_{i+1}}-p_{i}|}{x_{i+1}-x_{i}}$ on the $i^{th}$ boundary
segment. See Figure 4

|
---|---
Figure 4.  (a) A typical “general case” polygon $\Sigma^{2}$ with non-parallel
rays, and (b) a Taub-NUT style asymptotic model $\widetilde{\Sigma}{}^{2}$.

Our comparison manifold $\widetilde{M}{}^{4}$ will be any of the generalized
Taub-NUT manifolds; these have momentum functions

(37) $\displaystyle\tilde{\varphi}{}^{1}$
$\displaystyle\;=\;\frac{\alpha}{2}(y)^{2}+\frac{s_{d}}{2}\left(x+\sqrt{(x)^{2}+(y)^{2}}\right)$
$\displaystyle\tilde{\varphi}{}^{2}$
$\displaystyle\;=\;\frac{\beta}{2}(y)^{2}+\frac{s_{0}}{2}\left(-x+\sqrt{(x)^{2}+(y)^{2}}\right).$

The resulting generalized Taub-NUT manifolds were thoroughly studied in [23].
Their reductions are wedges in $\mathbb{R}^{2}$, which after possible affine
transformation is a quarter-plane. We must compute
$A=\left(\frac{\partial\\{\varphi^{1},\varphi^{2}\\}}{\\{x,y\\}}\right)$ and
$\widetilde{A}=\left(\frac{\partial\\{\tilde{\varphi}^{1},\tilde{\varphi}^{2}\\}}{\\{x,y\\}}\right)$.
From (37),

(38) $\displaystyle\frac{\partial\tilde{\varphi}^{1}}{\partial{}x}$
$\displaystyle\;=\;\frac{s_{d}}{2}\left(1+\cos\theta\right),\;\;\;\,\frac{\partial\tilde{\varphi}^{1}}{\partial{}y}\;=\;\left(\alpha{}r+\frac{s_{d}}{2}\right)\sin\theta,$
$\displaystyle\frac{\partial\tilde{\varphi}^{2}}{\partial{}x}$
$\displaystyle\;=\;\frac{s_{0}}{2}\left(-1+\cos\theta\right),\;\frac{\partial\tilde{\varphi}^{2}}{\partial{}y}\;=\;\left(\beta{}r+\frac{s_{0}}{2}\right)\sin\theta$

where we used $x=r\cos\theta$ and $y=r\sin\theta$; because $x$ and $y$ are in
the upper half-plane we have $\theta\in[0,\pi]$. Then we have

(39) $\displaystyle\widetilde{A}=\left(\begin{array}[]{cc}0&\alpha\\\
0&\beta\end{array}\right)y+\frac{1}{2}\left(\begin{array}[]{cc}s_{d}(1+\cos\theta)&s_{d}\sin\theta\\\
s_{0}(-1+\cos\theta)&s_{0}\sin\theta\end{array}\right)$
$\displaystyle\text{det}(\widetilde{A})=\frac{1}{2}\left(\alpha{}s_{0}+\beta{}s_{d}-(\alpha{}s_{0}-\beta{}s_{d})\cos\theta\right)y+\frac{1}{2}s_{0}s_{d}\sin\theta.$

For $A$ and $\text{det}(A)$, we use (36) to get

(40) $\displaystyle\frac{\partial\varphi^{1}}{\partial{}x}$
$\displaystyle\;=\;\frac{\partial\tilde{\varphi}^{1}}{\partial{}x}+\frac{s_{d}}{2}\left(\frac{x-x_{d}}{\sqrt{(x-x_{d})^{2}+(y)^{2}}}-\frac{x}{\sqrt{(x)^{2}+(y)^{2}}}\right)$
$\displaystyle\quad+\frac{1}{2}\sum_{i=1}^{d-1}\frac{m_{i+1}-m_{i}}{x_{i+1}-x_{i}}\left(\frac{x-x_{i}}{\sqrt{(x-x_{i})^{2}+(y)^{2}}}-\frac{x-x_{i+1}}{\sqrt{(x-x_{i+1})^{2}+(y)^{2}}}\right)$
$\displaystyle\frac{\partial\varphi^{1}}{\partial{}y}$
$\displaystyle\;=\;\frac{\partial\tilde{\varphi}^{1}}{\partial{}y}+\frac{s_{d}}{2}\left(\frac{y}{\sqrt{(x-x_{d})^{2}+(y)^{2}}}-\frac{y}{\sqrt{(x)^{2}+(y)^{2}}}\right)$
$\displaystyle\quad+\frac{1}{2}\sum_{i=1}^{d-1}\frac{m_{i+1}-m_{i}}{x_{i+1}-x_{i}}\left(\frac{y}{\sqrt{(x-x_{i})^{2}+(y)^{2}}}-\frac{y}{\sqrt{(x-x_{i+1})^{2}+(y)^{2}}}\right)$

and similarly

(41) $\displaystyle\frac{\partial\varphi^{2}}{\partial{}x}$
$\displaystyle\;=\;\frac{\partial\tilde{\varphi}^{2}}{\partial{}x}+\frac{s_{0}}{2}\left(\frac{x-x_{1}}{\sqrt{(x-x_{1})^{2}+(y)^{2}}}-\frac{x}{\sqrt{(x)^{2}+(y)^{2}}}\right)$
$\displaystyle\quad+\frac{1}{2}\sum_{i=1}^{d-1}\frac{n_{i+1}-n_{i}}{x_{i+1}-x_{i}}\left(\frac{x-x_{i}}{\sqrt{(x-x_{i})^{2}+(y)^{2}}}-\frac{x-x_{i+1}}{\sqrt{(x-x_{i+1})^{2}+(y)^{2}}}\right)$
$\displaystyle\frac{\partial\varphi^{2}}{\partial{}y}$
$\displaystyle\;=\;\frac{\partial\tilde{\varphi}^{2}}{\partial{}y}+\frac{s_{d}}{2}\left(\frac{y}{\sqrt{(x-x_{1})^{2}+(y)^{2}}}-\frac{y}{\sqrt{(x)^{2}+(y)^{2}}}\right)$
$\displaystyle\quad+\frac{1}{2}\sum_{i=1}^{d-1}\frac{n_{i+1}-n_{i}}{x_{i+1}-x_{i}}\left(\frac{y}{\sqrt{(x-x_{i})^{2}+(y)^{2}}}-\frac{y}{\sqrt{(x-x_{i+1})^{2}+(y)^{2}}}\right)$

We estimate these lengthy expressions using the binomial formula

(42)
$\frac{1}{\sqrt{1+t}}\;=\;\sum_{k=0}^{\infty}{{-1/2}\choose{k}}t^{k}\;=\;1-\frac{1}{2}t\,+\,\frac{3}{8}t^{2}-\frac{5}{16}t^{3}\pm\dots$

which converges when $t\in(-1,1]$. The idea is to use $t=\frac{1}{r}$, so
series of the form (42) converge for all sufficiently large $r$. We have

(43)
$\displaystyle\frac{x-x_{i}}{\sqrt{(x-x_{i})^{2}+(y)^{2}}}\;=\;\frac{x-x_{i}}{r}\frac{1}{\sqrt{1-2\frac{x_{i}}{r}\cos\theta+\left(\frac{x_{i}}{r}\right)^{2}}}$
$\displaystyle\quad\;=\;\left(\cos\theta-\frac{x_{i}}{r}\right)\sum_{k=0}^{\infty}{{-1/2}\choose{k}}\left(-2x_{i}r^{-1}\cos\theta+(x_{i})^{2}r^{-2}\right)^{k}$
$\displaystyle\quad\;=\;\cos\theta-\left(x_{i}\sin^{2}\theta\right)\frac{1}{r}-\frac{3}{2}\left(x_{i}^{2}\cos\theta\sin^{2}\theta\right)\frac{1}{r^{2}}+O(r^{-3})$

and

(44)
$\displaystyle\frac{y}{\sqrt{(x-x_{i})^{2}+(y)^{2}}}\;=\;\frac{y}{r}\frac{1}{\sqrt{1-2\frac{x_{i}}{r}\cos\theta+\left(\frac{x_{i}}{r}\right)^{2}}}$
$\displaystyle\quad\;=\;\sin\theta\sum_{k=0}^{\infty}{{-1/2}\choose{k}}\left(-2x_{i}r^{-1}\cos\theta+(x_{i})^{2}r^{-2}\right)^{k}$
$\displaystyle\quad\;=\;\sin\theta+\left(x_{i}\sin\theta\cos\theta\right)\frac{1}{r}+\frac{1}{4}(x_{i})^{2}\left(1+3\cos
2\theta\right)\sin\theta\frac{1}{r^{2}}+O(r^{-3})$

Then we have the series approximation for $A$

(45) $\displaystyle A=\left(\begin{array}[]{cc}0&\alpha\\\
0&\beta\end{array}\right)y+\frac{1}{2}\left(\begin{array}[]{cc}s_{d}(1+\cos\theta)&s_{d}\sin\theta\\\
s_{0}(-1+\cos\theta)&s_{0}\sin\theta\end{array}\right)$
$\displaystyle\quad\quad+\frac{1}{2}\left(\begin{array}[]{cc}(m_{d}-s_{d}x_{d})\sin^{2}\theta&-(m_{d}-s_{d}x_{d})\sin\theta\cos\theta\\\
-(n_{1}+s_{0}x_{1})\sin^{2}\theta&(n_{1}+s_{0}x_{1})\sin\theta\cos\theta\end{array}\right)\frac{1}{r}+O(r^{-2})$

and for $\text{det}(A)$ we find

(46)
$\displaystyle\text{det}(A)=\frac{1}{2}\left((s_{0}\alpha+s_{d}\beta)-(s_{0}\alpha-
s_{d}\beta)\cos\theta\right)y$ $\displaystyle\hskip
36.135pt+\frac{1}{2}\Big{(}(s_{0}s_{d}+\alpha(n_{1}+s_{0}x_{1})+\beta(m_{d}-s_{d}x_{d}))$
$\displaystyle\hskip
72.26999pt-(\alpha(n_{1}+s_{0}x_{1})+\beta(m_{d}-s_{d}x_{d}))\cos^{2}\theta\Big{)}\sin\theta+O(r^{-1}).$

Then because $\text{det}(A)$ and $\text{det}(\widetilde{A})$ have the same
leading term, we have therefore
$1-\text{det}(A)/\text{det}(\widetilde{A})=O(r^{-1})$. A computation gives

(47) $\displaystyle
1-\frac{\text{det}(A)}{\text{det}(\widetilde{A})}\;=\;\frac{(\alpha(n_{1}+s_{0}x_{1})+\beta(m_{d}-s_{d}x_{d}))\sin^{2}\theta}{-(\alpha{}s_{0}+\beta{}s_{d})+(\alpha{}s_{0}-\beta{}s_{d})\cos\theta}\frac{1}{r}+O(r^{-2}).$

To address the worry that the denominator in the leading term of (47) might be
zero, we point out that $s_{0}$ and $s_{d}$ have the same sign and
$\alpha,\beta\geq 0$. Therefore the denominator can only be zero if $\alpha=0$
or $\beta=0$ or both. When $\beta=0$ but $\alpha\neq 0$ or when $\alpha=0$ but
$\beta\neq 0$ (this is the “exceptional” case) we have

(48)
$1-\frac{\text{det}(A)}{\text{det}(\widetilde{A})}\;=\;-\frac{(n_{1}+s_{0}x_{1})(1+\cos\theta)}{s_{0}}\frac{1}{r}+O(r^{-2}).$

When both $\alpha=\beta=0$ the Taub-NUT is simply Euclidean space; the
estimate is

(49) $\displaystyle 1-\frac{\text{det}(A)}{\text{det}(\widetilde{A})}$
$\displaystyle\;=\;\frac{-(m_{d}s_{0}+s_{d}n_{1})-s_{0}s_{d}(x_{1}-x_{d})+(m_{d}s_{0}-s_{d}n_{1}-s_{0}s_{d}(x_{1}+x_{d}))\cos\theta}{2s_{0}s_{d}}\frac{1}{r}$
$\displaystyle\hskip 14.45377pt+O(r^{-2}).$

######  Lemma 5.2.

Assume $M^{4}$ has a closed reduction polygon $\Sigma^{2}$ with two boundary
rays that are not parallel. Then its asymptotic geometry is ALE, ALF, ALF-
like, or exceptional to order $1$.

###### Proof.

The comparison momenta $\tilde{\varphi}^{1}$, $\tilde{\varphi}^{2}$ produce
the generalized Taub-NUT metrics; see [23] for detailed information. When
$\alpha=\beta=0$ the “Taub-NUT” is flat Euclidean space (so metrics that are
asymptotically equivalent to this are ALE). When $\alpha=\beta>0$ then the
comparison metric is a multiple of the standard Ricci-flat Taub-NUT (so
metrics that are asymptotically equivalent to this are asymptotically ALF).
When $\alpha=0$ but $\beta\neq 0$ or vice-versa then the comparison metric is
an “exceptional Taub-NUT” (so metrics that are asymptotically equivalent to
this are asymptotically exceptional). And when $\alpha>0$, $\beta>0$ but
$\alpha\neq\beta$ then the comparison is a chiral Taub-NUTs (so metrics that
are asymptotically equivalent to this are asymptotically ALF-like).

From (47), (48), (49) we have that “general case” polygon metrics
$(\Sigma^{2},g_{\Sigma})$ are asymptotically equivalent to their corresponding
comparison metrics $(\widetilde{\Sigma}{}^{2},\tilde{g}_{\Sigma})$. Using
Corollary 3.3 of [23] we have that the coordinate distance function $r$ on
$\mathbb{C}$ is, to within a constant multiple, close to the distance function
$\tilde{\rho}$; therefore we have fast decay
$\|\tilde{g}_{\Sigma}-g_{\Sigma}\|=O(\tilde{\rho}{}^{-1})$.

The estimates (47), (48), (49) show the$(\Sigma^{2},g_{\Sigma})$ polygon
metric is close to the comparison polygon
$(\widetilde{\Sigma}{}^{2},\tilde{}g_{\Sigma})$, but we must also check that
the manifold metric $(M^{4},g)$ is asymptotically close to the comparison
metric $(\widetilde{M}{}^{2},\tilde{g})$. By (3) the action
$\varphi^{1}$-$\varphi^{2}$ components of the metric are orthogonal to the
angle $\theta_{1}$-$\theta_{2}$ components. The action components have already
been checked: this part of the metric is precisely the polygon metric. The
part perpendicular to this we denote $g^{\perp}$. By (3) we have

(50) $\displaystyle g^{\perp}\;=\;G^{ij}d\theta_{i}\otimes{}d\theta_{j}.$ (51)
$G^{ij}=g(\nabla\varphi^{i},\nabla\varphi^{j})=A^{is}A^{jt}g(dx^{i},\,dx^{j})=\delta_{st}A^{is}A^{jt}|dx|^{2}$

We have that $dx$ and $dy$ are orthogonal, so that
$\partial/\partial{}x=|dx|^{-2}\nabla{}x$. Then we have

(52) $\displaystyle g^{\perp}$
$\displaystyle\;=\;|dx|^{2}\delta_{st}(A^{is}d\theta_{i})\otimes(A^{jt}d\theta_{j})$
$\displaystyle\;=\;|dx|^{2}\delta_{st}\left(A^{is}J\frac{\partial}{\partial\varphi^{i}}\right)_{\flat}\otimes\left(A^{jt}J\frac{\partial}{\partial\varphi^{j}}\right)_{\flat}$
$\displaystyle\;=\;|dx|^{2}\delta_{st}\left(J\frac{\partial}{\partial{}x^{i}}\right)_{\flat}\otimes\left(J\frac{\partial}{\partial{}x^{j}}\right)_{\flat}$
$\displaystyle\;=\;|dx|^{-2}\left(Jdx\otimes{}Jdx+Jdy\otimes{}Jdy\right)$

where we used $\frac{\partial}{\partial\theta_{i}}=(Jd\varphi^{i})^{\sharp}$
and the dualization
$d\theta_{i}=-\left(J\frac{\partial}{\partial\varphi^{i}}\right)_{\flat}$. But
then $g^{\perp}=g_{\Sigma}(J\cdot,J\cdot)$ and because $J$ is orthogonal, we
have
$\|\tilde{g}^{\perp}-g^{\perp}\|_{\tilde{g}}=\|\tilde{g}_{\Sigma}-g_{\Sigma}\|_{\tilde{g}_{\Sigma}}$
which we proved decays like $\tilde{\rho}{}^{-1}$. ∎

###  5.2. The asymptotically $\mathbb{R}^{2}\times\mathbb{S}^{2}$ metrics

In this section we examine those metric polygons $(\Sigma^{2},g_{\sigma})$
that have parallel rays on $\partial\Sigma^{2}$, but no lines. We show they
are all asymptotically modeled by the class of polygons
$(\widetilde{\Sigma}{}^{2},g_{\Sigma})$ given by the following comparison
momentum functions

(53) $\displaystyle\tilde{\varphi}^{1}$
$\displaystyle=\frac{1}{2}(m_{1}+m_{d})-\frac{s_{0}}{2}(x-x_{1})+\frac{s_{d}}{2}(x-x_{d})$
$\displaystyle\quad+\left(\frac{s_{0}}{2}+\frac{m_{d}-m_{1}}{2(x_{d}-x_{1})}\right)\sqrt{(x-x_{1})^{2}+(y)^{2}}$
$\displaystyle\quad+\left(\frac{s_{d}}{2}-\frac{m_{d}-m_{1}}{2(x_{d}-x_{1})}\right)\sqrt{(x-x_{d})^{2}+(y)^{2}}+\frac{\alpha}{2}(y)^{2}$
$\displaystyle\tilde{\varphi}^{2}$
$\displaystyle=\frac{1}{2}(n_{1}+n_{d})+\frac{n_{d}-n_{1}}{x_{d}-x_{1}}\sqrt{(x-x_{1})^{2}+(y)^{2}}-\frac{n_{d}-n_{1}}{x_{d}-x_{1}}\sqrt{(x-x_{d})^{2}+(y)^{2}}$

where $x_{1}$, $x_{d}$ are the two Lipschitz points on the half-space
boundary, and $p_{1}=(m_{1},n_{1})$ and $p_{d}=(m_{d},n_{d})$ are the
corresponding polygon vertex points. For a depiction see Figure 5. From [22],
if the reduction $\Sigma^{2}$ of $M^{4}$ has parallel rays, then after
possible affine transformation of the $\varphi^{1}$-$\varphi^{2}$ plane its
momentum functions are

(54) $\displaystyle\varphi^{1}$
$\displaystyle=m_{1}+\frac{s_{0}}{2}\left(-(x-x_{1})+\sqrt{(x-x_{1})^{2}+(y)^{2}}\right)+\frac{s_{d}}{2}\left((x-x_{d})+\sqrt{(x-x_{d})^{2}+(y)^{2}}\right)$
$\displaystyle+\sum_{i-1}^{d-1}\frac{m_{i+1}-m_{i}}{2(x_{i+1}-x_{i})}\left(x_{i+1}-x_{i}+\sqrt{(x-x_{i})^{2}+(y)^{2}}-\sqrt{(x-x_{i+1})^{2}+(y)^{2}}\right)+\frac{\alpha}{2}(y)^{2}$
$\displaystyle\varphi^{1}$
$\displaystyle=n_{1}+\sum_{i=1}^{d-1}\frac{n_{i+1}-n_{i}}{2(x_{i+1}-x_{i})}\left(x_{i+1}-x_{i}+\sqrt{(x-x_{i})^{2}+(y)^{2}}-\sqrt{(x-x_{i+1})^{2}+(y)^{2}}\right)$

where $x_{1},\dots,x_{d}$ are the Lipschitz points on the upper half-space
boundary and $p_{i}=(m_{i},n_{i})$ are the corresponding polygon vertex
points. The parameter $\alpha\geq 0$ is a “free” parameter, meaning changing
this parameter does not change the polygon $\Sigma^{2}$, but does change the
metric $g_{\Sigma}$.

|
---|---
Figure 5.  (a) Depiction of a typical polygon $\Sigma^{2}$ with parallel rays,
and (b) the corresponding half-strip model $\widetilde{\Sigma}{}^{2}$.

As in the “generic” case above, we must compute the transition matrix
$\widetilde{A}$ for the model polygon and $A$ for the real polygon. Then using
(35) we compare the metrics by estimating
$1-\text{det}(A)/\text{det}(\widetilde{A})$.

The comparison functions are more complicated than they were in the generic
case, but we follow the same process for analyzing them. After taking partial
derivatives, we approximate the resulting terms using the expansion (42). For
$\widetilde{A}$

(55)
$\displaystyle\widetilde{A}\;=\;\left(\begin{array}[]{cc}\frac{\partial\tilde{\varphi}^{1}}{\partial{}x}&\frac{\partial\tilde{\varphi}^{1}}{\partial{}y}\\\
\frac{\partial\tilde{\varphi}^{2}}{\partial{}x}&\frac{\partial\tilde{\varphi}^{2}}{\partial{}y}\end{array}\right)$
$\displaystyle=\left(\begin{array}[]{cc}0&\alpha\,\sin\theta\\\
0&0\end{array}\right)\,r+\frac{1}{2}\left(\begin{array}[]{cc}-(s_{0}+s_{d})+(s_{0}+s_{d})\cos\theta&(s_{0}+s_{d})\sin\theta\\\
0&0\end{array}\right)$
$\displaystyle+\left(\begin{array}[]{cc}-\frac{1}{2}(m_{1}-m_{d}+s_{0}x_{1}+s_{d}x_{d})\sin^{2}\theta&\frac{1}{2}(m_{1}-m_{d}+s_{0}x_{1}+s_{d}x_{d})\sin\theta\cos\theta\\\
-\frac{1}{2}(n_{1}-n_{d})\sin^{2}\theta&\frac{1}{2}(n_{1}-n_{d})\sin\theta\cos\theta\end{array}\right)\frac{1}{r}$
$\displaystyle+O(r^{-2}).$

and therefore

(56) $\displaystyle\text{det}(\widetilde{A})$
$\displaystyle\;=\;\frac{1}{2}(n_{1}-n_{d})\alpha\sin^{3}\theta\;+\;O(r^{-1})$

The expressions for the $(\Sigma^{2},g_{\Sigma})$ metric can be arrived at
similarly, but even the first few terms are so large they seeem inexpressible
except at unreasonable length. We don’t record them, but fortunately the exact
expressions are unnecessary, and we need only record the fact that
$\text{det}(A)$ has the same constant term that $\text{det}(\widetilde{A})$
has. We have

(57) $\displaystyle\text{det}(A)$
$\displaystyle\;=\;\frac{1}{2}(n_{1}-n_{d})\alpha\sin^{3}\theta+O(r^{-1})$

Because $\text{det}(\widetilde{A})$ and $\text{det}(A)$ expansions have the
same constant terms we have

(58) $1-\frac{\text{det}(A)}{\text{det}(\widetilde{A})}=O(r^{-1})$

However unlike the “general case” above, we do not have any known relationship
between the function $r=\sqrt{(x)^{2}+(y)^{2}}$ on the upper half-space and
any distance function on $(\widetilde{M}{}^{4},\tilde{g})$. But because $r$ is
an exhaustion function (its sub-level are compact and their union equals
$\widetilde{\Sigma}{}^{2}$) so we know at least that $r\rightarrow\infty$ as
$\tilde{\rho}\rightarrow\infty$. Therefore $O(r^{-1})=o(1)$ on
$\widetilde{M}{}^{4}$. This is enough to establish the following theorem.

######  Lemma 5.3.

Assume $M^{4}$ has reduction $\Sigma^{2}$ so that $\Sigma^{2}$ has parallel
rays on its boundary. Then $(M^{4},g)$ is asymptotically modeled on
$(\widetilde{M},\tilde{g})$ to degree $0$, where $(\widetilde{M},\tilde{g})$
is given by the momentum functions (53).

###### Proof.

This a direct consequence of (57) which by (35) gives
$\|\pi^{*}g-\tilde{g}\|_{\tilde{g}}=o(1)$. ∎

###  5.3. Proof that the reductions are closed

In this section we prove that if $M^{4}$ has the equivariant asymptotics of
(1) or (2), then indeed its reduction $\Sigma^{2}$ is closed. First we
establish a way to compare the Killing fields on $M^{4}$ its model
$\widetilde{M}{}^{4}$.

######  Lemma 5.4.

Assume $M^{4}$ has Killing fields $\mathcal{X}^{1}$, $\mathcal{X}^{2}$ and
$\widetilde{M}{}^{4}$ has Killing fields $\widetilde{\mathcal{X}}{}^{1}$,
$\widetilde{\mathcal{X}}{}^{2}$. Assume $M^{4}$ is asymptotically
equivariantly modeled on $\widetilde{M}{}^{4}$ to order 0. Then
$\left|\|\mathcal{X}^{i}\|_{g}-\|\widetilde{\mathcal{X}}^{i}\|_{\tilde{g}}\right|=o(\|\widetilde{\mathcal{X}}^{i}\|_{\tilde{g}})$.

###### Proof.

Let $(\varphi^{1},\theta_{1},\varphi^{2},\theta_{2})$ and
$(\tilde{\varphi}^{1},\tilde{\theta}_{1},\tilde{\varphi}^{2},\tilde{\theta}_{2})$
be the action-angle coordinates on $M^{4}$ and $\widetilde{M}{}^{4}$,
respectively. Then $\mathcal{X}^{i}=\frac{\partial}{\partial\theta_{i}}$,
$\widetilde{\mathcal{X}}^{i}=\frac{\partial}{\partial\tilde{\theta}_{i}}$. Let
$\pi$ be the equivariant diffeomorphism that realizes the asymptotic modeling;
then
$\pi_{*}\frac{\partial}{\partial\tilde{\theta}_{i}}=\frac{\partial}{\partial\theta_{i}}$.
Thus

(59)
$\displaystyle{\left\|\frac{\partial}{\partial\theta_{i}}\right\|_{g}}^{2}$
$\displaystyle=g\left(\frac{\partial}{\partial\theta_{i}},\frac{\partial}{\partial\theta_{i}}\right)=g\left(\pi_{*}\frac{\partial}{\partial\tilde{\theta}_{i}},\pi_{*}\frac{\partial}{\partial\tilde{\theta}_{i}}\right)$
$\displaystyle=(\pi^{*}g-\tilde{g})\left(\frac{\partial}{\partial\tilde{\theta}_{i}},\frac{\partial}{\partial\tilde{\theta}_{i}}\right)+\tilde{g}\left(\frac{\partial}{\partial\tilde{\theta}_{i}},\frac{\partial}{\partial\tilde{\theta}_{i}}\right)$

so therefore

(60)
$\displaystyle\left|{\left\|\frac{\partial}{\partial\tilde{\theta}_{i}}\right\|_{\tilde{g}}}^{2}-{\left\|\frac{\partial}{\partial\theta_{i}}\right\|_{g}}^{2}\right|\;\leq\;\left|(\pi^{*}g-\tilde{g})\left(\frac{\partial}{\partial\tilde{\theta}_{i}},\frac{\partial}{\partial\tilde{\theta}_{i}}\right)\right|\;\leq\;\left\|(\pi^{*}g-\tilde{g})\right\|_{\tilde{g}}{\left\|\frac{\partial}{\partial\tilde{\theta}_{i}}\right\|_{\tilde{g}}}^{2}$

which simplifies to

(61)
$\displaystyle\left|\left\|\frac{\partial}{\partial\tilde{\theta}_{i}}\right\|_{\tilde{g}}-\left\|\frac{\partial}{\partial\theta_{i}}\right\|_{g}\right|\;\leq\;\left\|(\pi^{*}g-\tilde{g})\right\|_{\tilde{g}}\frac{\left\|\frac{\partial}{\partial\tilde{\theta}_{i}}\right\|_{\tilde{g}}}{1+\frac{\left\|\frac{\partial}{\partial\theta_{i}}\right\|_{g}}{\left\|\frac{\partial}{\partial\tilde{\theta}_{i}}\right\|_{\tilde{g}}}}\;\leq\;\left\|(\pi^{*}g-\tilde{g})\right\|_{\tilde{g}}\left\|\frac{\partial}{\partial\tilde{\theta}_{i}}\right\|_{\tilde{g}}.$

Because $\left\|(\pi^{*}g-\tilde{g})\right\|_{\tilde{g}}=o(1)$ by assumption,
the result follows. ∎

Next we prove asymptotic structure results for two types of model.

######  Lemma 5.5.

Let $\widetilde{M}{}^{4}$ be one of the generalized Taub-NUT metrics given by
(37). Then $\widetilde{M}{}^{4}$ has two unbounded polar submanifolds, and the
Killing field on each is asymptotically bounded away from zero.

###### Proof.

For the Taub-NUT momentum functions (37), the polygon
$\widetilde{\Sigma}{}^{2}$ is the first quadrant in the
$\tilde{\varphi}^{1}$-$\tilde{\varphi}^{2}$ plane. The map
$(\tilde{\varphi}^{1},\tilde{\varphi}^{2}):\overline{H}{}^{2}\rightarrow\widetilde{\Sigma}{}^{2}$
has just a single Lipschitz point $x_{0}=0$ on the upper half space boundary,
and $M^{4}$ has two polar submanifolds: one where $y=0$ and $x<0$ (which
projects to the positive $\tilde{\varphi}^{2}$-axis in the
$\tilde{\varphi}^{1}$-$\tilde{\varphi}^{2}$ plane) and one where $y=0$ and
$x>0$ (which projects to the positive $\tilde{\varphi}^{1}$-axis).

To measure the Killing fields we use the metric (33) We have
$d\tilde{\varphi}^{i}=J\widetilde{\mathcal{X}}^{i}$, so

(62)
$\small\|\widetilde{\mathcal{X}}^{i}\|^{2}=\|d\tilde{\varphi}^{i}\|^{2}=\left(\left(\frac{\partial\tilde{\varphi}^{i}}{\partial{}x}\right)^{2}+\left(\frac{\partial\tilde{\varphi}^{i}}{\partial{}y}\right)^{2}\right)\|dx\|_{\tilde{g}}^{2}=\left((\widetilde{A}^{i1})^{2}+(\widetilde{A}^{i2})^{2}\right)\|dx\|_{\tilde{g}}^{2}.$

From (33) we have $\|dx\|^{2}_{\tilde{g}}=y/\text{det}(\widetilde{A})$. From
(38), when $y=0$ we have

(63) $\displaystyle\widetilde{A}=\left(\begin{array}[]{cc}0&0\\\
-s_{0}&0\end{array}\right)\text{when $x<0$ and
}\left(\begin{array}[]{cc}s_{d}&0\\\ 0&0\end{array}\right)\text{when $x>0$,
and}$ $\displaystyle\lim_{y\rightarrow
0}\frac{1}{y}\text{det}(\widetilde{A})=s_{0}\left(-\frac{s_{d}}{2x}+\alpha\right)\text{when
$x<0$ and }s_{d}\left(\frac{s_{0}}{2x}+\beta\right)\text{when $x>0$.}$

The positive $\varphi^{2}$-axis is the projection of the polar submanifold
$x<0$, $y=0$ and the Killing field on this submanifold has norm

(64)
$\|\widetilde{\mathcal{X}}^{2}\|_{\tilde{g}}^{2}=\frac{-2x}{s_{d}-2\alpha{}x}\;\stackrel{{\scriptstyle
x\rightarrow\infty}}{{\longrightarrow}}\;\frac{1}{\alpha}.$

The positive $\varphi^{1}$-axis is the projection of the polar submanifold
$x>0$, $y=0$ and the Killing field on this submanifold has norm

(65)
$\|\widetilde{\mathcal{X}}^{2}\|_{\tilde{g}}^{2}=\frac{2x}{s_{0}+2\beta{}x}\;\stackrel{{\scriptstyle
x\rightarrow\infty}}{{\longrightarrow}}\;\frac{1}{\beta}.$

If $\alpha=0$ or $\beta=0$ we have unbounded growth. In any case we have
bounds away from zero. ∎

######  Lemma 5.6.

Let $\widetilde{M}{}^{4}$ be one of the $\mathbb{R}^{2}\times\mathbb{S}^{2}$
model metrics given by the momentum functions of (53). Then
$\widetilde{M}{}^{4}$ has two unbounded polar submanifolds, and the Killing
field on each grows unboundedly.

###### Proof.

This proof is almost the same as the proof of Lemma 5.5. Again polar
submanifolds occur exactly where $y=0$. The comparison polygon, as depicted in
Figure 5(b), has its terminal rays parallel to the $\varphi^{1}$-axis;
therefore the Killing field along either unbounded polar submanifold is
$\frac{\partial}{\partial\tilde{\theta}_{1}}$. Again using (33) we have

(66)
$\left\|\frac{\partial}{\partial\tilde{\theta}_{1}}\right\|_{\tilde{g}}^{2}\;=\;\left\|d\tilde{\varphi}^{1}\right\|_{\tilde{g}}^{2}\;=\;\frac{y}{\text{det}(\widetilde{A})}\left(\left(\frac{\partial\tilde{\varphi}^{1}}{\partial{}x}\right)^{2}+\left(\frac{\partial\tilde{\varphi}^{1}}{\partial{}y}\right)^{2}\right).$

The model momenta (53) allow exact computation of this quantity. When $y=0$,
we compute to the left of the first Lipschitz point (when $x<x_{1}$) that

(67)
$\left(\frac{\partial\tilde{\varphi}^{1}}{\partial{}x}\right)^{2}+\left(\frac{\partial\tilde{\varphi}^{1}}{\partial{}y}\right)^{2}=s_{0}{}^{2}\quad\text{and}\quad\lim_{y\rightarrow
0}\frac{y}{\text{det}(\widetilde{A})}=2\frac{(x-x_{1})(x-x_{d})}{(n_{1}-n_{d})s_{0}},$

and to the right of the second Lipschitz point (when $x>x_{d}$) that

(68)
$\left(\frac{\partial\tilde{\varphi}^{1}}{\partial{}x}\right)^{2}+\left(\frac{\partial\tilde{\varphi}^{1}}{\partial{}y}\right)^{2}=s_{d}{}^{2}\quad\text{and}\quad\lim_{y\rightarrow
0}\frac{y}{\text{det}(\widetilde{A})}=2\frac{(x-x_{1})(x-x_{d})}{(n_{1}-n_{d})s_{0}}.$

Therefore in both cases we obtain growth of
$\frac{\partial}{\partial\theta_{1}}$, approximately linearly with $x$.
Specifically, on the polygon boundary where $y=0$,

(69)
$\left\|\frac{\partial}{\partial\tilde{\theta}_{1}}\right\|_{\tilde{g}}=\begin{cases}\sqrt{\frac{2s_{0}}{n_{1}-n_{d}}}\sqrt{(x-x_{1})(x-x_{d})},&x<x_{1}\\\
\sqrt{\frac{2s_{d}}{n_{1}-n_{d}}}\sqrt{(x-x_{1})(x-x_{d})},&x>x_{d}.\end{cases}$

Currenty there is no known relationship between $x$ and the distance function
on the manifold, but certainly $x$ is an exhaustion function so
$x\rightarrow\infty$ implies $\tilde{\rho}\rightarrow\infty$. Therefore
$\|\widetilde{\mathcal{X}}{}^{1}\|_{\tilde{g}}\rightarrow\infty$ grows
unboundedly along either of the unbounded polar submanifolds. ∎

Proof of Lemma 5.1. Assume $M^{4}$ is equivariantly asymptotic to
$\widetilde{M}{}^{4}$ where $\widetilde{M}{}^{4}$ is a generalized Taub-NUT.
Because the diffeomorphism
$\pi:\widetilde{M}{}^{4}\setminus\widetilde{K}\rightarrow{}M^{4}$ is
equivariant the polar submanifolds map to polar submanifolds (as polar
submanifolds are zero-sets of Killing fields). By Lemma 5.5, the model
manifold $\widetilde{M}^{4}$ has polar submanifolds with Killing fields
bounded from below; therefore by Lemma 5.4 so does $M^{4}$. Therefore by
Theorem 1.5 the reduction $\Sigma^{2}$ of $M^{4}$ is closed.

Using Lemma 5.6, if $M^{4}$ is equivariantly asymptotically
$\mathbb{R}^{2}\times\mathbb{S}^{2}$, we reach the same conclusion. ∎

###  5.4. Proof of Theorem 1.7

Assume $(M^{4},g,J,\mathcal{X}^{1},\mathcal{X}^{2})$ is a scalar-flat Kähler
manifold with commuting holomorphic Kililng fields
$\\{\mathcal{X}^{1},\mathcal{X}^{2}\\}$. For the “$\Rightarrow$” direction,
assume its reduction $\Sigma^{2}$ is closed. If $\Sigma^{2}$ is compact, then
passing to $\Sigma^{2}$ the fact that $s=0$ on $M^{4}$ means
$\triangle_{\Sigma}\mathcal{V}^{1/2}=0$ on the precompact domain
$Int(\Sigma^{2})$. By the maximum principle $\mathcal{V}^{1/2}$ has no
interior maximum, but also $\mathcal{V}^{1/2}=0$ on $\partial\Sigma^{2}$, so
$\mathcal{V}$ is zero. This means $\mathcal{X}^{1}$, $\mathcal{X}^{2}$ are
everywhere linearly dependent, violating the assumption that the manifold is
toric.

If $\Sigma^{2}$ is $\mathbb{R}^{2}$, Theorem 1.6 states $M^{4}$ is flat. If
$\partial\Sigma^{2}$ has a single line then $\Sigma^{2}$ is a half-plane, so
by Theorem 1.5 of [22] then $M^{4}$ is an exceptional half-plane instanton (or
is flat). If $\partial\Sigma^{2}$ has non-parallel rays then Lemma 5.2 states
$M^{4}$ is equivariantly asymptotically “General” and if $\partial\Sigma^{2}$
has parallel rays then Lemma 5.3 states $M^{4}$ is equivariantly
asymptotically $\mathbb{R}^{2}\times\mathbb{S}^{2}$. These four possibilities,
along with the case that $\Sigma^{2}$ is the closed strip, exhausts the
possiblities for a closed reduction $\Sigma^{2}$.

For the “$\Leftarrow$” part of the proof, assume $M^{4}$ has the asymptotic
models given by any of (1)-(5) of Theorem 1.7. If $M^{4}$ is “equivariantly
asymptotically general” meaning it has the asymptotics of (1) or
“equivariantly asymptotically $\mathbb{R}^{2}\times\mathbb{S}^{2}$” meaning it
has the asymptotics of (2), the Lemma 5.1 states its reduction $\Sigma^{2}$ is
closed. The other three cases are a fortiori true: if $M^{4}$ is the
exceptional half-plane instanton or if the reduction of $M^{4}$ is the closed
half-strip then clearly $\Sigma^{2}$ is closed. Lastly if $M^{4}$ is flat
$\mathbb{C}^{2}$ then it has three possible reductions depending on which
commuting Killing fields are chosen. If two translational fields are chose
then $\Sigma^{2}$ is $\mathbb{R}^{2}$, if one translational and one rotational
field is chosen then $\Sigma^{2}$ is a half-plane in $\mathbb{R}^{2}$, and if
two rotational fields are chosen then $\Sigma^{2}$ is a quarter-plane in
$\mathbb{R}^{2}$. This concludes the proof of Theorem 1.7.

##  6\. Examples

We give two examples. The first is a complete 4-manifold with precompact but
non-polygonal reduction. The second is the case of the scalar-flat hyperbolic
space crossed with a sphere; this is the main example of the “equivariantly
asymptotically $\mathbb{R}^{2}\times\mathbb{S}^{2}$” model geometry.

|
---|---
Figure 6.  (a) A non-polygon reduction, and (b) the reduction of the cross
product of hyperbolic space with the unit sphere.

###  6.1. A non-polygon example

Consider the function

(70) $\displaystyle
G(\varphi^{1},\varphi^{2})\;=\;-\frac{1}{2}\log\left(1-\left(\varphi^{1}\right)^{2}-\left(\varphi^{2}\right)^{2}\right)$

defined on the open disk
$\Sigma^{2}=\\{\left(\varphi^{1}\right)^{2}+\left(\varphi^{2}\right)^{2}<1\\}$.
Interpreting this as a symplectic potential, we obtain metric
$g_{\Sigma,ij}=G_{ij}d\varphi^{i}\otimes{}d\varphi^{j}$ where
$G_{ij}\triangleq\frac{\partial^{2}{}G}{\partial\varphi^{i}\partial\varphi^{j}}$.
Explicitly,

(73) $\displaystyle
G_{ij}\;=\;\left(\begin{array}[]{cc}\frac{1+\left(\varphi^{1}\right)^{2}-\left(\varphi^{2}\right)^{2}}{\left(1-\left(\varphi^{1}\right)^{2}-\left(\varphi^{2}\right)^{2}\right)^{2}}&\frac{2\varphi^{1}\varphi^{2}}{\left(1-\left(\varphi^{1}\right)^{2}-\left(\varphi^{2}\right)^{2}\right)^{2}}\\\
\frac{2\varphi^{1}\varphi^{2}}{\left(1-\left(\varphi^{1}\right)^{2}-\left(\varphi^{2}\right)^{2}\right)^{2}}&\frac{1-\left(\varphi^{1}\right)^{2}+\left(\varphi^{2}\right)^{2}}{\left(1-\left(\varphi^{1}\right)^{2}-\left(\varphi^{2}\right)^{2}\right)^{2}}\end{array}\right).$

Then $(\Sigma^{2},g_{\Sigma})$ is a surface of revolution. See Figure 6(a).
Considering a path of the form $\gamma(t)=(a\,t,\;b\,t)$ where
$a^{2}+b^{2}=1$, then $|\dot{\gamma}|=\sqrt{2}\frac{\sqrt{1+t^{2}}}{1-t^{2}}$
and we see that $\int_{0}^{1}|\dot{\gamma}|dt=\infty$, so this surface is
complete. By 3, then $(\Sigma^{2},g_{\Sigma})$ is the Arnold-Liouville
reduction of a complete toric Kähler 4-manifold on
$\Sigma^{2}\times\mathbb{T}^{2}$ whose metric is
$g=G_{ij}d\varphi^{i}\otimes{}d\varphi^{j}+G^{ij}d\theta_{i}\otimes{}d\theta_{j}$.
By the usual formula for the curvature tensor in holomorphic coordinates,

(74)
$\Omega^{i}_{j}=\sqrt{-1}\partial\left(h^{i\bar{s}}\bar{\partial}{}h_{j\bar{s}}\right).$

Using the complex coordinates of (4) we have
$h^{i\bar{\jmath}}=\frac{1}{2}G^{ij}$ and
$\frac{\partial}{\partial{}z_{i}}=\frac{1}{2}\nabla\varphi^{i}$, so

(75)
$\Omega^{i}_{j}=\frac{1}{4}\sqrt{-1}\nabla\varphi^{k}(G^{is}\nabla\varphi^{l}(G_{js}))\;dz_{k}\wedge{}d\bar{z}_{l}.$

We can compute scalar curvature $s$ either using the Abreu equation (Equation
(10) of [1]), or else using (75), to find that

(76)
$R\;=\;4\frac{2-4(\boldsymbol{\varphi})^{2}-3(\boldsymbol{\varphi})^{4}-(\boldsymbol{\varphi})^{6}}{(1+(\boldsymbol{\varphi})^{2})^{3}}$

where we abbreviated
$\boldsymbol{\varphi}=\sqrt{(\varphi^{1})^{2}+(\varphi^{2})^{2}}$. We see $R$
is not signed; for example $R=+8$ at $\boldsymbol{\varphi}=0$ and decreases to
$-3$ asymptotically as $\boldsymbol{\varphi}\rightarrow 1$. A tedious but
straightforward check shows the norm of curvature is bounded. We find

(77)
$\small*(\Omega^{i}_{j}\wedge*\Omega^{j}_{i})=32\frac{3-10({\boldsymbol{\varphi}})^{2}+29({\boldsymbol{\varphi}})^{4}+24({\boldsymbol{\varphi}})^{6}+19({\boldsymbol{\varphi}})^{8}+6({\boldsymbol{\varphi}})^{10}+({\boldsymbol{\varphi}})^{12}}{(1+({\boldsymbol{\varphi}})^{2})^{6}}.$

###  6.2. The example of hyperbolic space crossed with the sphere

A typical scalar-flat metric on $\mathbb{R}^{2}\times\mathbb{S}^{2}$ is

(78)
$g=(dr^{1})^{2}+(\sinh(r^{1})d\theta_{1})^{2}+(dr^{2})^{2}+(\sin(r^{2})d\theta_{2})^{2}$

where $r^{1}\in[0,\infty)$, $r^{2}\in[0,\pi]$, and
$\theta_{1},\theta_{2}\in[0,2\pi)$. The complex structure is
$J\frac{\partial}{\partial\theta_{1}}=-\sinh(r_{1})\nabla{}r^{1}$ and
$J\frac{\partial}{\partial\theta_{2}}=-\sin(r^{2})\nabla{}r^{2}$; therefore
the momentum functions are $\varphi^{1}=\cosh(r^{1})$,
$\varphi^{2}=-\cos(r^{2})$. In action-angle coordinates we find

(79)
$g=\frac{1}{(\varphi^{1})^{2}-1}(d\varphi^{1})^{2}+\big{(}(\varphi^{1})^{2}-1\big{)}(d\theta^{1})^{2}+\frac{1}{1-(\varphi^{2})^{2}}(d\varphi^{2})^{2}+\big{(}1-(\varphi^{2})^{2}\big{)}(d\theta^{2})^{2}$

with ranges $\varphi^{1}\in[1,\infty)$ and $\varphi^{2}\in[-1,1]$. Therefore
$\Sigma^{2}$ is a closed half-strip with vertex points at $p_{1}=(1,1)$ and
$p_{d}=(1,-1)$; see Figure 6(b). The $\Sigma^{2}$ metric is

(80)
$g_{\Sigma}\;=\;\frac{1}{(\varphi^{1})^{2}-1}(d\varphi^{1})^{2}+\frac{1}{1-(\varphi^{2})^{2}}(d\varphi^{2})^{2}$

which is a flat metric on the closed half-strip $\varphi^{1}\in[1,\infty)$ and
$\varphi^{2}\in[-1,1]$. To compute the volumetric normal coordinates, it is
immediate that

(81)
$y\;=\;\sqrt{\mathcal{V}}\;=\;\sqrt{((\varphi^{1})^{2}-1)(1-(\varphi^{2})^{2})},$

and then because
$J_{\Sigma}{}d\varphi^{1}=\sqrt{\frac{1-(\varphi^{2})^{2}}{(\varphi^{1})^{2}-1}}d\varphi^{2}$
we find

(82) $x\;=\;\varphi^{1}\varphi^{2}.$

Solving for $\varphi^{1}$ and $\varphi^{2}$ in terms of $x$ and $y$, we find

(83)
$\displaystyle\varphi^{1}=\frac{1}{2}\left(\sqrt{(x-1)^{2}+(y)^{2}}+\sqrt{(x+1)^{2}+(y)^{2}}\right),$
$\displaystyle\varphi^{2}=\frac{1}{2}\left(-\sqrt{(x-1)^{2}+(y)^{2}}+\sqrt{(x+1)^{2}+(y)^{2}}\right).$

These are precisely the comparison momentum functions of
$(\tilde{\varphi}^{1},\tilde{\varphi}^{2})$ of (53) for $s_{0}=1$, $s_{d}=1$,
$x_{1}=-1$, $x_{d}=1$, $p_{1}=(m_{1},n_{1})=(1,1)$,
$p_{d}=(m_{d},n_{d})=(1,-1)$.

## References

  * [1] M. Abreu, _Kähler geometry of toric varieties and extremal metrics_ , International Journal of Mathematics, 9 (1998) 641–651
  * [2] M. Abreu and R. Sena-Dias, _Scalar-flat Kähler metrics on non-compact symplectic toric 4-manifolds_ , Annals of Global Analysis and Geometry, 41 No. 2 (2012) 209–239
  * [3] V. Arnold, _Mathematical Methods of Classical Mechanics_ , Graduate Texts in Mathematics, 60, Springer-Verlag, New York, 1978
  * [4] D. Calderbank, L. David, and P. Gauduchon, _The Guillemin formula and Kähler metrics on toric symplectic manifolds_ , Journal of Symplectic Geometry 4 No. 1 (2002) 767–784
  * [5] J. Cheeger and M. Gromov, _Collapsing Riemannian manifolds while keeping their curvature bounded. I._ Journal of Differential Geometry 23 No. 3 (1986)
  * [6] S.Y. Cheng and S.T. Yau, _Differential equations on Riemannian manifolds and their geometric applications_ , Communication on Pure and Applied Mathematics, 28 (1975) 333–354
  * [7] S. Cherkis and A. Kapustin, _Hyperkähler metrics from periodic monopoles_ , Physics Letters D 65 (2002) 084015
  * [8] T. Delzant, _Hamiltoniens périodique et images convexes de l’application moment_ , Bulletin de la Société Mathématique de France, 116 (1988) 315–339
  * [9] S. Donaldson, _A generalized Joyce construction for a family of nonlinear partial differential equations_ , Journal of Gökova Geometry/Topology Conferences, 3 (2009)
  * [10] S. Donaldson, _Constant scalar curvature metrics on toric surfaces_ , Geometric and Functional Analysis, 19 No. 1 (2009) 83–136
  * [11] G. Etesi, _The topology of asymptotically locally flat gravitational instantons_ , Physics Letters B 641 (2006) 461–465
  * [12] A. Grigor’yan and J. Masamune, _Parabolicity and stochastic completeness of manifolds in terms of the Green formula_ , Journal de Mathématiques Pures et Appliquées 9 (2013) 607–632
  * [13] V. Guillemin, _Kähler structures on toric varieties_ , Journal of Differential Geometry, 40 (1994) 285–309
  * [14] I. Holopainen and P. Koskela, _Volume growth and parabolicity_ , Proceedings of the American Mathematical Society, 129 No. 11 (2001) 3425–3435
  * [15] S. Hawking, _Gravitational instantons_. Physics Letters A 60 No. 2 (1977) 81–83.
  * [16] P. Li and L. Tam, _Positive harmonic functions on complete manifolds with non-negative curvature outside a compact set_ , Annals of Mathematics, 125 (1987) 171–207
  * [17] P. Li and L. Tam, _Harmonic functions and the structure of complete manifolds_ , Journal of Differential Geometry, 35 (1992) 359–383
  * [18] D. McDuff and D. Salamon. Introduction to symplectic topology. Vol. 27. Oxford University Press, 2017.
  * [19] R. Schoen and S.T. Yau, Lectures on Differential Geometry, International Press (1994)
  * [20] N. Trudinger amd X. Wang, ”Bernstein-Jörgens theorem for a fourth order partial differential equation.” Journal of Partial Differential Equations 15 No. 1 (2002) 78–88
  * [21] B. Weber, _Regularity and a Liouville theorem for a class of boundary-degenerate second order equations_ , Journal of Differential Equations. 281 (2021) 459–502
  * [22] B. Weber, _Analytic classification of toric Kähler instanton metrics in dimension 4_ , To Appear in Journal of Geometry
  * [23] B. Weber, _Generalized Kähler Taub-NUT Metrics and Two Exceptional Instantons_ , Communications in Analysis and Geometry 30 No. 7 (2022) 1575–1632