Hermitian, Ricci-flat, toric metrics]Hermitian, Ricci-flat toric metrics on non-compact surfaces à la Biquard-Gauduchon
Gonçalo Oliveira, Rosa Sena-Dias

Partially supported by the Fundação para a Ciência e a Tecnologia (FCT/Portugal) through project EXPL/MAT-PUR/1408/2021EKsta.

Centro de Análise Matemática, Geometria e Sistemas Dinâmicos, Departamento de Matemática, Instituto Superior Técnico

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In [5], Biquard-Gauduchon show that conformally Kähler, Ricci-flat, ALF toric metrics on the complement of toric divisors are: the Taub-NUT metric with reversed orientation, in the Kerr-Taub-bolt family or in the Chen-Teo family. In the same paper, Biquard-Gauduchon also give a unified construction for the above families relying on an axi-symmetric harmonic function on $\bbR^3$. In this work, we reverse this construction and use methods from [18] to show that all conformally Kähler, Ricci-flat, toric metrics on the complement of toric divisors, under some mild assumptions on the associated moment polytope, are among the families above. In particular all such metrics are ALF.

§ INTRODUCTION

Einstein metrics play an important role in geometry. In the Kähler setting a special subclass of such metrics has been much studied, namely Kähler-Einstein metrics. The existence problem for Kähler-Einstein metrics in the first Chern class of a Kähler manifold has been the subject of intense investigation. Chen-Donaldson-Sun (see [7] and subsequent articles by the same authors) have proved that such metrics exist if and only if the underlying Kähler class is  K-stable. In particular, there are Kähler $4$-manifolds such as $\CP^2\sharp\overline{\CP}^2$ or $\CP^2\sharp2\overline{\CP}^2$ which do not admit any Kähler-Einstein metric. It is then natural to look for Einstein metrics that are Hermitian i.e. for which the metric is preserved by the complex structure.  Page constructed one such metric on $\CP^2\sharp\overline{\CP}^2$.

It follows from the Riemannian Goldberg-Sachs Theorem and the work of Derdzinski that, in dimension four, all Hermitian-Einstein metrics are actually conformally Kähler [12, 4]. Derdzinski further showed that, under certain assumptions, extremal, Bach-flat, Kähler metrics on $4$-manifolds admit an Einstein metric in their conformal classes. Using Derdzinski's results, Chen-LeBrun-Weber (see [8]) proved the existence of a Hermitian-Einstein metric on $\CP^2\sharp2\overline{\CP}^2.$ It is natural to wonder wether Derdzinski's results could yield new Einstein metrics in the non-compact case and to what extent such metrics are unique when they exist.

In [5], Biquard and Gauduchon applied Derdzinski's methods to construct Hermitian, Ricci-flat, toric metrics on non-compact spaces having a Poincaré type behaviour at infinity. In fact, all of the metrics they construct this way had been previously written down. The metrics are: the Taub-NUT metric with reversed orientation on the complement of a divisor in $\cp^2$; in the Kerr family on the complement of a divisor in $\CP^2\sharp\overline{\CP}^2$; in the Taub-bolt family in the complement of a different divisor in $\CP^2\sharp\overline{\CP}^2$, or in the Chen-Teo family in the complement of a divisor in $\CP^2\sharp2\overline{\CP}^2.$ Although the smooth metrics appearing in [5] are not new, the authors give a new, unified construction for all the metrics in the above families. They also prove a classification result for Hermitian, Ricci-flat, toric metrics with ALF behaviour at infinity.

Let $(X,g)$ be a Hermitian, non-Kähler, Ricci-flat, ALF toric metric. Then $g$ is either one of the metrics constructed by Chen-Teo, one of the Kerr-Taub-bolt metrics or the Taub-NUT metric with opposite orientation.

From left to right, polytopes corresponding to the reversed Taub-Nut, Kerr, Taub-bolt and Chen–Teo families.

The Taub-NUT metric is a Kähler, Ricci-flat, toric metric on $\bbR^4$ which is also the complement of a torus invariant divisor in $\bbC^2$. There is a different complex structure in $\bbR^4$ with respect to which the Taub-NUT metric is no longer Kähler but rather Hermitian. It is asymptotic to $\bbR^2\times S^1$ in a precise manner at infinity, hence ALF. The Kerr-Taub-bolt metrics are other examples of Kähler Ricci-flat metric with a $\bbT^2$-symmetry. They are defined on the complement of a divisor in $\CP^2\sharp\overline{\CP}^2$ and include the famous Schwarzschild metric as a degeneration.
Many compact examples of Ricci-flat Kähler metrics arise via hyperKähler geometry. Ricci-flat, complete, non-compact metrics, with a certain assumption on the decay of the curvature at infinity, also known as instantons, have been well studied too, particularly Kähler gravitational instantons. We currently have classification result for such Kähler gravitational instantons thanks to Chen-Chen (see [6]). In particular their behaviour at infinity is known to be restricted.

In [18], the second author obtained a classification result for non-compact, Kähler, scalar-flat, toric metrics. As it turns out, Kähler, scalar-flat, toric metrics all arise from a construction in [3] which is based on an ansatz of Joyce that was translated into the language of action-angle coordinates on toric Kähler manifolds by Donaldson (see [10]). Such metrics are known as Donaldson generalised Taub-NUT metrics.

Let $(X,J,g)$ be a strictly unbounded toric Kähler surface. If $g$ is scalar-flat, then $g$ is equivariantly isometric to a Donaldson generalised Taub-NUT metric.

By using Joyce's ansatz, the theorem results from a classification of harmonic functions in Euclidean space with prescribed singularities. Toric geometry plays an important role and convexity is used to prove the positivity of an appropriate harmonic function. One of the main tools is a result due to Donaldson (see  [10]) that any Kähler, scalar-flat, toric metric can be written via Joyce's ansatz.

The goal of this note is to apply the methods in [18] to the setting of conformally Kähler, Ricci-flat, toric metrics. We shall use Biquard-Gauduchon's ansatz translated into the language of toric geometry in replacement of Joyce's ansatz. We will show that any conformally Kähler, Ricci-flat, toric metric can be written in terms of Biquard-Gauduchon's ansatz via a single axi-symmetric harmonic function on the upper half plane.

Our uniqueness result for such metrics is stronger than the result in [5] as it does not make use of the ALF hypothesis. In fact, contrary to most uniqueness results for special metrics on non-compact manifolds, our theorem holds without assumptions on asymptotics. To be more precise we prove the following.

Let $(M^4,J,\omega)$ be a compact toric Kähler surface. Let $D$ be a torus invariant divisor in $M.$ Let $X=M\setminus D$ and assume $X$ is endowed with two $J$-Hermitian, Ricci-flat,  torus invariant metrics $g$ and $\tilde{g}$ whose adapted Kähler forms[See the discussion following the statement of the Theorem, or Definition (<ref>) later in the text, for a precise definition of adapted Kähler form.] are cohomologous to $[\omega|_X]$. Then $g$ and $\tilde{g}$ are equivariantly isometric.

It follows from the Riemannian Goldberg-Sachs Theorem [12, 4] that such Hermitian and Einstein $4$-manifolds are necessarily conformally Kähler. Therefore, $g=e^{\sigma}g_K$ for a smooth function $\sigma$, which we can normalise to satisfy $\sigma(x)=0$ at some fixed $x \in X$, and a Kähler metric $g_K$ which together with $J$ determines a Kähler form $\omega_K$ which we call the adapted Kähler form. Notice that this is uniquely determined by imposing that $\sigma$ vanishes at $x \in X$. In our theorem, we require that the adapted forms for the two conformally Kähler, Ricci-flat, toric metrics be in the same cohomology class. The  Kähler metric $g_K$ together with $J$ determines a symplectic form $\omega.$ If $J$ and $g$ are torus invariant then, as we will see, so is  $g_K$ and therefore $\omega.$ The torus action must be Hamiltonian for $\omega$ and it therefore defines a moment polytope which is of Delzant type. The cohomology class of the adapted form fixes a moment polytope for the torus action on $(M,J)$, up to $SL(2,\bbZ)$ transformations and translations, because by Moser's trick cohomologous equivariant forms are equivariantly symplectomorphic and therefore have the same moment polytope, again up to $SL(2,\bbZ)$ transformations and translations. Furthermore, we remark that torus invariant divisors $D$ as in the statement, correspond to moment map pre-images of codimension-$1$ faces of the moment polytope.

A canonical quantity which is associated with $(M^4,D,J,\omega)$ is the extremal vector field of $X=M\setminus D$ which can be thought of as a vector in $\bbR^2\simeq \left(\text{Lie}(\bbT^2)\right)^*$. This is essentially the same as the Futaki invariant of $X=M/D.$ We will give its precise definition ahead. The important fact for now is that it depends only on the symplectic structure and can be calculated using the Delzant polytope.

Let $(M^4,J,\omega)$ be a compact toric Kähler surface and $D$ a torus invariant divisor in $M.$ Assume that  $X=M\setminus D$ carries a $J$-Hermitian, Ricci-flat, toric metric whose adapted Kähler form is cohomologous to $[\omega|_X]$. Further, suppose that the extremal vector field of the adapted Kähler form is not normal to exactly one of the non-compact edges of the moment polytope of $X$. Then, the Hermitian metric on $X$ is equivariantly isometric to one of the Biquard-Gauduchon metrics from Theorem (<ref>).

In particular, under the above assumptions all Hermitian, Ricci-flat, toric metrics are ALF.
Recall that such metrics are conformally Kähler. Then, the adapted Kähler form $\omega_K$ lies in the class $[\omega |_X]$ for $X=M\setminus D$, where $(M,J,\omega)$ is compact Kähler and $D \subset M$ is a torus invariant divisor. In particular, the moment polytope of $(X,\omega_K)$ is obtained by removing an edge from the moment polytope of $(M,\omega)$. On the other hand, there are conformally Kähler, Ricci-flat, toric metrics which are not ALF. Consider the Eguchi-Hanson metric on the total space of the bundle $\mathcal{O}(-2)$ over $\mathbb{CP}^1$. It is Kähler with respect to the standard complex structure on $\mathcal{O}(-2).$ Now $\mathcal{O}(-2)$ admits a different complex structure yielding the reversed orientation, which is biholomorphic to $\mathcal{O}(2)$. The Eguchi-Hanson metric is Hermitian with respect to this complex structure. Now $\mathcal{O}(2)$ is biholomorphic to $\cp^2$ minus a point which is biholomorphic to $\bbH_1\setminus D$ for a torus invariant divisor $D$ of self-intersection $2.$ But the adapted Kähler form associated to this reversed Eguchi-Hanson metric determines a polytope which is a $2$-simplex minus a vertex. This is not the moment polytope of a manifold of the form $X=M\setminus D$ for $(M^4,J)$ toric and $D$ a torus invariant divisor in $M.$ The Kähler form associated to the conformally Kähler metric is not equivariantly symplectomorphic to any symplectic form on $M\setminus D.$ Another way to think of this is to note that if the Eguchi-Hanson metric were conformal to a Kähler metric on $\bbH_1\setminus D,$ the corresponding cohomology class would integrate to zero on the removed divisor which is not possible.

Recently, Mingyang Li (see [15]) obtained uniqueness results for conformally Kähler, Ricci-flat, ALE metrics. Elsewhere (see [16]), Li also generalised Biquard-Gauduchon's  classification result to all ALF, conformally Kähler, Ricci-flat metrics by showing that such metrics are necessarily toric. Our results only apply to the toric setting but we make no assumption on asymptotic growth.

§ BACKGROUND

§.§ Conformally Kähler Einstein metrics

Since we make use of several results from [9], we will start by briefly stating those in this subsection. For more details and proofs see [9].

Let $(X,g)$ be any Riemannian manifold. There is a splitting induced by the Hodge star corresponding to metric on the space $\Lambda^2T^*X.$ This lets us define self-dual and anti-self-dual forms i.e. the $\pm1$ eigenspaces of the Hodge star on $\Lambda^2T^*X.$ According to this splitting, the self-dual part of the Weyl curvature $W$ which we denote by $W^+$ is the part of the endomorphism $W$ of  $\Lambda^2T^*X$ which takes $\Lambda^{2+}T^*X,$ to itself.  Derdzinski interprets $W^+$ at a point $x\in X$ as an endomorphism of $\Lambda^{2+}T_x^*X,$ the space of self-dual sections of $\Lambda^2T_x^*X.$ At each $x\in X$, $\Lambda^{2+}T_x^*X,$ has dimension $3$ and $W^+$ has a real spectrum.

The Riemannian manifold $(X,g)$ is said to be half-algebraically simple if at each point in $X,$ $W^+$ has at least one double eigenvalue. In this case we write $\sharp(\spec(W^+))\leq 2.$

When $X$ is a complex manifold we say that a metric $g$ on $X$ is Hermitian if $J$ preserves $g$ i.e.

g_x(JV_x,JW_x)=g_x(V_x,W_x), \, \forall x\in \, X, \, V_x, W_x\in T_xX.

When $(X,J, \omega, g_K)$ is Kähler, any metric conformal to $g_K$ is Hermitian. It follows from a Riemannian version of the Goldberg-Sachs Theorem [12, 4] that $4$-dimensional Hermitian Einstein metrics are also conformally Kähler.

We gather several results which are due to Derdzinski in a theorem below.

Let $(M,\omega, g_K)$ be a Kähler manifold.

* The Hermitian metric $\frac{g_K}{\scal^2(g_K)}$ is half-algebraically simple and satisfies $\delta W^+=0,$ where defined.

* If $e^\sigma g_K$ satisfies $\delta W^+=0,$ then $e^{-\sigma}$ is a constant multiple of $\scal^2(g_K)$ on the subset where $W^+\ne 0.$

* The Hermitian metric $\frac{g_K}{\scal^2(g_K)}$ is Einstein iff its Bach tensor vanishes identically.

* If $\frac{g_K}{\scal^2(g_K)}$ is Einstein, then $\nabla\scal(g_K)$ is holomorphic and the metric $g_K$ is extremal in the sense of Calabi.

Another result of Derdzinski's is the following.

Let $(X,g)$ be an oriented, half-algebraically simple, Einstein manifold. Let $\lambda$ be the only eigenvalue of $W^+$ which is not double and $\omega$ be the corresponding eingenform. Then $(X,\lambda^{2/3}g,\lambda^{2/3}\omega)$ is Kähler.

Next, we introduce the following notion which is convenient in stating our results.

Let $(X,g,J)$ be conformally Kähler and $x \in X$. Then, $g=e^{\sigma}g_K$ for a smooth function $\sigma$ satisfying $\sigma(x)=0$ and a Kähler metric $g_K$ which together with $J$ determines a Kähler form $\omega_K$. In this situation, we shall say that $\omega_K$ is the Kähler form adapted to $g$ and $x$.

In order to have uniqueness in the definition of the adapted Kähler form we have fixed $\sigma(x)=0$ in the previous definition. Otherwise, the adapted Kähler form would only be well defined up to scaling.

§.§ Toric Kähler surfaces

We give a very brief overview of toric Kähler geometry and action-angle coordinates mainly to set up notation. For more on this see [1].

A Kähler manifold $(X^4,\omega,g_K)$ is said to be toric if it admits an effective, isometric, Hamiltonian $\bbT^2$-action whose moment map is proper. Under such assumptions, the moment map

\bx=(x_1,x_2):X\rightarrow \bbR^2

has as image a convex polytope $P$ of the form

P=\{(x_1,x_2)\in \bbR^2: l_i(x_1,x_2):=\langle (x_1,x_2),\nu_i\rangle - \lambda_i\geq 0, i=1,\cdots, d+1\},

where $\nu_i$ are the primitive, interior facet normals to $P.$ At each vertex of $P$ the facet normals corresponding to the facets meeting at that vertex say $\nu_j$ and $\nu_{j+1}$ for $j=1,\cdots, d$ satisfy

\det(\nu_j,\nu_{j+1})=-1.

The pre-image via $\bx$ of each facet of $P$ is a $\bbT^2$-invariant $\cp^1$ i.e. a divisor.

On the pre-image of the interior of $P,$ the $\bbT^2$-action is free so that

\bx^{-1}(\interior(P))\simeq \interior(P)\times \bbT^2.

Note that this set is dense in $X.$ Let $(\theta_1,\theta_2)$ denote angle coordinates on $\bbT^2. $ Then $(x_1,x_2,\theta_1,\theta_2)$ are the so-called action-angle coordinates coordinates on $\bx^{-1}(\interior(P))$. Action-angle coordinates are Darboux coordinates for $\omega$ i.e.

\omega=dx_1\wedge d\theta_1+dx_2\wedge d\theta_2.

As for the metric $g_K,$ it is known that it can be expressed as

g_K=\sum_{i,j=1}^2 u_{ij}dx_i\otimes dx_j+u^{ij}d\theta_i\otimes d\theta_j

where $u:\interior(P)\mapsto \bbR$ is a smooth convex function, $u_{ij}$ denote the entries of the Hessian of $u$ and $u^{ij}$ denote the entries of the inverse of the Hessian of $u.$ This is explained in [18] for instance. The function $u$ is called the symplectic potential of the metric $g_K.$ The boundary behaviour of $u$ is determined by $P$. In particular,

is smooth up to the boundary of $P.$ Moreover, it has been shown that there is a smooth, positive function which we denote by $\delta$ such that,

\det(\Hess u)=\frac{\delta}{\prod_{j=1}^{d+1}l_j}.

We will also make use of the real sub-manifold of $X$ whose definition we quickly recall. For more details see [18].  Guillemin has shown that as complex manifolds, toric manifolds can be viewed as quotients of $\bbC^{d+1}$ by a subgroup of $(\bbC^*)^{d+1}$, where $\bbC^*$ acts on $\bbC$ in the usual way. Complex conjugation on $\bbC^{d+1}$ descends to $X$ and thus $X$ admits an isometric involution. The fixed point set of this involution is called the real sub-manifold of $X$ and we denote it by $X_\bbR$. The moment coordinates $(x_1,x_2)$ yield a coordinate system on the open dense subset of $X_\bbR$ given by $\bx^{-1}(\interior(P))\cap X_\bbR.$ The metric $g_K$ restricts to $X_\bbR$. In the coordinates  $(x_1,x_2)$ it is given by:

g_\bbR=\sum_{i,j=1}^2 u_{ij}dx_i\otimes dx_j.

Alternatively, it will be useful to think of $\bx$ as a Riemannian submersion onto $\interior(P)$ equipped with a metric with the same formula as $g_\bbR$ which we shall still denote by $g_\bbR$. Metric related quantities can be expressed in terms of $u$ and throughout the paper we will use the following formulas:

* Let $h:X\rightarrow \bbR$ be a smooth, $\bbT^2$-invariant function on $X$. Then $h$ can be thought of as a function on $P.$ In coordinates on the open set  $\bx^{-1}(\interior(P))$, $h$ depends only on $(x_1,x_2)$. The Laplacian of $h$ is given by:

\begin{equation}\label{lapgK}
\Lap h=\sum_{i,j=1}^2\frac{\partial}{\partial x_i}\left( u^{ij}\frac{\partial h}{\partial x_j}\right),
\end{equation}

on $\bx^{-1}(\interior(P)).$

* Such a smooth $\bbT^2$-invariant function $h:X\rightarrow \bbR$ is pulled back from $P$ and so we also take its Laplacian with respect to the metric $g_\bbR$ on $\interior(P)$ which we denote by $\Lap_\bbR h.$ It is given by

\Lap_\bbR h=\frac{1}{\sqrt{\det(\Hess u)}}\sum_{i,j=1}^2\frac{\partial}{\partial x_i}\left( \sqrt{\det(\Hess u)} u^{ij}\frac{\partial h}{\partial x_j}\right).

* The scalar curvature of $g_K$ is given by the Abreu formula,

\scal(g_K)=\sum_{i,j=1}^2\frac{\partial^2 u^{ij}}{\partial x_i \partial x_j}.

This can equally be written by making use of $\Lap_\bbR$ as

$$\scal(g_K)= \frac{\Lap_\bbR V}{V},$$

where $V=(\det(\Hess u))^{-\frac{1}{2}}$. See Proposition 3 in [17].

§.§ Extremal toric Kähler metrics

Let $(X^{2n},J,\omega,g_K)$ be a Kähler manifold. For a suitable class of functions $\varphi$ on $X,$ $\omega+\sqrt{-1}\partial\bar{\partial} \varphi$ defines another Kähler metric compatible with $J.$ In fact all Kähler metrics whose cohomology class is $[\omega]$ are of the form $\omega+\sqrt{-1}\partial\bar{\partial} \varphi$ for some $\varphi.$ Calabi defined a functional on the set $\{\varphi\in \mathcal{C}^\infty(X):\omega+\sqrt{-1}\partial\bar{\partial} \varphi \,\text{ is K\"ahler}\}$ by:

\mathcal{C}(\varphi)=\int_X\scal^2(\omega+\sqrt{-1}\partial\bar{\partial} \varphi)(\omega+\sqrt{-1}\partial\bar{\partial} \varphi)^n,

where $\scal(\omega+\sqrt{-1}\partial\bar{\partial} \varphi)$ means the scalar curvature of the Riemannian metric associated with $\omega+\sqrt{-1}\partial\bar{\partial} \varphi.$
A Kähler metric in $[\omega]$ is said to be extremal (in the sense of Calabi) if it is a critical point for $\mathcal{C}.$

It is well know that a Kähler metric $\omega$ is extremal in the sense of Calabi in its cohomology class iff $\nabla^{(1,0)} \scal(\omega)$, the $(1,0)$ part of the gradient of $ \scal(\omega)$, is a holomorphic vector field. A special class of extremal metrics is the class of constant scalar curvature Kähler metrics. In the toric setting, it can be shown that $(X^4,\omega,g_K)$ is extremal iff $\scal(g_K)$ is an affine function of the action coordinates $(x_1,x_2).$

We known that not all Kähler manifolds admit extremal metrics. For toric manifolds, independently of existence of extremal metrics, it is possible to calculate the only affine function that may be the scalar curvature of an extremal metric in case it exits. This is called the extremal affine function. We explain how to determine it for toric surfaces since it will be relevant for our purposes. Let $X$ be a toric Kähler manifold of real dimension $4$ with moment polytope $\Delta.$ As before write

\Delta=\{(x_1,x_2)\in \bbR^2: l_i(x_1,x_2):=\langle (x_1,x_2),\nu_i\rangle- \lambda_i\geq 0, i=1,\cdots, d\}.

Following [11], we are going to define a measure $d\sigma$ on the boundary of $\Delta.$ On the edge $l_i^{-1}(0)\cap \Delta$, which we denote by $E_i$ and whose interior normal is $\nu_i=(\nu_i^1,\nu_i^2),$ we define $d\sigma$ so that

d\sigma\wedge( \nu_i^1dx_1+\nu_i^2dx_2)=dx_1\wedge dx_2.

The extremal affine function $\alpha_0+\alpha_1x_1+\alpha_2x_2$ is the only affine function on $\Delta$ satisfying

\int_\Delta (\alpha_0+\alpha_1x_1+\alpha_2x_2)h(x_1,x_2)dx_1dx_2=\int_{\partial \Delta}h(x_1,x_2)d\sigma,

for all affine functions $h$ on $\Delta.$ This can be expressed as

\begin{pmatrix}
\int_\Delta 1 &\int_\Delta x_1&\int_\Delta x_2 \\
\int_\Delta x_1 &\int_\Delta (x_1)^2&\int_\Delta x_1x_2 \\
\int_\Delta x_2 &\int_\Delta x_1x_2&\int_\Delta (x_2)^2 \\
\end{pmatrix}
\begin{pmatrix}
\alpha_0\\
\alpha_1\\
\alpha_2
\end{pmatrix}=
\begin{pmatrix}
\int_{\partial \Delta} d\sigma  \\
\int_{\partial \Delta} x_1 d\sigma  \\
\int_{\partial \Delta} x_2 d\sigma   \\
\end{pmatrix}.

We can translate $\Delta$ so that $\alpha_0=0$ in which case we write $\fut=(\alpha_1,\alpha_2)$ and we call $\fut$ the extremal vector.

Let $(M,\omega,J)$ be a compact toric Kähler manifold with moment polytope $P$ and $D$ a torus invariant divisor. Let $E$ be the image of $D$ in $P$. Then, in this article we will be interested in using the setup of this section in the case where $X=M \setminus D$ and $\Delta =P\setminus E$. The extremal affine function of $M\setminus D$ is the extremal affine function of $M$ with weight zero along the edge $E.$

§.§ Biquard-Gauduchon's construction

In [5], Biquard-Gauduchon construct Hermitian, non-Kähler, Ricci-flat ALF toric metrics. Start with $f$ any function of $\bbR$ to $\bbR$ of the form

f(z)=A+\sum_{i=1}^d a_i|z-z_i|,

for some choice of $a_1,\cdots, a_d$ and $z_1<\cdots< z_d,$ and positive $A.$ There is a unique function $U^{BG}$ which is axi-symmetric harmonic in $\bbR^3$ and such that

\lim\limits_{\rho \to 0}\frac{U^{BG}(z,\rho)}{\log (\rho^2)}=f(z).

This can be explicitly written as

\begin{align}\nonumber
U^{BG}(z,\rho)& =2\sum_{i=1}^{d-1} a_i\left(\sqrt{\rho^2+(z-z_i)^2}-(z-z_i)\log\left(\frac{\sqrt{\rho^2+(z-z_i)^2}+z-z_i}{\rho}\right)\right) \\ \label{UBG}
& +A\log(\rho^2).
\end{align}

Here $z$ is the third coordinate in $\bbR^3$ and $\rho$ is the distance to the $z$-axis.
One of the results in [5] is the following.

Let $f$ be a function given by

f(z)=A+\sum_{i=1}^d a_i|z-z_i|,

for $A>0$ and ${a_1,\cdots,a_d>0}.$ Let $U^{BG}$ be the axi-symmetric harmonic function of $(z,\rho)$ defined on the whole upper half-plane and such that

\lim\limits_{\rho \to 0}\frac{U^{BG}(z,\rho)}{\log (\rho^2)}=f(z).

There is a local Kähler toric metric $g^{BG}_K$ with symplectic potential $u^{BG},$ such that:

\rho U^{BG}_{\rho}=\frac{1}{\scal(g^{BG}_K)}\qquad U^{BG}_z=\det(\triangledown u^{BG}, \fut),

where $\fut$ is the vector whose coefficients are the coefficients of the linear part of $\scal(g^{BG}_K)$. This metric is extremal and Bach-flat. Moreover, $g^{BG}=\frac{g^{BG}_K}{\scal^2(g^{BG}_K)}$ is Ricci-flat and locally toric.

There is a toric manifold of the form $M\setminus D,$ for a torus invariant divisor $D,$ such that the metric $g^{BG}_K$ extends smoothly to $M\setminus D$ only when $g^{BG}_K$ coincides with one of metrics in Theorem (<ref>).

The above is a consequence of Corollary (3.6), Corollary (5.2) and Lemma (8.1) in [5]. The metrics arising from the above theorem were all known but the theorem gives a new unified construction for all the metrics appearing in Theorem (<ref>).

§ AN EXTREMAL BACH-FLAT TORIC KÄHLER METRIC FROM $G$

Let $M$ be a toric Kähler manifold with moment polytope

P=\{(x_1,x_2)\in \bbR^2: l_i(x_1,x_2):=\langle (x_1,x_2),\nu_i\rangle- \lambda_i\geq 0, i=1,\cdots, d+1\}.

This is the image of the moment map $\bx:M\rightarrow \bbR^2$ and we shall denote its edges by

$$E_i=l_i^{-1}(0), \ \ i \in \lbrace 1, \ldots , d+1 \rbrace ,$$

and when no confusion may arise, we shall denote the last of these edges by $E:=E_{d+1}$. Let $D=\bx^{-1}(E)$ be the toric divisor corresponding this last edge of $P.$ Assume that $X=M\setminus D$ is endowed with a conformally Kähler, Ricci-flat, toric  metric $g.$ The goal of this section is to prove the following.

Let $g$ be a conformally Kähler, Ricci-flat, toric metric on $X.$ Then, there is a Kähler toric metric $g_K$ which is extremal and Bach-flat  to which $g$ is conformal.

The proof essentially follows from Theorem (<ref>) and Theorem (<ref>).
The first thing to note is that since $g$ is Ricci-flat it is Einstein and therefore it satisfies $\delta W^+=0.$ This is because

\nabla^p W_{pkij}=\nabla_i\mP_{kj}-\nabla_j\mP_{ki},

with $\mP=\frac{1}{2}\left(\text{Ric}-\frac{1}{6}\scal\right)$ being the Einstein tensor. In the Einstein case this is zero so that from the formula above $\delta W=0$. But

|\delta W|^2=|\delta W^+|^2+|\delta W^-|^2

and so $\delta W^+=0.$ Our metric $g$ on $X=M\setminus D$ is Ricci-flat and Hermitian and by the Riemannian Golberg–Sachs theorem [4, 12] it is half-algebraically simple. We can thus apply Theorem (<ref>) to $g$. Let $\lambda$ be the simple eigenvalue of $W^+(g)$ and $\omega$ the corresponding eigenform. Because $g$ is $\bbT^2$-invariant, so is $W^+$ and therefore $\lambda$ and $\omega$ are $\bbT^2$-invariant as well. The Kähler metric $g_K=\lambda^{2/3}g$ is therefore $\bbT^2$-invariant. We have $g=\lambda^{-2/3}g_K$ and therefore, combining the second bullet in Theorem (<ref>) $\lambda^{1/3}=\scal(g_K)$ and

for a toric Kähler metric $g_K.$ This metric must be extremal and Bach-flat from the third and fourth bullets in Theorem (<ref>).

§ ACTION-ANGLE COORDINATE SETUP

From the previous section there is a Kähler toric metric $g_K$ which is extremal and Bach-flat  to which the Ricci-flat metric $g$ is conformal. We can apply the toric Kähler framework to the metric $g_K.$ There is an open dense set in $X=M\setminus D$ and action-angle coordinates $(x_1,x_2,\theta_1,\theta_2)$ on that set and a symplectic potential $u$ for $g_K$ such that

g_K=\sum_{i,j=1}^2 \left( u_{ij}dx_i\otimes dx_j+u^{ij}d\theta_i\otimes d\theta_j \right),

and recall that we have an induced metric on the polytope which we denoted by $g_\bbR$. Because $g_K$ is extremal, $\scal(g_K)$ is an affine function of $(x_1,x_2).$ Without loss of generality we assume it vanishes at zero so that

\scal(g_K)=\alpha_1x_1+\alpha_2x_2

with $\fut=(\alpha_1,\alpha_2)$ being the extremal vector field.

\rho(x_1,x_2)=\frac{(\det(\Hess{u}))^{-1/2}}{\scal^2(g_K)}.

Then, both $\rho$ and $\scal(g_K)$ are $\bbT^2$-invariant functions, which are therefore pulled back from $P$, and we can consider taking their Laplacian with respect to the metric $g_\bbR$ on $\interior(P)$. This leads to the following lemma.

The function $\rho$ is harmonic for $g_\bbR$.

This proof is a really a calculation.

\Lap_\bbR \rho=\frac{1}{\sqrt{\det(\Hess u)}} \sum_{i,j=1}^2\frac{\partial}{\partial x_i}\left( \sqrt{\det(\Hess u)}u^{ij}\frac{\partial \rho}{\partial x_j}\right),

so that $\Lap_\bbR \rho=0$ iff

\sum_{i,j=1}^2\frac{\partial}{\partial x_i}\left( \sqrt{\det(\Hess u)}u^{ij}\frac{\partial \rho}{\partial x_j}\right)=0.

Now $-\sum_{i,j=1}^2\frac{\partial}{\partial x_i}\left( \sqrt{\det(\Hess u)}u^{ij}\frac{\partial \rho}{\partial x_j}\right)$ is given by
∑_i,j=1^2∂/∂x_i( √((u))u^ij( ((u))^-3/2/2^2 (g_K)∂(u)/∂x_j+ .
. 2((u))^-1/2/^3 (g_K)∂(g_K)/∂x_j))

=   ∑_i,j=1^2∂/∂x_i( u^ij( 1/2^2 (g_K)∂log((u))/∂x_j +2/^3 (g_K)∂(g_K)/∂x_j))
=   ∑_i,j=1^2∂/∂x_i( u^ij( 1/2^2 (g_K)∑_a,b=1^2u^abu_abj +2/^3 (g_K)∂(g_K)/∂x_j)),

u_{ijk}=\frac{\partial^3 u}{\partial x_i\partial x_j\partial x_k} , \qquad \forall i,j,k=1,2.

This is because for any smooth family of invertible matrices $A(t)$ we have

\frac{d\log\det A(t)}{dt}=\tr\left(A^{-1}(t)\frac{d A(t)}{dt}\right).

We also have

\frac{d A(t)^{-1}}{dt}=-A\frac{d A(t)}{dt}A^{-1},

\sum_{a,b,j=1}^2u^{ij}u^{ab}u_{abj}=-\sum_{k=1}^2u^{ik}_k

u^{ij}_k=\frac{\partial u^{ij}}{\partial x_l}, \qquad \forall i,j,k=1,2.

So $\Lap \rho=0$ iff

\sum_{i,j=1}^2\frac{\partial}{\partial x_i}\left(-\frac{\sum_{k=1}^2 u^{ik}_k}{2\scal^2(g_K)} +\frac{2u^{ij}}{\scal^3(g_K)}\frac{\partial \scal (g_K)}{\partial x_j}\right)=0.

This is equivalent to
-∑_i,k=1^2 u^ik_ik/2^2(g_K) +∑_i,k=1^2 u^ik_k/^3(g_K)∂(g_K)/∂x_i+
∑_j=1^2 ∑_i=1^2 2u^ij_i/^3(g_K)∂(g_K)/∂x_j

+∑_i,j=1^2 2u^ij/^3 (g_K)∂^2 (g_K)/∂x_i∂x_j-6∑_i,j=1^2 2u^ij/^4 (g_K)∂(g_K)/∂x_i∂(g_K)/∂x_j=0.

Because our metrics are extremal

\frac{\partial^2 \scal (g_K)}{\partial x_i\partial x_j}=0, \qquad \forall i,j=1,2.

From Abreu's equation

\sum_{i,k=1}^2 u^{ik}_{ik}=-\scal (g_K).

The above simplifies to yield

\begin{equation}\label{lap=0}
\frac{1}{2\scal(g_K)} +\frac{3\sum_{i,k=1}^2 u^{ik}_k}{\scal^3 (g_K)}\frac{\partial \scal (g_K)}{\partial x_i}-6\sum_{i,j=1}^2 \frac{2u^{ij}}{\scal^4 (g_K)}\frac{\partial \scal (g_K)}{\partial x_i}\frac{\partial \scal (g_K)}{\partial x_j}=0. \\
\end{equation}

Next we make use of the Bach-flatness condition. It is known that Bach-flatness in our setting is equivalent to

\scal \left(\frac{g_K}{\scal^2 (g_K)}\right)

being constant. Because $\frac{g_K}{\scal^2 (g_K)}$ is Ricci-flat, the constant is zero therefore Bach-flatness is equivalent to

\scal \left(\frac{g_K}{\scal^2 (g_K)}\right)=0

Now, using the notation $\Lap_{g_K}$ to denote the Laplacian on $X$ with respect to $g_K$ to avoid confusion, we have

\scal \left(\frac{g_K}{\scal^2 (g_K)}\right)=(\scal (g_K))^3\left(1-6\Lap_{g_K}\left(\frac{1}{\scal (g_K)}\right)\right),

and we get

\Lap_{g_K}\left(\frac{1}{\scal (g_K)}\right)=\frac{1}{6}.

Let us make this explicit by using Equation (<ref>).
1/6   =   ∑_i,j=1^2∂/∂x_i(u^ij∂/∂x_i(1/(g_K)))
=   -∑_i,j=1^2∂/∂x_i(u^ij/^2 (g_K)∂(g_K) /∂x_i)
=   -∑_j=1^2∑_i=1^2u^ij_i/^2 (g_K)∂(g_K) /∂x_i

-∑_i,j=1^2u^ij/^2 (g_K)∂^2 (g_K) /∂x_i∂x_j+∑_i,j=1^22u^ij/^3 (g_K)∂(g_K) /∂x_i∂(g_K) /∂x_j
=   -∑_j=1^2∑_i=1^2u^ij_i/^2 (g_K)∂(g_K) /∂x_i+∑_i,j=1^22u^ij/^3 (g_K)∂(g_K) /∂x_i∂(g_K) /∂x_j

for extremal metrics. Replacing in Equation (<ref>) we get

\begin{aligned}
&\frac{1}{2\scal(g_K)} +\frac{3\sum_{i,k=1}^2 u^{ik}_k}{\scal^3 (g_K)}\frac{\partial \scal (g_K)}{\partial x_i}-6\sum_{i,j=1}^2 \frac{2u^{ij}}{\scal^4 (g_K)}\frac{\partial \scal (g_K)}{\partial x_i}\frac{\partial \scal (g_K)}{\partial x_j}\\
&=\frac{1}{2\scal(g_K)} -\frac{1}{2\scal(g_K)} \\
\end{aligned}

and we are done.

Because $\rho$ is harmonic for $g_\bbR$ it admits a harmonic conjugate which we denote by $z$ so that

\sum_{i,j=1}^2u_{ij}dx_i\otimes dx_j =e^{2v}\left(d\rho^2+dz^2\right).

Next, we state an important but rather technical lemma. The proof goes exactly as the proof of Proposition 5.1 in [18] so we do not go over it here.

Let $M$ be a toric Kähler surface with moment polytope $P$ and $D$ be a torus invariant divisor in $X$ whose image via the moment map is $E.$ Let $g$ be a conformally Kähler toric metric conformal to an extremal Bach-flat toric Kähler metric $g_K$ with symplectic potential $u.$ Consider also the map $\bz=z+\sqrt{-1}\rho: P\setminus E\rightarrow \bbH$ where

\rho=\frac{(\det \Hess u)^{-1/2}}{\scal(g_K)}

and $z$ is the harmonic conjugate of $\rho$ for $g_K$ restricted to the real manifold. Then $\bz$ is a bijection.

§ THE FUNCTION $U$

As we have seen in Theorem (<ref>),  Biquard and Gauduchon make use of an axi-symmetric harmonic function on $\bbR^3$ in their construction for conformally Kähler, Ricci-flat metrics. This function, which we have denoted by $U^{BG},$ is given explicitly in their work by Equation (<ref>). In this section, our aim is to see that conversely, an axi-symmetric harmonic function $U$ can be associated to any conformally Kähler, Ricci-flat, toric metric on $M\setminus D.$  What we do here is very similar to what Donaldson does in [10] to show that every toric scalar-flat Kähler metric arises from Joyce's ansatz. The metric $g_K$ and its symplectic potential have specified behaviour at $\partial P\setminus E$ and this will imply a specific boundary behaviour for $U$ on $\partial \bbH$.

Consider the map $\bz=z+\sqrt{-1}\rho$ in Lemma (<ref>). Even though the functions ${\scal(g_K)}$ and $\det(\triangledown u, \fut)$ are functions on $P\setminus E,$ we interpret them as functions on $\bbH$ via Lemma (<ref>) i.e.

\begin{aligned}
\scal(g_K)(z+\sqrt{-1}\rho)&=\scal(g_K)\circ \bz^{-1}(z+\sqrt{-1}\rho),\\ \nonumber
\det(\triangledown u, \fut) (z+\sqrt{-1}\rho)&=\det(\triangledown u, \fut)\circ \bz^{-1} (z+\sqrt{-1}\rho)\\ \nonumber
\end{aligned}

We shall use Lemma (<ref>) to show the following proposition.

Let $M$ be a toric Kähler surface with moment polytope $P$ and $D$ be a torus invariant divisor in $X$ whose image via the moment map is an edge $E.$ Let $g$ be a Ricci-flat conformally Kähler, toric metric conformal to an extremal Bach-flat toric Kähler metric $g_K$ with symplectic potential $u.$ Let $P$ be normalised so that $\scal(g_K)=\alpha_1x_1+\alpha_2x_2$ for constants $\alpha_1,\alpha_2$. Set $\fut=(\alpha_1,\alpha_2).$  Then, there is a function $U:\bbH \rightarrow \bbR$ such that

\rho U_\rho=\frac{1}{\scal(g_K)(z+\sqrt{-1}\rho)},\qquad U_z=\det(\triangledown u, \fut) (z+\sqrt{-1}\rho).

The function $U$ is axi-symmetric harmonic i.e. it satisfies

\begin{equation}\label{Uharmonic}
\end{equation}

The function $\frac{U}{\log(\rho^2)}$ extends to $\partial \bbH$ and it is affine with slope $\det(\fut,\nu_i)$ when restricted to each segment of $\partial \bbH$ corresponding to the image via $\bz$ of the $i$th facet of $P\setminus E,$ for $i=1,\cdots,d.$

The first point is to show that the two equations are compatible which boils down to showing that

\frac{\partial}{\partial z }\left( \frac{1}{\rho \scal(g_K)}\right)=\frac{\partial \det(\triangledown u, \fut)}{\partial \rho },

where both functions are thought of as functions as $z+\sqrt{-1}\rho$ but we omit explicitly writing the argument. We start by relating $z+\sqrt{-1}\rho$ with toric complex coordinates. Recall that complex toric coordinates on $X$ are given by $\xi_i+\sqrt{-1}\theta_i$ for $i=1,2$ where $\bf{\xi}=(\xi_1,\xi_2)$ is

\bf{\xi}(\bf{x})=\triangledown u(\bf{x}).

We are going to follow an argument of Donaldson's from [10]. Consider the metric on the real manifold $g_\bbR.$ We shall express it using complex  coordinates.
∑_i,j=1^2 u_ijdx_i⊗dx_j   =   ∑_i=1^2 dξ_i⊗dx_i
=   (ξ_1,zdz+ξ_1,ρdρ)⊗(x_1,zdz+x_1,ρdρ)

=   (ξ_1,zx_1,z+ξ_2,zx_2,z)dz^2+ (ξ_1,ρx_1,ρ+ξ_2,ρx_2,ρ)dρ^2

We set

\mA=\begin{pmatrix}
x_{1,z}& x_{1,\rho}\\
\end{pmatrix},
\qquad
\mB=\begin{pmatrix}
\xi_{1,z}& \xi_{1,\rho}\\
\xi_{2,z}&\xi_{2,\rho}
\end{pmatrix}.

We start by noticing that $\mA^{T}\mB$ is a symmetric matrix. This is because

\mB=\frac{\partial \bf{\xi}}{\partial(z,\rho)}=\frac{\partial \bf{\xi}}{\partial \bf{x}}\frac{\partial \bf{x}}{\partial(z,\rho)}\\ \nonumber=\Hess (u) \mA

since $\bf{\xi}(\bf{x})=\triangledown u(\bf{x})$. Therefore
$$\mA^{T}\mB=\mA^{T}\Hess (u) \mA,$$
which is symmetric. On the other hand we have

\sum_{i,j=1}^2 u_{ij}dx_i\otimes dx_j =e^{2v}(dz^2+d\rho^2),

which implies
Now by taking determinants

so that $e^{2v}=\det(\mA)(\det(\Hess(u))^{1/2}$ and this implies that

\begin{pmatrix}
\xi_{2,\rho}& \xi_{1,\rho}\\
\xi_{2,z}&\xi_{1,z}
\end{pmatrix}=\det(\Hess(u))^{1/2}\begin{pmatrix}
x_{1,z}& -x_{2,z}\\
\end{pmatrix}.

We rewrite this as

\begin{equation}\label{xi&x}
\begin{cases}
\xi_{1,z}=\frac{1}{\rho}\frac{x_{2,\rho}}{\scal^2(g_K)}\\
\xi_{1,\rho}=-\frac{1}{\rho}\frac{x_{2,z}}{\scal^2(g_K)}\\
\end{cases}
\begin{cases}
\xi_{2,z}=-\frac{1}{\rho}\frac{x_{1,\rho}}{\scal^2(g_K)}\\
\xi_{2,\rho}=\frac{1}{\rho}\frac{x_{1,z}}{\scal^2(g_K)}\\
\end{cases},
\end{equation}

\begin{cases}
\xi_{1,z}=\frac{1}{\rho}\frac{x_{2,\rho}}{(\alpha_1x_1+\alpha_2x_2)^2}\\
\xi_{1,\rho}=-\frac{1}{\rho}\frac{x_{2,z}}{(\alpha_1x_1+\alpha_2x_2)^2}\\
\end{cases}
\begin{cases}
\xi_{2,z}=-\frac{1}{\rho}\frac{x_{1,\rho}}{(\alpha_1x_1+\alpha_2x_2)^2}\\
\xi_{2,\rho}=\frac{1}{\rho}\frac{x_{1,z}}{(\alpha_1x_1+\alpha_2x_2)^2}\\
\end{cases}.

Using this we have
-∂( ▿u,)/∂ρ   =   ∂(-α_2ξ_1+α_1ξ_2)/∂ρ
=   -α_2ξ_1,ρ+α_1ξ_2,ρ
=   α_2x_2,z/ρ(α_1x_1+α_2x_2)^2+α_1x_1,z/ρ(α_1x_1+α_2x_2)^2
=   ∂/∂z(α_1x_1+α_2x_2)/ρ(α_1x_1+α_2x_2)^2
=   -∂/∂z(1/ρ(α_1x_1+α_2x_2))
=   -∂/∂z(1/ρ(g_K))

which shows that $U$ exists as $\bbH$ is simply connected. Next we show $U$ satisfies Equation (<ref>).
U_ρρ+U_zz+U_ρ/ρ   =   ∂/∂ρ( 1/ρ(g_K))+∂( ▿u,)/∂z+ 1/ρ^2(g_K)
=    -1/ρ^2(g_K)- α_1x_1,ρ+α_2x_2,ρ/ρ^2(g_K) +α_2ξ_1,z-α_1ξ_2,z+1/ρ^2(g_K)
=    - α_1x_1,ρ+α_2x_2,ρ/ρ^2(g_K) +α_2ξ_1,z-α_1ξ_2,z
=   0

where we made use of Equation (<ref>) in the last step.
Next we will focus on the boundary behaviour of $U$. We have

\rho=\frac{(\det \Hess(u))^{-1/2}}{\scal^2 (g_K)}.

As we have seen

\det \Hess(u)(\bx)=\frac{\delta(\bx)}{\prod_{i=1}^d l_i(\bx)}, \qquad \bx \in P,

for a positive smooth function $\delta$ on $P\setminus E,$ so that $\rho$ vanishes at $\partial P\setminus E.$ Also near the $i$-the edge of $P\setminus E,$ $2\log(\rho)-\log(l_i)$ is smooth. What is more,

\rho U_{\rho}=\frac{1}{\scal(g_K)}=S_1(z)+ \rho \tilde{S}_2(z,\rho),

so that

U(z,\rho)=S_0(z)+S_1(z)\log(\rho)+ \rho S_2(z,\rho),

where $S_0, S_1, S_2 , \tilde{S}_2$ are functions of $z$, smooth in the interior of segments corresponding to images of facets or $P\setminus E$ via the map $\bf{z},$ and $S_2$ is smooth in a neighbourhood of those segments in $\bbH.$  In particular

\lim_{\rho\rightarrow 0}\frac{U(z,\rho)}{\log(\rho)}=S_1(z).

U_z= \det( \triangledown u,\fut)=S'_0(z)+S'_1(z)\log(\rho)+\rho \frac{\partial S_2}{\partial z}(z,\rho).

Because $u-\sum_{i=1}^dl_i\log(l_i)$ is smooth near the ith facet of $P\setminus E$ for $i=1,\cdots, d$,

\det(\triangledown u-\log(l_i)\nu_i,\fut)

is smooth as well. Near the ith facet $2\log(\rho)-\log(l_i)$ is smooth so

\det(\triangledown u-2\log(\rho)\nu_i,\fut)

is smooth and this implies that

S'_0(z)+S'_1(z)\log(\rho)+\rho \frac{\partial S_2}{\partial z}(z,\rho)-\log(\rho^2)\det (\nu_i,\fut)

is smooth therefore

S'_1(z)=-2\det (\nu_i,\fut)

is constant on the interior of the image of the $i$th edge and $S_1(z)$ is affine on the interior of the image of that edge as claimed.

Next we will show that the vanishing locus of $\scal(g_K)$ is highly restricted. The lemma below although simple plays a very important role in our results. In particular it explains the condition on the extremal vector field appearing in Theorem (<ref>).

In the setting of Proposition (<ref>) let $\fut$ be the extremal vector field associated with $P\setminus E.$ There are three possibilities for the vanishing locus of $\scal(g_K).$

* If $\det(\fut,\nu_1),\det(\fut,\nu_{d})\ne 0$ i.e. if $\fut$ is not perpendicular to either one of the non-compact edges of $P\setminus E$ then $\scal(g_K)$ vanishes along $D.$

* If exactly one of the two numbers $\det(\fut,\nu_1), \det(\fut,\nu_{d})$ is zero, then $\scal(g_K)$ vanishes at a single point in $D$ corresponding to either $E\cap E_d$ or $E\cap E_1$ via the moment map.

* If $\det(\fut,\nu_1)=0=\det(\fut,\nu_{d})$ then $\scal(g_K)$  is nowhere vanishing over $X.$

Let us use the same notation as in Proposition (<ref>). As we have just seen, the function

f(z):=\lim_{\rho\rightarrow 0}\frac{U(z,\rho)}{\log(\rho^2)}

is locally affine with slope $\det(\fut,\nu_i)$ over $]z_{i-1},z_i[$ where by convention $z_0=-\infty,$ $z_d=+\infty$ and $i=1,\cdots, d.$ On the other hand it follows from the proof above that

$$ f(z)=\frac{1}{2\scal(g_K)(z,0)}.$$

Note also that $\scal(g_K)$ can be thought of as affine function on $P \subset \bbR^2$ whose vanishing locus is a line normal to $\fut.$ Furthermore, the limits $z \to \pm \infty$ correspond to limits along the non-compact ends of the edges $E_1$ and $E_d$

* If $\det(\fut,\nu_1),\det(\fut,{\nu_{d}})\ne 0$ we see that

\lim_{z\rightarrow \pm\infty}\frac{1}{|\scal(g_K)|(z,0)}= \lim_{z\rightarrow \pm\infty}|f(z)| =\infty

so that

\lim_{z\rightarrow \pm\infty}{\scal(g_K)(z,0)}=0.

This shows that $\scal(g_K)$ vanishes at the vertices $E\cap E_1$ and $E\cap E_{d}$ so that it must vanish along $E$ because it is affine.

* If exactly one of the two numbers $\det(\fut,\nu_1), \det(\fut,\nu_{d})$ is zero then the same reasoning shows that $\scal(g_K)$ vanishes at exactly one the vertices $E\cap E_1,$   $E\cap E_{d}.$ The extremal vector field is proportional to either $\nu_{d-1}$ or $\nu_{1}$ so the line $\scal(g_K)=0$ is parallel to either $E_1$ if  $\det(\fut,\nu_1)=0$ or $E_{d}$ if $\det(\fut,\nu_d)=0$ and intercepts $\overline{P}$ at a single vertex.

* If $\det(\fut,\nu_1)=\det(\fut,\nu_{d})=0$ then the edges $E_1$ and $E_{d}$ must be parallel and both parallel to the line $\scal(g_K)=0$ in $\bbR^2.$ If this line were to intersect $\overline{P}$ then it would contain points in $P\setminus E$ and so $\scal(g_K)$ would vanish at points in $X$ and this would imply that $g=\frac{g_k}{\scal^2(g_K)}$ had singularities in $X$ which it does not.

In comparing the above proposition with [5], it is natural to study the boundary behaviour function of $U$ given by

\lim_{\rho\rightarrow 0}\frac{U(z,\rho)}{\log(\rho^2)}.

It follows from Theorem (<ref>) that such a function will yield a $U^{BG}$ if it is of the form

A+\sum_{i=1}^d a_i|z-z_i|,

for $A>0,$ $a_1,\cdots, a_d>0.$  Among those functions which are affine in segments and continuous, these are special.

Let $X$ be a toric Kähler surface with moment polytope $P$ so that

P=\{(x_1,x_2)\in \bbR^2: l_i(x_1,x_2):=\langle (x_1,x_2),\nu_i\rangle- \lambda_i\geq 0, i=1,\cdots, d+1\},

where $\nu_i$ are the primitive interior normals to the facets of $P.$ Let $D$ be a torus invariant divisor in $X$ whose image via the moment map is $E$ with normal $\nu_{d+1},$ the edge at infinity. Recall that the extremal vector field $\fut$ of $P\setminus E$ is of the form $\alpha_1\frac{\partial}{\partial x_1}+\alpha_2\frac{\partial}{\partial x_2}$ for constants $\alpha_1,\alpha_2.$

Consider a partition of $\partial \bbH$ into $d$ segments $]z_i,z_{i+1}[,$ $i=0,\cdots,d-1$ where we set $z_0=-\infty$ and $z_{d}=+\infty.$ Consider a function $f$ on $\partial \bbH$ which is continuous over $\bbR$ and affine in the interior of the above segments in $\partial \bbH$ and whose slope in $]z_i,z_{i+1}[$ is $\det(\fut, \nu_{i+1}).$ There are constants $a_1,\cdots, a_d,$ $A$ and $B$ such that

Also, $B=0$ iff $\det(\nu_1+\nu_d,\fut)=0.$

We start by justifying the last statement assuming the first. Notice that over $]-\infty,z_1[,$ $\sum_{i=1}^{d-1}a_i|z-z_i|$ has slope $-\sum_{i=1}^{d-1}a_i$ whereas it has slope $\sum_{i=1}^{d-1}a_i$ over $]z_{d-1},\infty[.$ Therefore, $f$ will be of the form $A+\sum_{i=1}^{d-1}a_i|z-z_i|$ if and only if its slope at $-\infty$ is the opposite of its slope at $+\infty$, i.e.

\det (\nu_1,\fut)=-\det( \nu_d,\fut).

In any case we have

\sum_{i=1}^{d-1}a_i|z-z_i|+A+Bz=
\begin{cases}
\left(-\sum_{i=1}^{d-1}a_i +B\right)z+A_1, \, \text{on} \, ]z_0,z_{1}[,\\
\left(-\sum_{i=1}^{d-1}a_i+2a_1+B\right) z+A_2, \, \text{on} \, ]z_1,z_{2}[,\\
\cdots \\
\left(-\sum_{i=1}^{d-1}a_i+2a_1+\cdots +2a_j+B\right)+A_j,\,  \text{on} \, ]z_j,z_{j+1}[,
\end{cases}

where $A_i$ are constants for $i=1,\cdots d-1,$ and

\begin{cases}
\det(\fut,\nu_1)z+C_1, \, \text{on} \, ]z_0,z_{1}[,\\
\det(\fut,\nu_2)z+C_2, \, \text{on} \, ]z_1,z_{2}[,\\
\cdots \\
\det(\fut,\nu_{j+1})z+C_j,\,  \text{on} \, ]z_j,z_{j+1}[.
\end{cases}

where $C_i$ are constants for $i=1,\cdots d-1.$ Set

2a_j={\det(\fut,\nu_{j+1}-\nu_{j})}, \qquad j=1,\cdots, d-1,

This choice for the $a_i$'s and $B$ will yield $f(z)=A+Bz+\sum_{i=1}^{d-1} a_i|z-z_i|$ for the right choice of $A.$

Our next lemma will be relevant for the proof of our main theorem.

In the setting of Proposition (<ref>), $\scal(g_K)>0$ over $X.$

As we have said, a zero of  $\scal(g_K)$ over $X$ is a singularity for $g=\frac{g_K}{\scal^2(g_K)}$ and therefore  $\scal(g_K)$ cannot vanish over $X$ so that its sign remains constant. We shall argue by contradiction and assume that $\scal(g_K)<0.$ Consider $\scal(g_K)$ as an affine function in $\bbR^2$ so that $\scal(g_K)<0$ on $P\setminus E$ means that $\scal(g_K)$ has a negative minimum on ${P}.$

We want to start by showing that this minimum must be achieved over $P\setminus E$ so that $\scal(g_K)$ has a negative minimum on $X$.

* If $\det(\fut,\nu_1),\det(\fut,\nu_{d})\ne 0$ then $\scal(g_K)$ vanishes along $E$ so its minimum on ${P}$ cannot be achieved on $E$ and must be achieved on $P\setminus E.$

* If exactly one of the two numbers $\det(\fut,\nu_1), \det(\fut,\nu_{d})$ is zero, say without loss of generality  $\det(\fut,\nu_1),$ then $\scal(g_K)$ vanishes along a line in $\bbR^2$ parallel to $E_1$ through the vertex $E\cap E_{d}.$ As $\scal(g_K)$ is affine, its minimum in $P$ is attained along a line parallel to this that intersects $P.$ All such lines intersect also $P\setminus E$ except perhaps the one through $E\cap E_{d}.$ But on this line $\scal(g_K)$ is zero and therefore not minimal.

* If $\det(\fut,\nu_1)=\det(\fut,\nu_{d})=0$ then $\scal(g_K)$ vanishes along a line in $\bbR^2$ parallel to $E_1$ and $E_{d}$ and also attains its minimum along one such line. But such a line cannot intercept ${P}$ without intersecting $P\setminus E.$ Therefore the minimum of $\scal(g_K)$ is attained over $X.$

The rest of the argument is due to LeBrun (see [13]). The fact that the curvature of the metric $\frac{g_K}{\scal^2(g_K)}$ is zero can be expressed as

0=-6\scal (g_K)\Lap\scal (g_K)-12|\grad \scal (g_K)|^2+(\scal (g_K))^3.

At a minimum of $\scal(g_K)$ over $X,$ $\grad \scal(g_K)=0$ and $\Lap\scal(g_K)<0$. But

and we get a contradiction.
Hence $\scal (g_K)$ is always positive on $X.$

The final result of this section is a consequence of the proofs of Lemmas (<ref>) and (<ref>).

If $\det(\fut,\nu_1),\det(\fut,\nu_{d})\ne 0$, then $2B=\det(\fut,\nu_1+\nu_d)=0$.

We saw in the proof of Lemma (<ref>) that under these hypothesis the scalar curvature vanishes along the edge $E=E_{d+1}$. Hence, $\eta$ must be proportional to $\nu_{d+1}$. On the other hand, the smoothness of $M$ implies that

and so

as claimed.

§ PROOF OF THE MAIN THEOREMS

We are now in a position to prove Theorems (<ref>) and (<ref>).

§.§ Proof of Theorem (<ref>)

As before, consider $M$ a compact toric Kähler surface with moment polytope $P.$  Let $D$ be a torus invariant divisor in $M$ whose moment map image is a facet $E$ in $P$ which we may assume is the $d+1$ facet.  Assume that $X=M\setminus D$ is endowed with a conformally Kähler, Ricci-flat, toric metric $g$ determining a function $U$ as in Proposition (<ref>) and a function

f(z)= \lim_{\rho\rightarrow 0}\frac{U(z,\rho)}{\log(\rho^2)}.

From Lemma (<ref>) we know that there are numbers $a_1,\cdots,a_d,$ constants $A,B$ and $z_1,\cdots z_{d-1}$ points on $\partial \bbH$ such that

Set $U^{ref}$ to be

\begin{equation}\label{explicitUBG}
\begin{aligned}
U^{ref}=&2\sum_{i=1}^{d-1} a_i\left(\sqrt{\rho^2+(z-z_i)^2}-(z-z_i)\log\frac{\sqrt{\rho^2+(z-z_i)^2}+z-z_i}{\rho}\right)\\
\end{aligned}
\end{equation}

and notice that this too, satisfies

\lim_{\rho\rightarrow 0}\frac{U^{ref}(z,\rho)}{\log(\rho^2)} = f(z).

By Theorem (<ref>) this will yield a conformally Kähler, Ricci-flat, toric metric if $A>0$ and $a_1,\cdots, a_d>0$ and $B=0$ but we do not assume these conditions for now.

In the setup described above, we have

Regardless of the value of the constants, the function $U^{ref}$ is axi-symmetric harmonic i.e. it satisfies Equation (<ref>) and

\lim_{\rho\rightarrow 0}\frac{U^{ref}(z,\rho)}{\log(\rho^2)}=f(z).

The main idea of the proof is very similar to the main idea in [18] and goes back to work of Dominic Wright (see [19]). Consider the function

\gamma=\frac{U_\rho^{ref}-U_\rho}{\rho}.

* We start by proving that $\gamma$ is axi-symmetric harmonic on $\bbR^5$ i.e. it satisfies

\gamma_{\rho\rho}+\gamma_{zz}+\frac{3\gamma_\rho}{\rho}=0.

This is a consequence of the fact that both $U$ and $U^{ref}$ satisfy Equation (<ref>). We have
∂/∂ρ(U_ρ/ρ)   =   U_ρρ/ρ-U_ρ/ρ^2,
∂^2 /∂ρ^2(U_ρ/ρ)   =   U_ρρρ/ρ-2U_ρρ/ρ^2+2U_ρ/ρ^3,
∂^2 /∂z^2(U_ρ/ρ)   =   U_ρzz/ρ.

∂^2 /∂ρ^2(U_ρ/ρ)+∂^2 /∂z^2(U_ρ/ρ)   =   U_ρρρ+U_ρzz/ρ-2U_ρρ/ρ^2+2U_ρ/ρ^3

=   ∂/∂ρ(U_ρρ+U_zz)/ρ-2U_ρρ/ρ^2+2U_ρ/ρ^3

=   -∂/∂ρ(U_ρ/ρ )/ρ-2U_ρρ/ρ^2+2U_ρ/ρ^3

=   -U_ρρ/ρ^2+U_ρ/ρ^3-2U_ρρ/ρ^2+2U_ρ/ρ^3

=   -3U_ρρ/ρ^2+3U_ρ/ρ^3

=   -3∂/∂ρ(U_ρ/ρ).

Since the above equalities are simply a consequence of the fact $U$ and $U^{ref}$ both satisfy Equation (<ref>). By linearity, the string of equalities holds when we replace $U$ by $U^{ref}$ and $\gamma$ satisfies the claimed PDE.

* Next we shall show that $\gamma$ extends smoothly over $\partial \bbH.$  We know from the proof of Proposition (<ref>) that

U(z,\rho)=S_0(z)+S_1(z)\log \rho +\rho S_2(z,\rho),

where $S_0$ is a function of $z$, smooth in the segments $\cup_{i} ] z_i , z_{i+1} [$, $S_1(z)=2f(z)$, and $S_2$ is smooth in a neighbourhood of these segments in $\bbH.$ Similarly we have

U^{ref}(z,\rho)=S^{ref}_0(z)+S^{ref}_1(z)\log(\rho)+\rho S^{ref}_2(z,\rho),

where again $S^{ref}_1(z)=2f(z)$. Therefore

\begin{align*}
\gamma(z,\rho) & =  \left(\frac{\partial S^{ref}_2}{\partial \rho} (z,\rho) - \frac{\partial S_2}{\partial \rho} (z,\rho) \right) \\
& \ \ \ \ + \frac{S^{ref}_2(z,\rho)-S_2(z,\rho)}{\rho} .
\end{align*}

Hence, $\gamma$ will continuously extend to $\partial \mathbb{H}$ if

$$S^{ref}_2(z,\rho)-S_2(z,\rho) = O(\rho).$$

It turns out that both $S^{ref}_2(z,\rho)$ and $S_2(z,\rho)$ are $O(\rho)$ as it can be seen from the fact that both $U^{ref}$ and $U$ satisfy Equation (<ref>). Indeed, using $f''(z)=0$ and our expansion near $\rho=0$, we find that
0   =   U_ρρ+U_zz+U_ρ/ρ

=   S_2(z,ρ)/ρ + S_0”(z) + 3 ∂S_2/∂ρ + 2 ρ∂^2 S_2/∂ρ^2,

and recall that $S_2$ is smooth in a neighbourhood of the relevant segments in $\bbH.$ It therefore follows that we must have $S_2(z,\rho)=O(\rho)$ for the right hand side to remain bounded.

Consider the function

\frac{U_\rho}{\rho}=\frac{\rho U_\rho}{\rho^2 }=\frac{1}{\rho^2\scal (g_K) } .

From Lemma (<ref>), $\scal (g_K)>0$ so that

The rest of the arguments is also inspired on [18]. We have

\gamma\leq \frac{U^{ref}_\rho}{\rho}.

On the other hand, from the explicit formula for $U^{ref}$ in Equation (<ref>) we can see that

\frac{U^{ref}_\rho}{\rho}=\frac{2\left(A+{Bz}+\sum a_i \sqrt{(z-z_i)^2+\rho^2}\right)}{\rho^2}.

Hence there is a constant $C$ so that

\frac{U^{ref}_\rho}{\rho}\leq\frac{CR}{\rho^2},

where $R=\sqrt{z^2+\rho^2}.$ As in [18] this implies $\gamma$ is bounded. The argument is roughly as follows. Let $w$ be a point in $\bbR^5$. Because $\gamma$ is harmonic,

\gamma(w)=\frac{1}{aR^4}\int_{\partial B(w,R)}\gamma(w_0)dw_0,

for a universal constant $a$, so that, using the bound on $\gamma$ we get

\gamma(w)\leq\frac{C'}{R^3}\int_{\partial B(w,R)}\frac{dw_0}{\rho^2},

for a constant $C'.$ As one can see from direct calculations

\int_{\partial B(0,R)}\frac{dw_0}{\rho^2}\leq C''R^2,

and this shows $\gamma$ is bounded. Since it is smooth on $\bbR^5$ and harmonic, it must be constant, say $\gamma=k$ for $k \in \mathbb{R}$. Hence, from the equation

we conclude that there is a function $K(z)$ such that

However, from the fact that

\lim_{\rho\rightarrow 0}\frac{U^{ref}(z,\rho)}{\log(\rho^2)} = f(z) = \lim_{\rho\rightarrow 0}\frac{U(z,\rho)}{\log(\rho^2)} ,

we conclude that $\frac{k\rho^2}{2}-K(z)$ must vanish and therefore $U=U^{ref}$ as we wanted to show.

Assume that $X=M\setminus D$ is endowed with two conformally Kähler, Ricci-flat, toric metrics $g$ and $\tilde{g}$ whose adaptated Kähler forms are cohomologous. Now let $g_K$ and $\tilde{g}_K$ be extremal Kähler metrics conformal to $g$ and $\tilde{g}$ as in Proposition (<ref>). Together with the complex structure on $X$ each determines a Kähler form $\omega_K$ and $\tilde{\omega}_K$ and $[\omega_K]=[\tilde{\omega}_K].$
By Moser's trick there is an equivariant diffeomorphism $\Psi$ of $X$ such that $\Psi^*\omega_K=\tilde{\omega}_K.$  This diffeomorphism does not a priori preserve the complex structure. Because $(X,\omega_K)\simeq(X, \tilde{\omega}_K)$ the moment polytopes of the two manifolds are the same up to a translation and an $SL(2,\bbZ)$ transformation. Both metrics determine functions $U$ and $\tilde{U}$ as in Proposition (<ref>) and the corresponding $f$ and $\tilde{f}$ defined via

f(z)=\lim_{\rho\rightarrow 0}\frac{U(z,\rho)}{\log(\rho^2)}\quad \text{and } \tilde{f}(z)=\lim_{\rho\rightarrow 0}\frac{\tilde{U}(z,\rho)}{\log(\rho^2)}.

Now $f$ and $\tilde{f}$ are locally affine functions with the same slopes. We must have

f(z)=\sum_{i=1}^{d-1}a_i|z-z_i|+A+Bz \quad  \text{and } \tilde{f}(z)=\sum_{i=1}^{d-1}a_i|z-\tilde{z}_i|+A+Bz.

We will show that $z_i=\tilde{z}_i$ for $i=1,\cdots, d-1$ so that  $f$ and $\tilde{f}$ must coincide.

Let $X=M\setminus D$ be endowed with two conformally Kähler, Ricci-flat, toric metric $g$ and $\tilde{g}$  whose adapted Kähler forms are cohomologous, determining functions $U$ and $\tilde{U}$ as in Proposition (<ref>) and the corresponding $f$ and $\tilde{f}.$ Then $z_i=\tilde{z}_i$ so that $f=\tilde{f}.$

Consider $E_i$, the $i$th edge on $P\setminus E.$ Its pre-image $S_i$ via the moment map in $X$ is a $\mathbb{CP}^1$ whose volume is given by

\text{Vol}(S_i)=\int_{S_i}\omega=\int_{S_i}dx_1\wedge d\theta_1+dx_2\wedge d\theta_2.

On $E_i,$ the direction in $\bbT^2$ which is not collapsed is perpendicular to $\nu_i=(\nu_i^1,\nu_i^2)$ so that we can write $(\theta_1,\theta_2)=t(\nu_i^2,-\nu_i^1)$ for $t\in ]0,2\pi[$ and

\text{Vol}(S_i)=\int_{S_i}\omega=2\pi\int_{E_i}\left(\nu_i^2dx_1-\nu_i^1dx_2\right).

Now we want to express the integral above in $(z,\rho)$ coordinates, in which

E_i=\{(z,\rho): \rho=0,\, z\in [z_{i-1},z_i]\}.

Assume that $\scal(g_K)$ is not constant along $E_i$ which happens iff $\det(\fut,\nu_i)\ne 0.$ Then, the easiest way to express $(x_1,x_2)$ via $(z,\rho)$ is through the scalar curvature of $g_K.$
On $E_i,$ $\nu_i^1dx_1+\nu_i^2dx_2=0$ so that
for a coordinate $x$ and

\nu_i^2dx_1-\nu_i^1dx_2=|\nu_i|^2dx, \quad d\scal(g_K)=\det(\fut,\nu_i)dx.

If $\det(\fut,\nu_i)\ne 0$

\text{Vol}(S_i)=2\pi\frac{|\nu_i|^2}{\det(\fut,\nu_i)}\int_{E_i}d\scal(g_K).

Near $E_i,$

\rho U_{\rho}=\frac{1}{\scal(g_K)}=2f(z)+\rho S_2(z,\rho),

so that

\begin{align*}
\text{Vol}(S_i) & = 2\pi\frac{|\nu_i|^2}{\det(\fut,\nu_i)}\left(\scal(g_K)(z_i,0)-\scal(g_K)(z_{i-1},0)\right), \\
& = \frac{\pi|\nu_i|^2}{\det(\fut,\nu_i)}\left(\frac{1}{f(z_i)}-\frac{1}{f(z_{i-1})}\right).
\end{align*}

We consider a different moment coordinate, namely $\mu$, to follows the notation of [5] (Proposition 6.1 in [5]) given by

\mu=-2\left(z+\frac{\rho H_\rho-2H}{H_z}\right)=2\left(\frac{\rho^2 U_z+2H}{\rho U_\rho}-z\right),

where $H$ is the harmonic conjugate of $U$ that is
$$H_z=\rho U_\rho, \quad H_\rho=-\rho U_z.$$

Near $E_i,$ the asymptotic behaviour of $U$ yields

\begin{aligned}
&H_\rho=-\rho U_z=-2\det(\fut,\nu_i) \rho\log(\rho)+O(\rho\log(\rho)),\\
&H_z=\rho U_\rho=2\det(\fut,\nu_i)z+{2}C_i+O\left(1\right),\\
\end{aligned}

where $C_i$ are as in the proof of Lemma (<ref>) and $K_i$ are constants. This gives

\mu=\frac{C_iz+2K_i}{f(z)},

As before $(dx_1,dx_2)=(\nu_i^2,-\nu_i^1)dx,$ and we can assume $dx=Cd\mu,$ so

\begin{aligned}
\text{Vol}(S_i)=&2C\pi{|\nu_i|^2}\int_{E_i}d\mu\\
\end{aligned}

When $\fut$ is not perpendicular to $E_i,$ we see that $\text{Vol}(S_i)$ determines

\left(\frac{z_i}{f(z_i)}-\frac{z_{i-1}}{f(z_{i-1})}\right) \, \text{and} \left(\frac{1}{f(z_i)}-\frac{1}{f(z_{i-1})}\right).

When $E_i$ is a compact edge in $P\setminus E,$ that is $1<i<d-2,$ then $\text{Vol}(S_i)$ is $2\pi$ times the euclidean length of $E_i$ so that this length determines $z_{i-1}$ and $z_i$ for $1<i<{d-1}$ when $E_i$ is not perpendicular to $\fut$. But there is at most one compact edge perpendicular to $\fut$ and in this case we can rely on adjacent edges to determine the missing $z_i.$

This Lemma finishes the proof of Theorem (<ref>) as it implies that $U=U^{ref}$ and $\tilde{U}=U^{ref}$ so $U$ and $\tilde{U}$ also differ by a constant and the metrics are isometric.

§.§ Proof of Theorem (<ref>)

By assuming that the extremal vector field is not normal to either one of the non-compact edges in the moment polytope of $X$ we may conclude that $\det(\fut,\nu_1)\ne 0$ and $\det(\fut,\nu_{d-1})\ne 0.$ The conformally Kähler, Ricci-flat, toric metric determines a Kähler metric $g_K,$ a function $U$ as in Proposition (<ref>) and a function

f(z)= \lim_{\rho\rightarrow 0}\frac{U(z,\rho)}{\log(\rho^2)}.

From Lemma (<ref>) and Corollary (<ref>), we know that there are numbers $a_1,\cdots, a_d,$ a constant $A$ and $z_1,\cdots, z_{d-1},$ points on $\partial \bbH$ such that

where $2a_i=\det(\fut,\nu_{i+1}-\nu_{i})$ for $i=1,\cdots, d-1$. We have shown in Lemma (<ref>) that $\scal(g_K)>0$ and this implies that $U_\rho>0$ since $\scal{g_K}=\frac{1}{\rho U_\rho}.$ We also know that the quantity

V=-\left(\rho U_\rho+\frac{U^2_\rho U_{zz}}{U^2_{\rho z}+U^2_{zz}}\right),

is positive so that $U_{zz}<0.$ A maximum principle for $U$ then implies that the function
$$z\mapsto U(z,\rho)$$
is concave and $f$ is convex. These arguments appear in the proof of Lemma 4.2 in [5]. The convexity of $f$ implies the inequality $a_1,\cdots, a_d>0.$
As above set $U^{ref}$ to be

\begin{equation}
\begin{aligned}
U^{ref}=&2\sum_{i=1}^{d-1} a_i\left(\sqrt{\rho^2+(z-z_i)^2}-(z-z_i)\log\frac{\sqrt{\rho^2+(z-z_i)^2}+z-z_i}{\rho}\right)\\
\end{aligned}
\end{equation}

As in the proof of Theorem (<ref>) above we can show that $U=U^{ref}.$ Now

V=-\left(\rho U_\rho^{ref}+\frac{(U^{ref}_\rho)^2U^{ref}_{zz}}{(U^{ref}_{\rho\rho})^2+(U^{ref}_{zz})^2}\right),

is given by $1+\frac{2A}{R}$ as one can see by direct inspection. On the other hand, because $U=U^{ref},$ this must be positive which can only happen if $A>0.$ We are in the setting of Theorem (<ref>) and $U^{ref}=U^{BG}$ so that our metric coincides with a Biquard-Gauduchon metric. This finishes the proof of Theorem (<ref>).

[1]
M. Abreu
Kähler geometry of toric manifolds in symplectic coordinates, Symplectic and contact topology: interactions and perspectives , Fields Inst. Commun., 35 (2003), Amer. Math. Soc., Providence, RI, 1–24.

[3]
M. Abreu, R. Sena-Dias
Scalar-flat Kähler metrics on non-compact symplectic toric 4-manifolds, Ann. Global Anal. Geom. 41 (2012), no. 2, 209–239.

[4]
A. Vestislav, P. Gauduchon
The Riemannian Goldberg–Sachs theorem, International Journal of Mathematics 8 (1997), no. 4, 421-439.

[5]
O. Biquard, P. Gauduchon
On Toric Hermitian ALF Gravitational Instantons,  Comm. Math. Phys. 399 (2023), no. 1, 389–422.

[6]
G. Chen, X. Chen
Gravitational instantons with faster than quadratic curvature decay. I, Acta Math. 227 (2021), (2), 263–307.

[7]
X. Chen, S. Donaldson, S. Sun
Kähler-Einstein metrics on Fano manifolds. I: Approximation of metrics with cone singularities, J. Amer. Math. Soc. 28 (2015), no. 1, 183–197.

[8]
X. Chen, C. Lebrun, B. Weber
On conformally Kähler, Einstein manifolds, J. Amer. Math. Soc. 21 (2008), no. 4, 137–1168.

[9]
A. Derdzinski
Self-Dual Kähler Manifolds and Einstein Manifolds of Dimension Four, Comp. Math. 49 (1983) 405–433.

[10]
S. Donaldson
A generalised Joyce construction for a family of nonlinear partial differential equations, J. Gökova Geom. Topol. GGT 3 (2009), 1–8.

[11]
S. Donaldson
Extremal metrics on toric surfaces: a continuity method, J. Differential Geom. 79 (2008), no. 3, 389–432.

[12]
C. Lebrun
Einstein metrics on complex surfaces, Geometry and physics (Aarhus, 1995), Lecture Notes in Pure and Appl. Math 184, 167–176.

[13]
C. Lebrun
Bach-Flat Kähler Surfaces, J. Geom. Analysis 30 (2020) 2491–2514.

[15]
M. Li
Classification results for Hermitian non-Kähler gravitational instantons,  arXiv:2304.01609.

[16]
M. Li
On 4-dimensional Ricci-flat ALE manifolds,  arXiv:2310.13197.

[17]
G. Oliveira, R. Sena-Dias
Minimal Lagrangian tori and action-angle coordinates, Transactions of the American Mathematical Society, 374 (2021), 11, 7715–7742.

[18]
R. Sena-Dias
Uniqueness among scalar-flat Kähler metrics on non-compact toric 4-manifolds, J. Lond. Math. Soc. 103 (2021), no. 2, 372–397.

[19]
D. Wright
The geometry of anti-self-dual orbifolds, Ph.D thesis, Imperial College, London, 2009.


# Isometric group of $(\alpha,\beta)$-type Finsler space and the symmetry of
Very Special Relativity

Xin Li1,4<EMAIL_ADDRESS>Zhe Chang2,4<EMAIL_ADDRESS>Xiaohuan Mo3
<EMAIL_ADDRESS>1Institute of Theoretical Physics, Chinese Academy of
Sciences, 100190 Beijing, China
2Institute of High Energy Physics, Chinese Academy of Sciences, 100049
Beijing, China
3Key Laboratory of Pure and Applied Mathematics School of Mathematical
Sciences, Peking University, Beijing 100871, China
4Theoretical Physics Center for Science Facilities, Chinese Academy of
Sciences

###### Abstract

The Killing equation for a general Finsler space is set up. It is showed that
the Killing equation of $(\alpha,\beta)$ space can be divided into two parts.
One is the same with Killing equation of a Riemannian metric, another equation
can be regarded as a constraint. The solutions of Killing equations present
explicitly the isometric symmetry of Finsler space. We find that the isometric
group of a special case of $(\alpha,\beta)$ space is the same with the
symmetry of Very Special Relativity (VSR). The Killing vectors of Finsler-Funk
space are given. Unlike Riemannian constant curvature space, the 4 dimensional
Funk space with constant curvature just have 6 independent Killing vectors.

###### pacs:

02.40.-k,11.30.-j

##  I Introduction

In the past few years, two interesting theories of investigating the violation
of Lorentz Invariance (LI) are proposed. One is the so called Doubly Special
Relativity (DSR) Amelino1 ; Amelino2 ; Amelino3 ; Smolin1 ; Smolin2 . This
theory takes Planck-scale effects into account by introducing an invariant
Planckian parameter in the theory of special relativity. Another is the so
called Very Special Relativity (VSR) developed by Cohen and Glashow Glashow .
This theory suggested that the exact symmetry group of nature may be
isomorphic to a subgroup SIM(2) of the Poincare group. And the SIM(2) group
semi-direct product with the spacetime translation group gives an
8-dimensional subgroup of the Poincare group called ISIM(2) Kogut . Under the
symmetry of ISIM(2), the CPT symmetry is preserved and many empirical
successes of special relativity are still functioned.

Recently, Physicists found that the two theories mentioned above are related
with Finsler geometry. Girelli, Liberati and Sindoni Girelli  showed that the
Modified dispersion relation (MDR) in DSR can be incorporated into the
framework of Finsler geometry. The symmetry of the MDR was described in the
Hamiltonian formalism. Also, Gibbons, Gomis and Pope Gibbons  showed that the
Finslerian line element
$ds=(\eta_{\mu\nu}dx^{\mu}dx^{\nu})^{(1-b)/2}(n_{\rho}dx^{\rho})^{b}$ is
invariant under the transformations of the group DISIM${}_{b}(2)$ (1-parameter
family of deformations of ISIM(2)).

Finsler geometry as a natural generation of Riemann geometry could provide new
sight on modern physics. the model based on Finsler geometry could explain the
recent astronomical observations which Einstein’s gravity could not. An
incomplete list includes: the flat rotation curves of spiral galaxies can be
deduced naturally without invoking dark matter Finsler DM ; the anomalous
accelerationAnderson  in solar system observed by Pioneer 10 and 11
spacecrafts could account for a special Finsler space-Randers space Finsler PA
; the secular trend in the astronomical unitKrasinsky ; Standish  and the
anomalous secular eccentricity variation of the Moon’s orbitWilliams  could
account for effect of the length change of unit circle in Finsler
geometryFinsler AU .

Thus, the symmetry of Finslerian spacetime is worth investigating. The way of
describing spacetime symmetry in a covariant language (the symmetry should not
depend on any particular choice of coordinate system) involves the concept of
isometric transformation. In fact, the symmetry of spacetime is described by
the so called isometric group. The generators of isometric group is directly
connected with the Killing vectorsKilling . In this paper, we use solutions of
the Killing equation to establish the symmetry of a class of Finslerian
spacetime. In particular, we show that the isometric group of a special kind
of $(\alpha,\beta)$ space is equivalent to the symmetry of the VSR.

##  II Killing vector in Riemann space

In this section, we give a brief review of the Killing vectors in Riemann
space (further material can be found, for example, in Weinberg ). Under a
given coordinate transformation $x\rightarrow\bar{x}$, the Riemannian metric
$g_{\mu\nu}(x)$ transforms as

$\bar{g}_{\mu\nu}(\bar{x})=\frac{\partial
x^{\rho}}{\partial\bar{x}^{\mu}}\frac{\partial
x^{\sigma}}{\partial\bar{x}^{\nu}}g_{\rho\sigma}(x).$ (1)

Any transformation $x\rightarrow\bar{x}$ is called isometry if and only if the
transformation of the metric $g_{\mu\nu}(x)$ satisfies

$g_{\mu\nu}(\bar{x})=\frac{\partial
x^{\rho}}{\partial\bar{x}^{\mu}}\frac{\partial
x^{\sigma}}{\partial\bar{x}^{\nu}}g_{\rho\sigma}(x).$ (2)

One can check that the isometric transformations do form a group. It is
convenient to investigate the isometric transformation under the infinitesimal
coordinate transformation

$\bar{x}^{\mu}=x^{\mu}+\epsilon V^{\mu},$ (3)

where $|\epsilon|\ll 1$. To first order in $|\epsilon|$, the equation (2)
reads

$V^{\kappa}\frac{\partial g_{\mu\nu}}{\partial
x^{\kappa}}+g_{\kappa\mu}\frac{\partial V^{\kappa}}{\partial
x^{\nu}}+g_{\kappa\nu}\frac{\partial V^{\kappa}}{\partial x^{\mu}}=0.$ (4)

By making use of the covariant derivatives with respect to Riemannian
connection, we can write the above equation as

$V_{\mu|\nu}+V_{\nu|\mu}=0,$ (5)

where $``|"$ denotes the covariant derivative. Any vector field $V_{\mu}$
satisfies equation (5) is called Killing vector. Thus, the problem of finding
all isometries of a given metric $g_{\mu\nu}(x)$ is reduced to find the
dimension of the linear space formed by Killing vectors.

In Riemann geometry, by making use of the covariant derivative, one could
obtain the Ricci identities or interchange formula

$V_{\rho|\mu|\nu}-V_{\rho|\nu|\mu}=-V_{\sigma}R^{~{}\sigma}_{\rho~{}\nu\mu},$
(6)

where $R^{~{}\sigma}_{\rho~{}\nu\mu}$ is the Riemannian curvature tensor. And
the first Bianchi identity for the Riemannian curvature tensor gives

$R^{~{}\sigma}_{\rho~{}\nu\mu}+R^{~{}\sigma}_{\nu~{}\mu\rho}+R^{~{}\sigma}_{\mu~{}\rho\nu}=0.$
(7)

Deducing from the equation (6) and (7), we obtain

$V_{\rho|\mu|\nu}=V_{\sigma}R^{~{}\sigma}_{\nu~{}\mu\rho}.$ (8)

Thus, all the derivatives of $V_{\mu}$ will be determined by the linear
combinations of $V_{\mu}$ and $V_{\mu|\nu}$. Once the $V_{\mu}$ and
$V_{\mu|\nu}$ at an arbitrary point of Riemannian space is given, then
$V_{\mu}$ and $V_{\mu|\nu}$ at any other point is determined by integration of
the system of ordinary differential equations. Therefore, the dimension of
linear space formed by Killing vector can be at most $\frac{n(n+1)}{2}$ in $n$
dimensional Riemannian space. If a metric admits that the maximum number
$\frac{n(n+1)}{2}$ of Killing vectors, its Riemann space must homogeneous and
isotropic (or the space is isotropic for every point). Such space is called
maximally symmetry space. In Riemann geometry, the Schur’s lemma tells us that
a Riemannian space with at least 3 dimension is maximally symmetry space if
and only if its sectional curvature is constant. Also, one can check that a 2
dimensional Riemannian space is maximally symmetry space if and only if its
sectional curvature is constant. Thus, the maximal symmetry of a given metric
is an intrinsic property, and not depending on the choice of coordinate
system.

One special maximally symmetry space is the Minkowskian space. The Killing
equation (5) of a given Minkowskian metric $\eta_{\mu\nu}(x)$ reduces to

$\frac{\partial V_{\mu}}{\partial x^{\nu}}+\frac{\partial V_{\nu}}{\partial
x^{\mu}}=0.$ (9)

The solution of the above equation is

$V^{\mu}=Q^{\mu}_{~{}\nu}x^{\nu}+C^{\mu},$ (10)

where $Q_{\mu\nu}=\eta_{\rho\mu}Q^{\rho}_{~{}\nu}$ is an arbitrary constant
skew-symmetric matrix and $C^{\mu}$ is an arbitrary constant vector. Thus,
substituting the solution (10) into the coordinate transformation (3) we
obtain

$\bar{x}^{\mu}=(\delta^{\mu}_{\nu}+\epsilon Q^{\mu}_{~{}\nu})x^{\nu}+\epsilon
C^{\mu}.$ (11)

One should find that the term $\delta^{\mu}_{\nu}+\epsilon Q^{\mu}_{~{}\nu}$
in above equation is just the Lorentz transformation matrix and the term
$\epsilon C^{\mu}$ is related to the spacetime translation. Expanding the
matrix $\delta^{\mu}_{\nu}+\epsilon Q^{\mu}_{~{}\nu}$ and the vector $\epsilon
C^{\mu}$ near identity, one could obtain the famous Poincare algebra.

Other two types of maximally symmetry space are spherical and hyperbolic case.
Without loss of generality, we set its constant sectional curvature to be $\pm
1$ for spherical and hyperbolic case respectively. The length element of
spherical and hyperbolic case is given in a unified form

$ds^{2}=\frac{\sqrt{(1+k(x\cdot x))(dx\cdot dx)-k(x\cdot dx)^{2}}}{1+k(x\cdot
x)},$ (12)

where the $\cdot$ denotes the inner product with respect to Minkowskian metric
and $k=\pm 1$ for spherical and hyperbolic case respectively. The metric is
given as

$g_{\mu\nu}=\left(\frac{\eta_{\mu\nu}}{1+k(x\cdot
x)}-k\frac{x_{\mu}x_{\nu}}{(1+k(x\cdot x))^{2}}\right),$ (13)

where $x_{\mu}\equiv\eta_{\mu\nu}x^{\nu}$. The Christoffel symbols of the
above length element is given as

$\gamma^{\rho}_{\mu\nu}=-k\frac{x_{\mu}\delta^{\rho}_{\nu}+x_{\nu}\delta^{\rho}_{\mu}}{1+k(x\cdot
x)}.$ (14)

Thus, the Killing equation (5) now reads

$\frac{\partial V_{\mu}}{\partial x^{\nu}}+\frac{\partial V_{\nu}}{\partial
x^{\mu}}+\frac{2k}{1+k(x\cdot x)}(x_{\mu}V_{\nu}+X_{\nu}V_{\mu})=0.$ (15)

The solution of the above equation is

$V^{\mu}\equiv g^{\mu\nu}V_{\nu}=Q^{\mu}_{~{}\nu}x^{\nu}+C^{\mu}+k(x\cdot
C)x^{\mu},$ (16)

where the index of $Q$ and $C$ are raise and lower by Minkowskian metric
$\eta^{\mu\nu}$ and its matrix reverse $\eta_{\mu\nu}$.

##  III Killing vectors in Finsler space

Instead of defining an inner product structure over the tangent bundle in
Riemann geometry, Finsler geometry is base on the so called Finsler structure
$F$ with the property $F(x,\lambda y)=\lambda F(x,y)$ for all $\lambda>0$,
where $x$ represents position and $y\equiv\frac{dx}{d\tau}$ represents
velocity. The Finsler metric is given asBook by Bao

$g_{\mu\nu}\equiv\frac{\partial}{\partial y^{\mu}}\frac{\partial}{\partial
y^{\nu}}\left(\frac{1}{2}F^{2}\right).$ (17)

Finsler geometry has its genesis in integrals of the form

$\int^{r}_{s}F(x^{1},\cdots,x^{n};\frac{dx^{1}}{d\tau},\cdots,\frac{dx^{n}}{d\tau})d\tau~{}.$
(18)

So the Finsler structure represents the length element of Finsler space.

Like Riemannian case, to investigate the Killing vector we should construct
the isometric transformation of Finsler structure. Let us consider the
coordinate transformation (3) together with the corresponding transformation
for $y$

$\bar{y}^{\mu}=y^{\mu}+\epsilon\frac{\partial V^{\mu}}{\partial
x^{\nu}}y^{\nu}.$ (19)

Under the coordinate transformation (3) and (19), to first order in
$|\epsilon|$, we obtain the expansion of the Finsler structure,

$\bar{F}(\bar{x},\bar{y})=\bar{F}(x,y)+\epsilon V^{\mu}\frac{\partial
F}{\partial x^{\mu}}+\epsilon y^{\nu}\frac{\partial V^{\mu}}{\partial
x^{\nu}}\frac{\partial F}{\partial y^{\mu}},$ (20)

where $\bar{F}(\bar{x},\bar{y})$ should equal $F(x,y)$. Under the
transformation (3) and (19), a Finsler structure is called isometry if and
only if

$F(x,y)=\bar{F}(x,y).$ (21)

Then, deducing from the (20) we obtain the Killing equation $K_{V}(F)$ in
Finsler space

$K_{V}(F)\equiv V^{\mu}\frac{\partial F}{\partial
x^{\mu}}+y^{\nu}\frac{\partial V^{\mu}}{\partial x^{\nu}}\frac{\partial
F}{\partial y^{\mu}}=0.$ (22)

Searching the Killing vectors for general Finsler structure is difficult.
Here, we give the Killing vectors for a class of Finsler
space-$(\alpha,\beta)$ spaceShen  with metric defining as

$\displaystyle F=\alpha\phi(s),~{}~{}~{}s=\frac{\beta}{\alpha},$ (23)
$\displaystyle\alpha=\sqrt{g_{\mu\nu}y^{\mu}y^{\nu}}~{}~{}{\rm
and}~{}~{}\beta=b_{\mu}(x)y^{\mu},$ (24)

where $\phi(s)$ is a smooth function, $\alpha$ is a Riemannian metric and
$\beta$ is one form. Then, the Killing equation (22) in $(\alpha,\beta)$ space
reads

$\displaystyle 0$ $\displaystyle=$ $\displaystyle K_{V}(\alpha)\phi(s)+\alpha
K_{V}(\phi(s))$ (25) $\displaystyle=$
$\displaystyle\left(\phi(s)-s\frac{\partial\phi(s)}{\partial
s}\right)K_{V}(\alpha)+\frac{\partial\phi(s)}{\partial s}K_{V}(\beta).$

And by making use of the Killing equation (22), we obtain

$\displaystyle K_{V}(\alpha)$ $\displaystyle=$
$\displaystyle\frac{1}{2\alpha}(V_{\mu|\nu}+V_{\nu|\mu})y^{\mu}y^{\nu},$ (26)
$\displaystyle K_{V}(\beta)$ $\displaystyle=$
$\displaystyle\left(V^{\mu}\frac{\partial b_{\nu}}{\partial
x^{\mu}}+b_{\mu}\frac{\partial V^{\mu}}{\partial x^{\nu}}\right)y^{\nu},$ (27)

where $``|"$ denotes the covariant derivative with respect to the Riemannian
metric $\alpha$. The solutions of the Killing equation (22) have three viable
scenarios. The first one is

$\phi(s)-s\frac{\partial\phi(s)}{\partial s}=0~{}~{}{\rm
and}~{}~{}K_{V}(\beta)=0,$ (28)

which implies $F=\lambda\beta$ for all $\lambda\in\mathbb{R}$. The second one
is

$\frac{\partial\phi(s)}{\partial s}=0~{}~{}{\rm and}~{}~{}K_{V}(\alpha)=0,$
(29)

which implies $F=\lambda\alpha$ for all $\lambda\in\mathbb{R}$. The above two
scenarios are just trivial space, we will not consider in this section. Here
we mainly consider the case that $\phi(s)-s\frac{\partial\phi(s)}{\partial
s}\neq 0$ and $\frac{\partial\phi(s)}{\partial s}\neq 0$. This will induce the
last scenario. Apparently, in the last scenario we have such solutions

$\displaystyle V_{\mu|\nu}+V_{\nu|\mu}$ $\displaystyle=$ $\displaystyle 0,$
(30) $\displaystyle V^{\mu}\frac{\partial b_{\nu}}{\partial
x^{\mu}}+b_{\mu}\frac{\partial V^{\mu}}{\partial x^{\nu}}$ $\displaystyle=$
$\displaystyle 0,$ (31)

while $a_{0}\neq 0$. The first equation (30) is no other than the Riemannian
Killing equation (5). The second equation (31) can be regarded as the
constraint for the Killing vectors that satisfy the Killing equation (30).
Therefore, in general the dimension of the linear space formed by Killing
vectors of $(\alpha,\beta)$ space is lower than the Riemannian one. Additional
solutions of (22) should be obtained from (22), it will be discussed in next
section.

##  IV Symmetry of VSR

One important physical example of $(\alpha,\beta)$ space is VSR. While we take
$\phi(s)=s^{m}$ and $m$ is an arbitrary constant, the Finsler structure takes
the form proposed by Gibbons et al.Gibbons

$\displaystyle F$ $\displaystyle=$ $\displaystyle\alpha^{1-m}\beta^{m}$ (32)
$\displaystyle=$
$\displaystyle(\eta_{\mu\nu}y^{\mu}y^{\nu})^{(1-m)/2}(b_{\rho}y^{\rho})^{m}.$

We denote it by VSR metric. In this example, $\eta_{\mu\nu}$ is Minkowskian
metric and $b_{\rho}$ is a constant vector. One immediately obtain from the
first Killing equation (30) of $(\alpha,\beta)$ space,

$V^{\mu}=Q^{\mu}_{~{}\nu}x^{\nu}+C^{\mu}.$ (33)

And the second Killing equation (31) gives the constraint for Killing vector
$V^{\mu}$,

$b_{\mu}Q^{\mu}_{~{}\nu}=0.$ (34)

Taking the light cone coordinate Kogut  $\alpha=\sqrt{2y^{+}y^{-}-y^{i}y^{i}}$
(with i ranging over the values 1 and 2) and supposing
$b_{\mu}=\\{0,0,0,b_{-}\\}$, we know that in general $Q^{-}_{~{}\mu}\neq 0$.
It means the Killing vectors of VSR metric (32) do not have components
$Q_{+-}$ and $Q_{+i}$. In another word, the generators of the group that
leaves the metric (32) invariant are just four spacetime translation
generators $(P_{+},P_{i},P_{-})$ and three subgenerators of Lorentz algebra
$(M_{-i},M_{ij})$. This result implies that the VSR metric is invariant under
the E(2) group (the group of two-dimensional Euclidean motion).

The above investigation and the Killing equations (30) and (31) obtained in
section 3 are under the premise that the direction of $y^{\mu}$ is arbitrary.
It means that no preferred direction exists in spacetime. If the spacetime
does have a special direction, the Killing equation (25) will have a special
solution. The VSR metric is first suggested by Bogoslovsky Bogoslovsky . He
assumed that the spacetime has a preferred direction. Following the assumption
and taking the null direction to be the preferred direction, we deduce from
Killing equation (25) that

$\displaystyle 0$ $\displaystyle=$ $\displaystyle
s^{m}\left(\frac{1-n}{2\alpha}\left(\frac{\partial V_{\mu}}{\partial
x^{\nu}}+\frac{\partial V_{\nu}}{\partial
x^{\mu}}\right)y^{\mu}y^{\nu}+ms^{-1}b_{\rho}\frac{\partial V^{\rho}}{\partial
x^{\kappa}}y^{\kappa}\right)$ (35) $\displaystyle=$ $\displaystyle
s^{m}\frac{1}{\alpha\beta}\Bigg{(}\frac{1-n}{2}\left(\frac{\partial
V_{\mu}}{\partial x^{\nu}}+\frac{\partial V_{\nu}}{\partial
x^{\mu}}\right)b_{\kappa}$ $\displaystyle+m\eta_{\mu\nu}b^{\rho}\frac{\partial
V_{\rho}}{\partial x^{\kappa}}\Bigg{)}y^{\mu}y^{\nu}y^{\kappa}.$

The above equations has a special solution

$V_{+}=(Q_{+-}+m\eta_{+-})x^{-}+C_{+},$ (36)

where $Q_{+-}$ is not only an antisymmetrical matrix, but also satisfying the
property

$b_{-}=-b^{+}Q_{+-}.$ (37)

It implies that the Lorentz transformation for $b^{+}$ is

$\left(\delta^{+}_{+}+\epsilon(m\delta^{+}_{+}+Q^{+}_{~{}+})\right)b^{+}=\left(1+\epsilon(n+1)\right)b^{+},$
(38)

which means the null direction $b^{+}$ (or $b_{-}$) is invariant under the
Lorentz transformation. Therefore, if the spacetime has a preferred direction
in null direction, the symmetry corresponded to $Q_{+-}$ is restored. In such
case the VSR metric is invariant under the the transformations of the group
DISIM${}_{b}(2)$ proposed by Gibbons et al.Gibbons .

Another important physical example of $(\alpha,\beta)$ space is Randers
spaceRanders . While we set $\phi(s)=1+s$, the Finsler structure takes the
form

$F=\alpha+\beta.$ (39)

Then, in Randers space the Killing equation (25) reads

$K_{V}(\alpha)+K_{V}(\beta)=0.$ (40)

Since the $K_{V}(\alpha)$ contains irrational term of $y^{\mu}$ and
$K_{V}(\beta)$ only contains rational term of $y^{\mu}$, therefore the
equation (40) satisfies if and only if $K_{V}(\alpha)=0$ and $K_{V}(\beta)=0$.
If Randers space is flat, its Killing vectors satisfy the same Killing
equations with VSR metric. It implies the flat Randers metric is invariant
under the E(2) group semi-direct product with the spacetime translation. Thus,
let us focus on the non-flat Randers space. In section 2, we know that a
Riemannian space is maximally symmetry space if and only if it is a constant
sectional curvature space. Due to the first Killing equation (30) of
Finslerian space is the same with the Riemannian one and the Killing vectors
of Finslerian space must satisfy the constraint (31), we know that a Finsler
space with constant flag curvature (the notion of flag curvature is the
counterpart of sectional curvature) has less independent Killing vectors than
its Riemannian counterpart, in general.

Here, we just investigate a special Randers space with constant flag curvature
$-\frac{1}{4}$. It is called Funk metricFunk . This metric space can be
regarded as a flat Minkowskian space which influenced by a radial “wind”
(vector)Gibbons1 . For Randers space with other types of constant flag
curvature, its $\alpha$ term is not a Riemannian constant sectional curvature
space. Then, its independent Killing vectors should less than Funk one in
general. The Funk metric is given by

$F=\frac{\sqrt{(1-x\cdot x)(y\cdot y)+(x\cdot y)^{2}}}{1-x\cdot
x}+\frac{x\cdot y}{1-x\cdot x},$ (41)

where the $\cdot$ denotes the inner product with respect to Minkowskian
metric. By making use of the result (16) obtained in section 2, we know that
the first Killing equation $K_{V}(\alpha)=0$ (30) implies

$V^{\mu}=Q^{\mu}_{~{}\nu}x^{\nu}+C^{\mu}-(x\cdot C)x^{\mu}.$ (42)

And the Funk metric implies $b_{\mu}(x)=\frac{x_{\mu}}{1-x\cdot x}$. Then, we
obtain the partial derivative for $b_{\mu}(x)$,

$\frac{\partial b_{\nu}}{\partial x^{\mu}}=\frac{\eta_{\mu\nu}}{1-x\cdot
x}+\frac{2x_{\mu}x_{\nu}}{(1-x\cdot x)^{2}},$ (43)

and the partial derivative for Killing vectors $V^{\mu}$,

$\frac{\partial V^{\mu}}{\partial
x^{\nu}}=Q^{\mu}_{~{}\nu}-\delta^{\mu}_{\nu}(x\cdot C)-x^{\mu}C_{\nu}.$ (44)

By making use of the equation (42), (43) and (44), and deducing from the
second Killing equation (31) we obtain

$\frac{2Q_{\nu\mu}x^{\mu}}{1-x\cdot x}+C_{\nu}=0.$ (45)

Substituting the equation (45) into (42), we get

$V^{\mu}=Q^{\mu}_{~{}\nu}x^{\nu}\left(\frac{3-x\cdot x}{1-x\cdot x}\right).$
(46)

Therefore, the dimension of the linear space formed by the Killing vectors of
Funk metric is $6$. And the spacetime translation generators corresponded to
$C^{\mu}$ depend on the generators of Lorentz group corresponded to
$Q^{\mu}_{~{}\nu}$.

##  V Conclusion

In the past few years, two interesting theories which investigated the
violation of Lorentz Invariance (LI) are proposed. We showed that the Killing
vectors satisfy the same Killing equation (30) of a Riemannian metric, and the
major difference is the Killing vectors of $(\alpha,\beta)$ metric need to
satisfy the constraint (31). We proved that the isometric group of a flat
$(\alpha,\beta)$ space is just the symmetry in VSR proposed by Cohen and
Glashow. If the space do not have a preferred direction, we showed that the
generators induced by such Killing vectors are just isomorphic to the E(2)
group semidirect the spacetime translation. While a preferred direction has
chosen, it is showed that the symmetry of such space isomorphic to the SIM(2)
group semidirect the spacetime translation. For non flat case, we considered
the Funk metric with constant flag curvature. It is showed that the numbers of
independent Killing vectors of Funk metric is just $6$. The determination for
the maximal number of independent Killing vectors of $(\alpha,\beta)$ space
(or general Finsler space) still is a open problem. We hope it could be solved
in the future.

###### Acknowledgements.

We would like to thank Prof. C. J. Zhu for useful discussions. The work was
supported by the NSF of China under Grant No. 10525522, 10875129 and 10771004.

## References

  * (1) G. Amelino-Camelia, Phys. Lett. B 510, 255 (2001).
  * (2) G. Amelino-Camelia, Int. J. Mod. Phys. D 11, 35 (2002).
  * (3) G. Amelino-Camelia, Nature 418, 34 (2002).
  * (4) J. Magueijo and L. Smolin, Phys. Rev. Lett. 88, 190403 (2002).
  * (5) J. Magueijo and L. Smolin, Phys. Rev. D 67, 044017 (2003).
  * (6) A. G. Cohen and S.L. Glashow, Phys. Rev. Lett. 97, 021601 (2006).
  * (7) J. B. Kogut and D. E. Soper, Phys. Rev. D 1, 2901 (1970).
  * (8) F. Girelli, S. Liberati and L. Sindoni, Phys. Rev. D 75, 064015 (2007).
  * (9) G. W. Gibbons, J. Gomis and C. N. Pope, Phys. Rev. D 76, 081701 (2007).
  * (10) Z. Chang and X. Li, Phys. Lett. B 668, 453 (2008).
  * (11) J. D. Anderson et al., Phys. Rev. Lett. 81 2858, (1998); Phys. Rev. D 65 082004, (2002); Mod. Phys. Lett. A 17 875, (2002).
  * (12) X. Li and Z. Chang, arXiv:gr-qc/0909.3713.
  * (13) G. A. Krasinsky and V. A. Brumberg, Celest. Mech. Dyn. Astrn. 90, 267 (2004).
  * (14) E. M. Standish, Proc. IAU Colloq. 196, 163 (2005).
  * (15) J. G. Williams and D. H. Boggs, in Proceedings of 16th International Workshop on Laser Ranging ed. S. Schillak, (Space Research Centre, Polish Academy of Sciences), 2009.
  * (16) X. Li and Z. Chang, arXiv:gr-qc/0911.1890.
  * (17) W. Killing, J. f. d. reine u. angew. Math. (Crelle), 109, 121 (1892).
  * (18) S. Weinberg, Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity, Wiley, New York, 1972.
  * (19) D. Bao, S. S. Chern and Z. Shen, An Introduction to Riemann–Finsler Geometry, Graduate Texts in Mathmatics 200, Springer, New York, 2000.
  * (20) Z. Shen, Some perspectives in Finsler geometry, MSRI Publication Series. Cambridge: Cambridge university press, 2004.
  * (21) G. Bogoslovsky, arXiv:gr-qc/0706.2621.
  * (22) G. Randers, Phys. Rev. 59, 195 (1941).
  * (23) P. Funk, Math. Ann. 101, 226 (1929).
  * (24) G. W. Gibbons, et al., Phys. Rev. D 79, 044022 (2009).