# Electronic structure and resonant inelastic x-ray scattering in Ca3Ru2O7

V.N. Antonov  G. V. Kurdyumov Institute for Metal Physics of the N.A.S. of
Ukraine, 36 Academician Vernadsky Boulevard, UA-03142 Kyiv, Ukraine  Max-
Planck-Institute for Solid State Research, Heisenbergstrasse 1, 70569
Stuttgart, Germany  D.A. Kukusta  G. V. Kurdyumov Institute for Metal Physics
of the N.A.S. of Ukraine, 36 Academician Vernadsky Boulevard, UA-03142 Kyiv,
Ukraine  L.V. Bekenov  G. V. Kurdyumov Institute for Metal Physics of the
N.A.S. of Ukraine, 36 Academician Vernadsky Boulevard, UA-03142 Kyiv, Ukraine

###### Abstract

We have investigated the electronic structure of the transition metal oxide
Ca3Ru2O7 within density-functional theory using the generalized gradient
approximation while considering strong Coulomb correlations in the framework
of the fully relativistic spin-polarized Dirac linear muffin-tin orbital band-
structure method. Ca3Ru2O7 can be classified as a Mott insulator since it was
expected to be metallic from band structure calculations. We found that the
magnetic ground state of Ca3Ru2O7 possesses an AFM-$b$ magnetic structure with
the Ru spin moments ordered antiferromagnetically along the $b$ axis. We have
investigated the resonant inelastic x-ray scattering (RIXS) spectra at the Ru
and Ca $K$, $L_{3}$, and $M_{3}$ edges as well as at the O $K$ edge. The
experimentally measured RIXS spectrum of Ca3Ru2O7 at the Ru $L_{3}$ edge
possesses a sharp feature $\leq$2 eV corresponding to transitions within the
Ru $t_{2g}$ levels. The excitation located from 2 eV to 4 eV is due to
$t_{2g}$ $\rightarrow$ $e_{g}$ transitions. The third wide structure situated
at 4.5$-$11 eV appears due to transitions between the Ru 4$d_{\rm{O}}$ states
derived from the tails of oxygen 2$p$ states and $e_{g}$ and $t_{2g}$ states.
The RIXS spectra at the Ru $L_{3}$ and $M_{3}$ edges are very similar.
However, the corresponding RIXS spectra at the Ca site quite differ from each
other due to the significant difference in the widths of core-levels. The RIXS
spectrum at the O $K$ edge consists of three major inelastic excitations. We
found that the first two low energy features $\leq$2.5 eV are due to interband
transitions between occupied and empty O${}_{t_{2g}}$ states which appear due
to the strong hybridization between oxygen 2$p$ and Ru $t_{2g}$ states in
close vicinity to the Fermi level. The next major peak between 2.5 and 8 eV
reflects interband transitions from occupied O 2$p$ to empty O${}_{t_{2g}}$
states. A wide energy interval between 4 and 11 eV is occupied by rather weak
O2p $\rightarrow$ O${}_{e_{g}}$ transitions.

###### pacs:

75.50.Cc, 71.20.Lp, 71.15.Rf

##  I Introduction

Ruddlesden-Popper (RP) type ruthenates An+1RunO3n+1 (A = Sr or Ca), where $n$
is a number of Ru-O layers per unit cell, have attracted much attention due to
their fascinated physical properties. For example, Sr2RuO4 ($n$ = 1) possesses
a unique $p$-wave superconductivity Maeno et al. (1994, 2001), however,
Ca2RuO4 is a typical Mott insulator Cao et al. (1997a). In the $n$ = $\infty$
family SrRuO3 is a ferromagnetic metal Cao et al. (1997a), whereas CaRuO3 does
not show any magnetic ordering Klein et al. (1999). Double-layered Sr3Ru2O7
($n$ = 2) has an orthorhombic crystal structure (space group $Bbcb$, No. 68)
Shaked et al. (2000), where RuO6 octahedra rotate around the $c$ axis. The
ionic radius of Ca2+ is smaller than that of Sr2+, therefore, Ca3Ru2O7 has a
more distorted crystal structure, namely, the orthorhombic symmetry with
$Bb2_{1}m$ space group (No. 36) with rotation and tilting of RuO6 octahedra
Cao et al. (2000); Yoshida et al. (2005). Sr3Ru2O7 is a paramagnetic metal
with unusual large quasiparticle masses Borzi et al. (2008), it shows the
Fermi-liquid behavior, and ferromagnetic instability in the ground state Ikeda
et al. (2000). On the other hand, Ca3Ru2O7 is a Mott-like antiferromagnetic
(AFM) insulator, where ferromagnetic bilayers stack antiparallel along the $c$
axis (N$\acute{\rm{e}}$el temperature $T_{N}$ = 56 K) Cao et al. (1997b); Ke
et al. (2011). This ruthenate possesses the first-order metal-to-insulator
transition at $T_{MI}$ = 48 K that distorts the RuO6 octahedra and coincides
with an upturn in the out-of-plane resistivity Ohmichi et al. (2004); Yoshida
et al. (2005). A spin rotation from the $a$ axis to the $b$ axis (AFM-$b$)
occurs at $T_{MI}$ that appears to be mediated by an incommensurate spin state
Yoshida et al. (2004); Dashwood et al. (2020). The system also shows complex
magnetic field dependence Bao et al. (2008); Sokolov et al. (2019). Ca3Ru2O7
stimulated much research effort due to its colossal magnetoresistance and
anisotropic magnetic behavior Cao et al. (1997b); McCall et al. (1998); Cao et
al. (2003); Ohmichi et al. (2004); Cao et al. (2004). This bilayer compound
also retains highly anisotropic electric resistivity down to the lowest
temperatures Yoshida et al. (2004). Specific-heat and photoemission studies
Puchkov et al. (1998); Cao et al. (1997c) indicate unusual behavior for the
density of states (DOSs) and bandwidth of Ca3Ru2O7 at low temperatures,
reflecting strong electron correlation effects in the 4$d$ band. The optical
conductivity shows a small energy gap of 25 meV at low temperature Lee et al.
(2007). The Raman spectra measurements by Liu et al. Liu et al. (1999) reveal
a charge energy gap of $\Delta_{c}$ $\sim$ 96 meV in Ca3Ru2O7. The large gap
ratio $\Delta_{c}/k_{B}T_{MI}$ = 23 associated with this gap is indicative of
strong electronic correlations, perhaps pointing to a Mott-Hubbard-type
transition. It is believed that the wide spread of physical phenomena in these
ruthenates are due to a delicate balance of competing interactions, such as,
orbital degrees of freedom, crystal structure distortion, electron
correlation, and spin-orbit coupling (SOC) Mazin and Singh (1997); Mizokawa et
al. (2001); Fang et al. (2004); Liu et al. (2008); Haverkort et al. (2008);
Bertinshaw et al. (2021).

There are several band-structure calculations of Ca3Ru2O7 based on the
density-functional theory (DFT) Singh and Auluck (2006); Liu (2011); Jin and
Ku (2018); Markovi et al. (2018); Puggioni et al. (2020). Singh and Auluck
provided the band structure calculations of Ca3Ru2O7 and found that the local
spin-density approximation (LSDA) failed to reproduce the insulating ground
state Singh and Auluck (2006). Liu reported an electronic structure study on
Ca3Ru2O7 using first-principles calculations including the SOC and on-site
Coulomb interaction Liu (2011). He found that the observed insulating ground
state of Ca3Ru2O7 can be reproduced if the magnetic moments are aligned along
the $b$ axis.

In the present study, we focus on the electronic structure and resonant
inelastic x-ray scattering (RIXS) spectra of Ca3Ru2O7. The RIXS measurements
at the Ru $L_{3}$ edge for the Ca3Ru2O7 ruthenate have been successfully
performed recently by Bertinshaw et al. Bertinshaw et al. (2021). They capture
the in-plane magnon across the entire Brillouin zone, giving a spin-wave gap
of $\sim$8 meV. At a higher energy loss, they observe $dd$-type excitations
between the $t_{2g}$ and $e_{g}$ bands. The measurements of the oxygen $K$
RIXS spectrum have been provided in Ref. von Arx et al. (2020). The authors
resolve two intra-$t_{2g}$ excitations in Ca3Ru2O7. The lowest lying
excitation is interpreted as a magnetic transverse mode with multiparticle
character. They also observe a very intensive peak at $\sim$3.3 eV in the O
$K$ RIXS spectrum.

In this paper, we report a detailed theoretical DFT study of the electronic
structure and RIXS spectra of the Ca3Ru2O7 oxide at the Ru and Ca $K$,
$L_{3}$, and $M_{3}$ edges as well as at the O $K$ edge to investigate the
influence of the SOC and Coulomb correlations. The energy band structure of
this RP ruthenate was calculated using the fully relativistic spin-polarized
Dirac linear muffin-tin orbital band-structure method. We used both the
generalized gradient approximation (GGA) and the GGA+$U$ approach to assess
the sensitivity of the RIXS results to different treatment of the correlated
electrons.

This paper is organized as follows. The crystal structures of Ca3Ru2O7 and
computational details are presented in Sec. II. Section III presents the
electronic and magnetic structures of this ruthenate. In Sec. IV, the
theoretical investigations of the RIXS spectra of Ca3Ru2O7 at the Ru and Ca
$K$, $L_{3}$ edges and at the O $K$ edge are presented. The theoretical
results are compared with available experimental data. Finally, the results
are summarized in Sec. V.

##  II Computational details

###  II.1 Resonant inelastic x-ray scattering.

In the direct RIXS process Ament et al. (2011), the incoming photon with
energy $\hbar\omega_{\mathbf{k}}$, momentum $\hbar\mathbf{k}$, and
polarization $\bm{\epsilon}$ excites the solid from the ground state $|{\rm
g}\rangle$ with energy $E_{\rm g}$ to the intermediate state $|{\rm I}\rangle$
with energy $E_{\rm I}$. During relaxation, an outgoing photon with energy
$\hbar\omega_{\mathbf{k}^{\prime}}$, momentum $\hbar\mathbf{k}^{\prime}$, and
polarization $\bm{\epsilon}^{\prime}$ is emitted, and the solid is in state
$|f\rangle$ with energy $E_{\rm f}$. As a result, an excitation with energy
$\hbar\omega=\hbar\omega_{\mathbf{k}}-\hbar\omega_{\mathbf{k}^{\prime}}$ and
momentum $\hbar\mathbf{q}$ = $\hbar\mathbf{k}-\hbar\mathbf{k}^{\prime}$ is
created. Our implementation of the code for the calculation of the RIXS
intensity uses Dirac four-component basis functions Nemoshkalenko et al.
(1983) in the perturbative approach Arola et al. (1997). RIXS is the second-
order process, and its intensity is given by

$\displaystyle
I(\omega,\mathbf{k},\mathbf{k}^{\prime},\bm{\epsilon},\bm{\epsilon}^{\prime})$
$\displaystyle\propto$ $\displaystyle\sum_{\rm f}\left|\sum_{\rm
I}{\langle{\rm
f}|\hat{H}^{\prime}_{\mathbf{k}^{\prime}\bm{\epsilon}^{\prime}}|{\rm
I}\rangle\langle{\rm I}|\hat{H}^{\prime}_{\mathbf{k}\bm{\epsilon}}|{\rm
g}\rangle\over E_{\rm g}-E_{\rm I}}\right|^{2}$ (1)
$\displaystyle\times\delta(E_{\rm f}-E_{\rm g}-\hbar\omega),$

where the RIXS perturbation operator in the dipole approximation is given by
the lattice sum
$\hat{H}^{\prime}_{\mathbf{k}\bm{\epsilon}}=\sum_{\mathbf{R}}\hat{\bm{\alpha}}\bm{\epsilon}\exp(-{\rm
i}\mathbf{k}\mathbf{R})$, where $\alpha$ are the Dirac matrices. The sum over
the intermediate states $|{\rm I}\rangle$ includes the contributions from
different spin-split core states at the given absorption edge. The matrix
elements of the RIXS process in the frame of the fully relativistic Dirac LMTO
method were presented in our previous publication Antonov et al. (2022).

###  II.2 X-ray magnetic circular dichroism

Magneto-optical (MO) effects refer to various changes in the polarization
state of light upon interaction with materials possessing a net magnetic
moment, including rotation of the plane of linearly polarized light (Faraday,
Kerr rotation), and the complementary differential absorption of left and
right circularly polarized light (circular dichroism). In the near visible
spectral range these effects result from excitation of electrons in the
conduction band. Near x-ray absorption edges, or resonances, MO effects can be
enhanced by transitions from well-defined atomic core levels to transition
symmetry selected valence states.

Within the one-particle approximation, the absorption coefficient
$\mu^{\lambda}_{j}(\omega)$ for incident x-ray polarization $\lambda$ and
photon energy $\hbar\omega$ can be determined as the probability of electronic
transitions from initial core states with the total angular momentum $j$ to
final unoccupied Bloch states

$\displaystyle\mu_{j}^{\lambda}(\omega)$ $\displaystyle=$
$\displaystyle\sum_{m_{j}}\sum_{n\bf k}|\langle\Psi_{n\bf
k}|\Pi_{\lambda}|\Psi_{jm_{j}}\rangle|^{2}\delta(E_{n\bf
k}-E_{jm_{j}}-\hbar\omega)$ (2) $\displaystyle\times\theta(E_{n\bf
k}-E_{F})\,,$

where $\Psi_{jm_{j}}$ and $E_{jm_{j}}$ are the wave function and the energy of
a core state with the projection of the total angular momentum $m_{j}$;
$\Psi_{n\bf k}$ and $E_{n\bf k}$ are the wave function and the energy of a
valence state in the $n$-th band with the wave vector k; and $E_{F}$ is the
Fermi energy.

Here, $\Pi_{\lambda}$ is the electron-photon interaction operator in the
dipole approximation

$\Pi_{\lambda}=-e\mbox{\boldmath$\alpha$}\bf{a_{\lambda}},$ (3)

where $\bm{\alpha}$ are the Dirac matrices and $\bf{a_{\lambda}}$ is the
$\lambda$ polarization unit vector of the photon vector potential, with
$a_{\pm}=1/\sqrt{2}(1,\pm i,0)$, $a_{\parallel}=(0,0,1)$. Here, $+$ and $-$
denote, respectively, left and right circular photon polarizations with
respect to the magnetization direction in the solid. Then, x-ray magnetic
circular and linear dichroisms are given by $\mu_{+}-\mu_{-}$ and
$\mu_{\parallel}-(\mu_{+}+\mu_{-})/2$, respectively. More detailed expressions
of the matrix elements in the electric dipole approximation may be found in
Refs. Guo et al. (1994); Antonov et al. (2004); Arola et al. (2004). The
matrix elements due to magnetic dipole and electric quadrupole corrections are
presented in Ref. Arola et al. (2004).

###  II.3 Crystal structure.

Ca3Ru2O7 has the orthorhombic symmetry with $Bb2_{1}m$ space group (No. 36)
with the rotation and tilting of RuO6 octahedra Cao et al. (2000); Yoshida et
al. (2005) (see Table 1 and Fig. 1). The structure of Ca3Ru2O7 consists of a
RuO6 double layer and CaO layer per unit cell. The Ru4+ cations are surrounded
by oxygen octahedra with the interatomic distances Ru-O1, Ru-O2, Ru-O3, and
Ru-O4 of 1.9872, 1.9873, 2.0033, and 1.9949 Å , respectively. The inter-atomic
distances between Ru and Ca1 and Ca2 ions are equal to 3.1263 and 3.0355 Å ,
respectively.

Table 1: The atomic as well as Wyckoff positions (WP) of Ca3Ru2O7 with $Bb2_{1}/m$ lattice symmetry (lattice constants $a$ = 5.3677 Å, $b$ = 5.5356 Å and $c$ = 19.5212 Å) at temperature of 8 K Yoshida et al. (2005).  Atom | WP | $x$ | $y$ | $z$
---|---|---|---|---
Ca1 | 4$a$ | 0.7382 | 0.1949 | 0
Ca2 | 8$b$ | 0.2443 | 0.3036 | 0.3112
Ru | 8$b$ | 0.2533 | 0.7512 | 0.4011
O1 | 8$b$ | 0.8188 | 0.2309 | 0.6987
O2 | 4$a$ | 0.3370 | 0.7702 | 0.5
O3 | 8$b$ | 0.4474 | 0.9500 | 0.0819
O4 | 8$b$ | 0.9481 | 0.0458 | 0.1162

The neutron diffraction measurements show that the lattice constants $a$ and
$c$ of Ca3Ru2O7 shorten on cooling from room temperature (RT), while $b$
elongates Yoshida et al. (2005). At the first-order transition temperature of
48 K the lattice parameters jump, though, the space group symmetry $Bb2_{1}m$
is not affected throughout the temperature range measured Yoshida et al.
(2005). The $c$ constant shortens by about 0.1%, while both $a$ and $b$
lengthen by about 0.07% at $T_{MI}$.

Figure 1: (Color online) (upper panel) The crystal structure of Ca3Ru2O7
(space group $Bb2_{1}m$, No. 36) Yoshida et al. (2005) with Ru3+ ions in
octahedral oxygen coordination.

The lengths of Ru-O1 and Ru-O3 bonds increase with decreasing temperature. On
the other hand, those of Ru-O2 and Ru-O4 decrease with decreasing temperature.
The angles between the RuO6 octahedra are almost temperature independent above
$T_{N}$ and little affected by the first-order phase transition. However, some
characteristic temperature dependences can be observed $\leq$40 K Yoshida et
al. (2005). The angle of Ru-O1-Ca2 decreases with decreasing temperature. On
the other hand, the angle of Ru-O2-Ru increases with decreasing temperature
Yoshida et al. (2005).

###  II.4 Calculation details

The details of the computational method are described in our previous papers
Antonov et al. (2006, 2007, 2010, 2022) and here we only mention several
aspects. The band structure calculations were performed using the fully
relativistic linear muffin-tin orbital (LMTO) method Andersen (1975); Antonov
et al. (2004). This implementation of the LMTO method uses four-component
basis functions constructed by solving the Dirac equation inside an atomic
sphere Nemoshkalenko et al. (1983). The exchange-correlation functional of the
GGA type was used in the version of Perdew, Burke, and Ernzerhof Perdew et al.
(1996). The Brillouin zone integration was performed using the improved
tetrahedron method Blöchl et al. (1994). The basis consisted of Ca and Ru $s$,
$p$, $d$, and $f$, and O $s$, $p$, and $d$ LMTOs.

To consider the electron-electron correlation effects, we use in this work the
relativistic generalization of the rotationally invariant version of the
LSDA+$U$ method Yaresko et al. (2003) which takes into account that, in the
presence of spin-orbit coupling, the occupation matrix of localized electrons
becomes non-diagonal in spin indexes. Hubbard $U$ was considered an external
parameter and varied from 0.8 to 3.8 eV for the 4$d$ Ru site. The values of
exchange Hund coupling used in our calculations were obtained from constrained
LSDA calculations Dederichs et al. (1984); Pickett et al. (1998) and equal to
$J_{H}$=0.8 eV for Ru. Thus, the parameter $U_{eff}=U-J_{H}$, which roughly
determines the splitting between the lower and upper Hubbard bands, varied
between 0 and 3.0 eV for the 4$d$ metal site. We adjusted the value of $U$ to
achieve the best agreement with the experiment.

The x-ray absorption spectroscopy (XAS), x-ray magnetic circular dichroism
(XMCD), and RIXS spectra were calculated taking into account the exchange
splitting of core levels. The finite lifetime of a core hole was accounted for
by folding the spectra with a Lorentzian. The widths of core levels $\Gamma$
for Ca, Ru, and O were taken from Ref. Campbell and Parr (2001). The finite
experimental resolution of the spectrometer was accounted for by a Gaussian of
0.6 eV (the $s$ coefficient of the Gaussian function).

##  III Electronic and magnetic structures

Figure 2 presents the energy band structure of Ca3Ru2O7 calculated in the
fully relativistic Dirac GGA+SO approximation (the upper panel) and
considering Coulomb correlations in the GGA+SO+$U$ approach (the lower panel).
The GGA+SO approximation produces a metallic ground state in Ca3Ru2O7 in
contradiction with the experimental data which show that Ca3Ru2O7 is a Mott-
like AFM insulator below $T_{MI}$ = 48 K Cao et al. (1997b); Ohmichi et al.
(2004); Yoshida et al. (2005); Ke et al. (2011). To produce the correct ground
state one has to take into account strong Coulomb correlations in Ca3Ru2O7.
The GGA+SO+$U$ approach shifts the occupied and empty $t_{2g}$ bands downward
and upward, respectively, by $U_{eff}$/2 producing a direct energy gap of
0.215 eV and an indirect one of 0.053 eV for $U_{eff}$ = 1.3 eV. We found that
this value of Hubbard $U$ applied for the Ru site produces the best agreement
between the calculated and experimentally measured RIXS spectra at the Ru
$L_{3}$ edge in Ca3Ru2O7. The energy gap is increased with increasing Hubbard
$U$.

Figure 2: (Color online) The energy band structure and total density of states
(DOS) [in states/(cell eV)] of Ca3Ru2O7 calculated in the GGA+SO approximation
(the upper panel) and GGA+SO+$U$ approach (the lower panel).  Figure 3: (Color
online) The partial density of states (DOS) in Ca3Ru2O7 calculated in the
GGA+SO+$U$ approach with $U_{eff}$ = 1.3 eV.

Figure 3 presents the partial density of states (DOS) in Ca3Ru2O7 calculated
in the GGA+SO+$U$ approach. Four electrons occupy the $t_{2g}$-type low energy
band (LEB) manifold in the energy interval from $-$1.7 eV to $-$0.6 eV and
between $-$0.35 eV and $E_{F}$ in Ca3Ru2O7. The empty $t_{2g}$ states [the
upper energy band (UEB)] occupy the energy range from 0.1 eV to 0.9 eV. The
$e_{g}$-type states of Ru are distributed far above the Fermi level from 1.3
eV to 5.0 eV. The occupation number of 4$d$ electrons in the Ru atomic sphere
is equal to 5.66, which is much larger than the expected value of four
$t_{2g}$ electrons in Ru4+. The excessive charge is provided by the tails of O
2$p$ states inside the Ru atomic spheres. These charge transfer 4$d_{\rm{O}}$
states, which are located from $-$7.6 to approximately $-$5.0 eV, play an
essential role for the RIXS spectrum at the Ru $L_{2,3}$ edges (see Section
IV). The Ca 3$d$ states are mostly empty and located from 5.0 to 8.2 eV above
the Fermi level. The oxygen 2$p$ states are strongly hybridized with the Ru
4$d$ states due to small interatomic O-Ru distances and delocalized nature of
O 2$p$ states. They occupy the energy interval between $-$7.6 eV and $-$2.5
eV. The small peaks in the vicinity of the Fermi level are due to the
hybridization between O 2$p$ and Ru $t_{2g}$ LEB and UEB.

Figure 4: (Color online) A half of the Ca3Ru2O7 unit cell with AFM-ordered Ru
moments parallel to the $b$ axis (the AFM-$b$ order).

We found that in the ground state Ca3Ru2O7 possesses an AFM-$b$ magnetic
structure with the Ru spin moments ordered antiferromagnetically along the $b$
axis (see Fig. 4). It is in consistence with experimental data Yoshida et al.
(2004); Dashwood et al. (2020) and previous band structure calculations Liu
(2011).

Table 2:  The theoretically calculated in GGA+SO+$U$ ($U_{eff}$ = 1.3 eV) spin $M_{s}$, orbital $M_{l}$, and total $M_{tot}$ magnetic moments (in $\mu_{\text{B}}$) in Ca3Ru2O7.  Atom | $M_{s}$ | $M_{l}$ | $M_{total}$
---|---|---|---
Ca1 | 0.0002 | 0.0007 | 0.0009
Ca2 | 0.0165 | 0.0014 | 0.0179
Ru | 1.4783 | 0.1370 | 1.6153
O1 | 0.0625 | 0.0072 | 0.0697
O2 | 0.0120 |  $-$0.0065 | 0.0055
O3 | 0.1008 | 0.0097 | 0.1105
O4 | 0.1141 | 0.0096 | 0.1237

Table 2 presents the theoretically calculated spin $M_{s}$, orbital $M_{l}$,
and total $M_{tot}$ magnetic moments in Ca3Ru2O7. The Ru spin and orbital
moments are parallel in Ca3Ru2O7 in accordance with Hund’s third rule. The
spin and orbital parts of the magnetic moments for the Ca ions in Ca3Ru2O7 are
found to be very small. The spin and orbital magnetic moments at the oxygen
sites are also relatively small with the largest moments at the O4 site and
the smallest ones at the O2 site.

##  IV XAS, XMCD, and RIXS spectra

###  IV.1 Ru $L_{3}$ and $M_{3}$ XAS, XMCD, and RIXS spectra

The experimental RIXS spectrum at the Ru $L_{3}$ edge was measured by
Bertinshaw et al. Bertinshaw et al. (2021) in the energy range up to 6.5 eV.
In addition to an elastic peak centered close to zero energy loss, the
spectrum consists of a peak at $\sim$0.5 eV with a high energy shoulder at 1.3
eV and two peaks at 3 and 5.8 eV. We found that the two peak structure
situated $\leq$2 eV corresponds to intra-$t_{2g}$ excitations. These peaks are
very sensitive to the value of the energy gap in Ca3Ru2O7 and the relative
position of the $t_{2g}$ LEB and UEB bands (Fig. 2). Figure 5 shows the
experimental RIXS spectrum measured by Bertinshaw et al. Bertinshaw et al.
(2021) (open magenta circles) in comparison with the theoretical spectra
calculated for $t_{2g}\to t_{2g}$ transitions with different values of
$U_{eff}$. The best agreement was found for the GGA+SO+$U$ approach with
$U_{eff}$ = 1.3 eV. The calculations with larger $U_{eff}$ shift the RIXS
spectrum towards higher energies.

Figure 5: (Color online) The experimental RIXS spectrum measured by Bertinshaw
et al. Bertinshaw et al. (2021) (open magenta circles) at the Ru $L_{3}$ edge
of Ca3Ru2O7 compared with the theoretical spectra calculated for
$t_{2g}\rightarrow t_{2g}$ transitions with different $U_{eff}$.  Figure 6:
(Color online) The experimental RIXS spectrum (open magenta circles) measured
by Bertinshaw et al. Bertinshaw et al. (2021) at the Ru $L_{3}$ edge of
Ca3Ru2O7 compared with the theoretically calculated partial contributions to
the RIXS spectrum from different interband transitions (the lower panel). The
upper panel presents the theoretically calculated partial contributions to the
RIXS spectrum from different interband transitions at the $M_{3}$ edge. The
calculations were carried out in the GGA+SO+$U$ approach with $U_{eff}$= 1.3
eV.

Figure 6 (the lower panel) presents the experimental RIXS spectrum (open
magenta circles) at the Ru $L_{3}$ edges for Ca3Ru2O7 Bertinshaw et al. (2021)
in a wide energy interval in comparison with the theoretically calculated
partial contributions from different interband transitions in the GGA+SO+$U$
approach. As we mentioned above, the two peaks situated $\leq$2 eV correspond
to intra-$t_{2g}$ excitations (dashed and dotted blue curves in Fig. 6). We
divided the occupied part of the $t_{2g}$ energy band into two groups: four
bands in close vicinity to the Fermi level (blue curves in Fig. 2) and the
other 12 bands situated at lower energy (red curves in Fig. 2); and calculated
the interband transitions from these two groups to empty $t_{2g}$ states
(magenta curves in Fig. 2) separately. We found that the small number of
occupied bands in close vicinity to the Fermi level produce a larger RIXS
signal (peak $\alpha$) than the large number of low energy bands (peak
$\beta$) due to corresponding matrix elements. The peak $\gamma$ located at
$\sim$3 eV (the red curve in the lower panel of Fig. 6) was found to be due to
$t_{2g}\rightarrow e_{g}$ transitions. The strong fine structure $\delta$ at
$\sim$5.8 eV (the green curve) is due to 4$d_{\rm{O}}$ $\rightarrow$ $t_{2g}$
transitions. The structure which extends from 7 up to 11 eV (the magenta
curve) presents 4$d_{\rm{O}}$ $\rightarrow$ $e_{g}$ transitions. The
theoretical calculations are in good agreement with the experimental data.

Figure 6 (the upper panel) shows the theoretically calculated RIXS spectrum at
the Ru $M_{3}$ edge presented as partial contributions from different
interband transitions. The shape of the spectrum is very similar to the
corresponding spectrum at the Ru $L_{3}$ edge, however, with approximately one
order of magnitude smaller intensity.

Figure 7: (Color online) The theoretically calculated XAS (the upper panel)
and XMCD (the lower panel) spectra at the Ru $L_{3}$ and $M_{3}$ edges of
Ca3Ru2O7 calculated in the GGA+SO+$U$ approach with $U_{eff}$= 1.3 eV.

Figure 7 shows the theoretically calculated XAS (the upper panel) and XMCD
(the lower panel) spectra at the Ru $L_{3}$ and $M_{3}$ edges of Ca3Ru2O7
calculated in the GGA+SO+$U$ approach. The widths of the 2$p_{3/2}$ and
3$p_{3/2}$ core levels are close ($\Gamma_{2p_{3/2}}$ = 1.87 eV and
$\Gamma_{3p_{3/2}}$ = 2.2 eV Campbell and Parr (2001)). Therefore, both the Ru
$L_{3}$ and $M_{3}$ XAS spectra possess very similar structures with a major
peak at $\sim$3 eV, a low energy shoulder at 0.7 eV, and an additional high
energy peak at 15 eV. The last fine structure is due to the hybridization of
Ru 4$d$ states with the oxygen 2$p$ states which have a corresponding peak at
15 eV above the Fermi level (not shown). Two low energy fine structures in the
Ru $L_{3}$ ($M_{3}$) XAS spectrum $\leq$5 eV are due to transitions from the
core 2$p_{3/2}$ (3$p_{3/2}$) level to empty $t_{2g}$ (the low energy shoulder)
and $e_{g}$ (the major peak) states. The transitions to $e_{g}$ states have
larger intensity in comparison with the corresponding transitions into empty
$t_{2g}$ states. The XMCD spectra also have a two-peak structure (the lower
panel of Fig. 7) but these spectra are dominated by transitions into empty
$t_{2g}$ states with a small contribution from empty $e_{g}$ orbitals at
higher energy. The XMCD spectrum at the $M_{3}$ edge is much smaller than at
the $L_{3}$ edge.

###  IV.2 Ca $L_{3}$ and $M_{3}$ XAS, XMCD, and RIXS spectra

Figure 8: (Color online) The theoretically calculated XAS (the upper panel)
and XMCD (the lower panel) spectra at the Ca $L_{3}$ and $M_{3}$ edges of
Ca3Ru2O7 calculated in the GGA+SO+$U$ approach with $U_{eff}$= 1.3 eV.

Figure 8 shows the theoretically calculated XAS (the upper panel) and XMCD
(the lower panel) spectra at the Ca $L_{3}$ and $M_{3}$ edges of Ca3Ru2O7. The
Ca 3$d$ partial DOS is situated far above the Fermi level (see Fig. 3).
Therefore, both the Ca $L_{3}$ and $M_{3}$ XAS spectra possess one-peak
structures at $\sim$7 eV above the Fermi level. The Ca $M_{3}$ XAS spectrum is
much wider than the corresponding $L_{3}$ spectrum due to a significantly
larger width of the 3$p_{3/2}$ core level in comparison with the 2$p_{3/2}$
one Campbell and Parr (2001). The Ca $L_{3}$ XMCD spectrum has a complicated
shape with several positive and negative peaks. It is more than one order of
magnitude smaller than the Ru $L_{3}$ XMCD spectrum (the lower panel of Fig.
7) due to smaller SOC, spin and orbital magnetic moments at the Ca site in
comparison with the Ru site (Table 2). The dichroism at the Ca $M_{3}$ edge is
very small even compared to the Ca $L_{3}$ XMCD spectrum.

Figure 9: (Color online) The theoretically calculated RIXS spectra at the Ca
$L_{3}$ and $M_{3}$ edges in Ca3Ru2O7 (the upper panel) calculated in the
GGA+SO+$U$ approach ($U_{eff}$ = 1.3 eV) and partial contributions to the RIXS
spectra from different interband transitions at the Ca $M_{3}$ (the middle
panel) and Ca $L_{3}$ edges (the lower panel).

Figure 9 show the theoretically calculated RIXS spectra at the Ca $L_{3}$ and
$M_{3}$ edges in Ca3Ru2O7 (the upper panel) and partial contributions to the
RIXS spectra from different interband transitions at the Ca $M_{3}$ (the
middle panel) and Ca $L_{3}$ edges (the lower panel). The low energy two-peak
fine structure $\alpha$ $\leq$2.5 eV (the blue curve) was found to be due to
Ca${}_{t_{2g}}\rightarrow$ Ca${}_{t_{2g}}$ transitions. More precisely,
between Ca 3$d$ occupied and empty states, which are located in the energy
region of Ru $t_{2g}$ states from $-$1.7 eV to $E_{F}$ and from 0.1 to 0.9 eV,
respectively (Fig. 3). The next fine structure $\beta$ located between 2.5 and
6.5 eV arises from interband transitions between 4$d_{O}$ states (which are
derived from the tails of oxygen 2$p$ states inside the Ca atomic spheres) and
Ca${}_{t_{2g}}$ states. The peak $\gamma$ is due to interband transitions from
Ca${}_{t_{2g}}$ states to the Ca empty 4$d$ states. The large high energy peak
$\delta$ is due to $4d_{O}\rightarrow$ Cad transitions.

Figure 10: (Color online) The theoretically calculated RIXS spectra at the Ca
$M_{3}$ edge in Ca3Ru2O7 obtained for different parameters $\Gamma$.

The RIXS spectra at the $L_{3}$ and $M_{3}$ edges have to possess similar
shapes because they reflect interband transitions between $d_{5/2}$ band
states and the 2$p_{3/2}$ and 3$p_{3/2}$ core levels, respectively. This is
true indeed in the case of the Ru $L_{3}$ and $M_{3}$ RIXS spectra (see Fig.
6). However, the $L_{3}$ and $M_{3}$ RIXS spectra at the Ca site differ from
each other (the upper panel of Fig. 9). There is strong suppression of the low
energy peaks $\alpha$ and $\beta$ in the Ca $M_{3}$ RIXS spectrum in
comparison with the Ca $L_{3}$ spectrum. Such a puzzle can be explained by the
significant difference in the widths of corresponding core-levels
($\Gamma_{3p_{3/2}}$ = 1.2 eV and $\Gamma_{2p_{3/2}}$ = 0.21 eV Campbell and
Parr (2001)). Figure 10 shows the theoretically calculated RIXS spectra at the
Ca $M_{3}$ edge in Ca3Ru2O7 obtained for different core-level parameters
$\Gamma$. Under decreasing the parameter $\Gamma$ from 1.2 eV to 0.2 eV the
intensity of the low energy peaks $\alpha$ and $\beta$ is increased and the
shape of the Ca $M_{3}$ RIXS spectrum is transformed to the $L_{3}$ one. The
high energy peak $\delta$ does not change its shape. On the other hand, when
we changed the parameter $\Gamma$ to a greater extent from 1 to 6 eV we found
that the intensity of the peak $\delta$ monotonously decreased.

It should be mentioned that the x-ray absorption and RIXS processes have
different nature. The matrix elements of the RIXS process are more complicated
than the x-ray absorption ones Arola et al. (1997); Antonov et al. (2004,
2022). As a result, the XAS spectra possess linear dependence on the core
level width: the larger $\Gamma_{core}$ the wider the XAS spectrum without a
shape change (see the upper panel of Fig. 8). The RIXS spectra show nonlinear
dependence on $\Gamma_{core}$, which strongly affects the spectrum shape.

Experimental measurements of the XAS, XMCD, and RIXS spectra at the Ca
$L_{3}$, $M_{3}$, and Ru $M_{3}$ edges are highly desirable.

###  IV.3 O, Ca, and Ru $K$ RIXS spectra

The RIXS spectra at the O $K$ edge in Ca3Ru2O7 were measured by Arx et al. von
Arx et al. (2020) up to 5 eV. The O $K$ RIXS spectrum consists of a peak
centered at zero energy loss, which comprises the elastic line and other low-
energy features such as phonons, magnons, etc., two inelastic excitations at
0.5 eV, 1.8 eV, and a major peak at $\sim$3.3. We found that the first two low
energy features are due to interband transitions between occupied and empty
O${}_{t_{2g}}$ states, which appear as a result of the strong hybridization
between oxygen 2$p$ states with Ru $t_{2g}$ LEB and UEB in the vicinity of the
Fermi level (see Fig. 3). Therefore, the oxygen $K$ RIXS spectroscopy can be
used for the estimation of the energy band gap and positions of Ru 4$d$
Hubbard bands. The major peak at 3.3 eV reflects interband transitions from
occupied O 2$p$ states to the empty oxygen states which originate from the
hybridization with Ru $t_{2g}$ states. We found that the theory reproduces
well the shape and energy position of the two low energy features. The energy
position of the major peak at 3.3 eV is also reproduced well by theory,
although our calculations give a wider major peak in comparison with the
experiment. The interband transitions O${}_{2p}\rightarrow$ O${}_{e_{g}}$ are
relatively small and occupy a wide energy interval from 4.1 to 11 eV.

Figure 11: (Color online) The theoretically calculated RIXS spectrum at the O
$K$ edge in Ca3Ru2O7 obtained in the GGA+SO+$U$ approach ($U_{eff}$ = 1.3 eV)
and partial contributions to the RIXS spectrum from different interband
transitions in comparison with the experiment von Arx et al. (2020).

Let us consider now the XAS, XMCD, and RIXS spectra at the Ru and Ca $K$
edges. For that we first present the partial Ca 4$p$ and Ru 5$p$ DOS in Fig.
12 in a wide energy interval from $-$20 to 40 eV. We distinguish several
groups of the bands. The group $a$ derives from the hybridizations of the $p$
states with corresponding Ca 4$s$ and Ru 5$s$ states. The group $b$ is due to
the hybridization with oxygen 2$p$ states. The groups $c$ and $d$ are from the
hybridization with Ru $t_{2g}$ LEB and UEB, respectively. The group $f$ comes
from the hybridization with Ru $e_{g}$ states. At last, the very wide
structure $g$, which we divide into several subgroups, is the Ca 4$p$ and Ru
5$p$ bands themselves.

Figure 12: (Color online) The partial Ca 4$p$ and Ru 5$p$ DOS [in states/(atom
eV)] in Ca3Ru2O7 calculated in the GGA+SO+$U$ ($U_{eff}$= 1.3 eV) approach.

Figure 13 presents the theoretically calculated XAS (the upper panel) and XMCD
(the lower panel) at the Ca and Ru $K$ edges. The four fine structures $f$,
$g_{1}$, $g_{2}$, and $g_{3}$ in the Ca $K$ XAS spectrum can be easily
distinguished as transitions from the Ca 1$s$ core level into corresponding
groups of the Ca 4$p$ states with the same labels. The width of the Ru 1$s$
core level is much larger than the Ca one ($\Gamma^{\rm{Ru}}_{1s}$ = 5.33 eV
and $\Gamma^{\rm{Ca}}_{1s}$ = 0.77 eV Campbell and Parr (2001)), therefore,
the Ru $K$ XAS spectrum possesses less structured features. The XMCD spectra
at both Ca and Ru $K$ edges are at least three order of magnitude smaller than
the corresponding XAS spectra.

Figure 13: (Color online) The theoretically calculated XAS (the upper panel)
and XMCD spectra (the lower panel) at the Ca and Ru $K$ edges in Ca3Ru2O7
calculated in the GGA+SO+$U$ approach ($U_{eff}$ = 1.3 eV).  Figure 14: (Color
online) The theoretically calculated RIXS spectra at the Ca and Ru $K$ edges
(the upper panel) and partial contributions to the RIXS spectra from different
interband transitions (the lower panel) in Ca3Ru2O7 calculated in the
GGA+SO+$U$ approach ($U_{eff}$ = 1.3 eV).

Figure 14 (the upper panel) presents the theoretically calculated Ca and Ru
$K$ RIXS spectra in Ca3Ru2O7. The spectra significantly differ from each
other. The partial contributions from different interband transitions are also
presented in Fig. 14 at the Ru (the lower panel) and Ca (the middle panel) $K$
edges. The $c$ $\rightarrow$ $d$ ($t_{2g}$ $\rightarrow$ $t_{2g}$) transitions
are visible at the Ca site but they are very small at the Ru one. The
transitions to empty $t_{2g}$ UEB states from the $p$ states which are
hybridized with corresponding oxygen 2$p$ and 2$s$ states ($b$ $\rightarrow$
$d$ and $a$ $\rightarrow$ $d$, respectively) are very sharp and intense at the
Ca site and suppressed at the Ru one due to the very large width of the Ru
1$s$ core level. The largest contribution to the Ru $K$ RIXS spectrum comes
from $b$ $\rightarrow$ $g$ transitions. They are also shifted towards higher
energy in comparison with the Ca $K$ spectrum due to the corresponding shift
of empty Ru 5$p$ states in respect to Ca 4$p$ states (see Fig. 12).

Experimental measurements of the XAS, XMCD, and RIXS spectra at the Ca and Ru
$K$ edges are also highly desirable.

##  V Conclusions

Our calculations show that Ca3Ru2O7 has a novel Mott insulating state, which
is induced by the interplay of SO coupling and Coulomb repulsion.

The electronic and magnetic properties of Ca3Ru2O7 oxide have been
investigated theoretically using first-principle calculations in the frame of
the fully relativistic spin-polarized Dirac LMTO band-structure method in
order to understand the importance of Coulomb interaction and spin-orbit
coupling. The GGA approximation produces a metallic ground state in Ca3Ru2O7
in contradiction with experimental data which show that this oxide is a Mott-
like AFM insulator. To produce the correct ground state one has to take into
account strong Coulomb correlations. The GGA+SO+$U$ approach shifts the
occupied and empty $t_{2g}$ bands downward and upward, respectively, by
$U_{eff}$/2 producing a dielectric ground state in Ca3Ru2O7 with a direct
energy gap of 0.215 eV and an indirect one of 0.053 eV. The energy gap is
increased with increasing Hubbard $U$. We found that in the ground state
Ca3Ru2O7 possesses an AFM-$b$ magnetic structure with the Ru spin moments
ordered antiferromgnetically along the $b$ axis in consistence with
experimental data. The Ru spin and orbital moments are parallel in Ca3Ru2O7 in
accordance with Hund’s third rule. The spin and orbital parts of the magnetic
moments for the Ca ions in Ca3Ru2O7 are found to be very small. The magnetic
moments at the oxygen sites are also relatively small with the largest spin
moments at the O4 site and the smallest ones at the O2 site.

We have investigated theoretically the XAS, XMCD, and RIXS spectra at the Ru
and Ca $L_{3}$, $M_{3}$, and $K$ edges as well as at the O $K$ edge of the
Ca3Ru2O7 oxide. The calculated results are in good agreement with available
experimental data. We have found that the best agreement between the theory
and experiment in the RIXS spectra at the Ru $L_{3}$ edge can be achieved for
$U_{eff}$ = 1.3 eV. The calculations with larger values of $U_{eff}$ shift the
RIXS spectrum towards higher energies.

The experimentally measured RIXS spectrum of Ca3Ru2O7 at the Ru $L_{3}$ edge
possesses a sharp feature below 2 eV corresponding to transitions within Ru
$t_{2g}$ levels. The excitation located from 2 to 4 eV is due to $t_{2g}$
$\rightarrow$ $e_{g}$ transitions. The third wide structure situated at
4.5$-$11 eV appears due to transitions between Ru 4$d_{\rm{O}}$ states and
$e_{g}$ and $t_{2g}$ states. The RIXS spectra at the $L_{3}$ and $M_{3}$ edges
have to possess similar shapes because they reflect similar interband
transitions. This is true indeed in the case of the Ru $L_{3}$ and $M_{3}$
RIXS spectra. However, the shapes of the $L_{3}$ and $M_{3}$ RIXS spectra at
the Ca site quite differ from each other. Such a puzzle can be explained by
the difference in the widths of corresponding core-levels. The significantly
larger width of the 3$p_{3/2}$ core-level leads to the suppression of the low
energy fine structures in the Ca $M_{3}$ RIXS spectrum.

The RIXS spectrum at the O $K$ edge consists of three major inelastic
excitations. We found that the two low energy features $\leq$2.5 eV are due to
interband transitions between occupied and empty O${}_{t_{2g}}$ states which
appear due to the strong hybridization between oxygen 2$p$ and Ru $t_{2g}$
states in close vicinity to the Fermi level. The next major peak at 2.5$-$8 eV
reflects interband transitions between occupied O 2$p$ and empty oxygen states
which originate from the hybridization with Ru $t_{2g}$ states. Rather weak
O2p $\rightarrow$ O${}_{e_{g}}$ transitions occupy a wide energy interval
between 4 and 11 eV. Although the partial Ca 4$p$ and Ru 5$p$ densities of
states possess similar energy band structures the RIXS spectra at the Ca and
Ru $K$ edges significantly differ from each other due to the difference in the
widths of corresponding 1$s$ core levels.

## Acknowledgments

We are thankful to Dr. Alexander Yaresko from the Max Planck Institute FKF in
Stuttgart for helpful discussions.

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# A Product Model for Generalizing Poincaré-Type Kähler Metrics

Ethan Lane Addison  Department of Mathematics, University of Notre Dame, Notre
Dame, IN 46556<EMAIL_ADDRESS>
###### Abstract.

We begin by defining a type of Kähler metric near the zero section of a
trivial holomorphic open disk bundle $N$ over a compact Kähler manifold $X$ by
incorporating flows generated by holomorphic vector fields on $X$. These
metrics are then shown to deviate exponentially from Poincaré-type metrics on
$N\setminus X$ in terms of the log-polar distance from $X$ in $N$. Lastly we
see that they arise naturally when perturbing classes containing Poincaré-type
Kähler metrics of constant scalar curvature to obtain nearby cscK metrics even
when the perturbed class on $X$ does not admit a cscK metric.

##  1\. Introduction

A persistent theme in complex geometry is the search for preferred or
canonical choices of metrics on complex manifolds. In the case that the
manifold is Kähler, one can asked more pointed questions such as whether or
not a canonical metric can be found in a given Kähler class. By seeking
minimizers of the $L^{2}$-norm of the curvature tensor in such a class, E.
Calabi introduced in [3] the notion of an extremal metric, a generalization of
Kähler metrics of constant scalar curvature, respectively abbreviated to extK
and cscK metrics. Examples of the longevity of the search for canonical
metrics stretch back to the classical Uniformization Theorem, up through Yau’s
Theorem [14] which resolved a conjecture of Calabi, continuing to the modern
day with the replete understanding of the Kähler-Einstein setting elucidated
by Chen-Donaldson-Sun [4].

A fundamental property of compact Kähler manifolds shown by LeBrun-Simanca in
[8] is the openness in the Kähler cone of the set of Kähler classes which
contain extremal metrics. In fact, if the manifold has discrete automorphism
group, then this openness is upheld even in the case of cscK metrics. When one
relaxes compact Kähler to complete Kähler, properties like this LeBrun-Simanca
openness are less clear, an issue compounded by restricting attention to only
specific types of metrics, such as those defined by their asymptotic behavior
near the ends. In cases wherein the ends are composed of the components of a
complex hypersurface of some compactified manifold — as arise in studying
quasiprojective varieties or are conjectured to emerge [6] in seeking a
minimizing sequence for the Calabi functional when a minimizer does not exist
— one can define complete metrics with cusp-like singularities along the
hypersurface.

Let $(X,\sigma)$ be a compact cscK manifold with complex dimension $n$. After
removing the zero section, the open unit disk bundle $N=\mathbb{D}\times X$
admits such a complete cscK metric $\omega$ by taking the product of $\sigma$
and the standard Poincaré metric on the punctured disk. Following the
definition given by Auvray [2], other metrics which are quasi-isometric to
$\omega$ and have bounded derivatives at all orders with respect to $\omega$
are considered metrics of Poincaré-type, abbreviated here as PT. Despite
initially seeming like a loose categorization, Auvray also illustrates that
the asymptotic properties of such metrics are tightly controlled near $X$
(which we hereafter identify with the zero section of $N$) when the metric is
extremal; we note here that recent investigations of these metrics tend toward
the extK and cscK setting since the special case of Kähler-Einstein PT metrics
has been well documented by Yau [13], Kobayashi [7], and Wu [12].

The asymptotics imply that if $\tilde{\sigma}$ is a smooth real $(1,1)$-form
on all of $N$ such that $\tilde{\sigma}+d\beta$ is PT, then this PT metric is
extK (respectively cscK) only if it induces an extK (respectively cscK) metric
on $X$ in the class $[\tilde{\sigma}|_{X}]$. Though PT metrics enjoy many of
the same properties of Kähler metrics on compact manifolds, this above
restriction belies a sensitivity to the geometry of the hypersurface $X$ that
prevents perturbations of cscK PT metrics from being cscK. From LeBrun-Simanca
openness, if the class $[\sigma]$ on $X$ is perturbed to another nearby class,
then the resulting class will at least admit an extremal metric. However, if
this resulting metric is not cscK, which could be the case if $X$ admits
nontrivial holomorphic vector fields, then the perturbation of $[\sigma]$
cannot be extended to a cscK PT metric on the punctured disk bundle in a way
that yields another cscK PT metric, as Sektnan in [11] shows the extremal
vector field of an extK PT metric must be holomorphically extendable across
$X$.. One can infer then that PT metrics, while rich in examples and desirable
properties, may be insufficient on their own to establish a consummate theory
of complete extK metrics, as suggested by the non-PT complete extremal metrics
constructed by Apostolov-Auvray-Sektnan [1] in the toric setting.

By incorporating the vector fields directly into a Kähler distortion potential
on $\breve{N}=N\setminus X$, one can compensate for extra elements in the
kernel of the Lichnerowicz operator for the PT metric and define a new
complete metric which is not PT but still well-behaved and which can moreover
be chosen to be cscK when one is starting from cscK initial data. The
construction of such metrics is the basis of this article; given an
appropriate vector field $V$ on $X$, the associated potential which we refer
to as the _gnarl_ associated to $V$ and $\sigma$ will be notated $\Psi_{V}$.
The result of adding a Kähler metric to the complex Hessian of $\Psi_{V}$ will
be referred to as a _gnarled_ metric. Using the harmonic $(1,1)$-forms on $X$
as the parameter of perturbation, our main result can be stated in these
terms:

######  Theorem 1.1.

There is a neighborhood $U\subset\mathcal{H}^{1,1}(X)$ such that for every
$\eta\in U$ there exists a vector field $V_{\eta}\in H^{0}(X,T_{X})$ and a
cscK gnarled metric $\varpi_{V_{\eta}}$ associated to $V_{\eta}$ and
$\sigma+\eta$ on $\breve{N}$ such that for every nonzero section $s$ of $N$
whose fiberwise value is constant, the cohomology class of the pullback metric
is $[s^{*}\varpi_{V_{\eta}}]=[\sigma+\eta]$.

This result is in a similar vein to the compact case explored by LeBrun-
Simanca, and while one can consider it a local result, the gnarl can be
included into any manifold for which $X$ is a divisor with holomorphically
trivial normal bundle with the use of a suitable cutoff. The following two
sections serve to explicate the construction of the gnarled metric in this
simple product case and then to provide a proof of the main result by the
introduction of a modified scalar curvature operator which we show can be
inverted to solve the necessary equation guiding constant curvature for our
gnarled metric.

##  2\. Construction of the Metric

In order to construct the gnarl, we take $z$ as the standard coordinate on
$\mathbb{D}$ and define a “logarithmic-polar” coordinate system on
$\mathbb{D}^{*}=\mathbb{D}\setminus\\{0\\}$ by using the standard angular
coordinate $\theta$ along with a radial function $\tau=\ln(-\ln\lvert
z\rvert)$ which goes to $\infty$ near $X$. The cusp metric we use for
$\mathbb{D}^{*}$ has $\tau$ as a Kähler potential and will be written
$dd^{c}\tau$, where we define the operator $d^{c}=JdJ^{-1}$. Here the complex
structure $J$ is a real endomorphism acting on covariant tensors by
precomposition, thus on $k$-forms, $J^{-1}=(-1)^{k}J$. Under the decomposition
$d=\partial+\bar{\partial}$, we have that $d^{c}=i(\partial-\bar{\partial})$,
and therefore the complex Hessian operator can be written as
$dd^{c}=2i\bar{\partial}\partial$. From the definition of the new coordinates,
we can see $d^{c}\tau=e^{-\tau}d\theta$ as well as the “cusp” nature of the
volume form $e^{-\tau}d\theta\wedge d\tau$ upon approach to the zero section
$X$.

Since any obstruction to LeBrun-Simanca openness for PT metrics in this
context will arise from holomorphic vector fields on $X$, we now note that by
Matsushima [10] and Lichnerowicz [9] the following fact that $X$ admitting a
cscK metric implies that the space of holomorphic vector fields on $X$ is a
complex Lie algebra which splits into

$H^{0}(X,T_{X})=\mathfrak{a}\oplus\mathfrak{h}\oplus J\mathfrak{h}.$

Here $\mathfrak{a}$ represent fields parallel with respect to the Levi-Civita
connection while $J\mathfrak{h}$ is a subspace of real vector fields which are
Hamiltonian with respect to the symplectic form $\sigma$. From the
relationship between the symplectic form and the metric tensor on $X$, each
$V\in\mathfrak{h}$ is therefore a gradient vector field, i.e.
$V=\text{grad}f_{V}$ for a real function $f_{V}$, which has an associated
time-$t$ flow notated by $F_{tV}$. These are the only vector fields we need to
consider since the kernel of the Lichnerowicz operator contains only
holomorphy potentials like $f_{V}$, and the nontrivial parallel fields cannot
be gradients because they do not have zeroes on $X$.

Starting from such a $V$, we can define a function
$\psi_{tV}\in\mathcal{C}^{\infty}(\mathbb{R}\times X)$ as the solution to the
differential equation:

(2.1) $\frac{d}{dt}\psi_{tV}=F^{*}_{tV}f_{V},\text{ and }\psi_{0}=0.$

Since the flow is a path in Aut${}_{0}(X)$, $F^{*}_{tV}\sigma$ remains in the
same de Rham and Dolbeault class of $\sigma$, and so by the $dd^{c}$-lemma, we
see that the $\psi_{tV}$ satisfies

(2.2) $dd^{c}\psi_{tV}=F^{*}_{tV}\sigma-\sigma.$

This function serves as the starting point of the gnarl. Recalling that we
consider $N$ as a bundle over $X$, we set $\nu:N\twoheadrightarrow X$ as the
projection map. Then $\tau\times\nu$ is a map from $\breve{N}$ to
$\mathbb{R}\times X$, so $\psi_{tV}$ can be pulled back to the punctured disk
bundle.

######  Definition 2.1.

Given $V\in\mathfrak{h}$, the gnarl associated to $V$ is the function given by
$\Psi_{V}=(\tau\times\nu)^{*}\psi_{tV}$. For a given Kähler metric $\varpi$ on
$\breve{N}$, we set $\varpi_{V}=\varpi+dd^{c}\Psi_{V}$.

Recalling that the product metric on $\breve{N}$ is given by
$\omega=dd^{c}\tau+\nu^{*}\sigma$, where we now suppress the pullback by $\nu$
unless necessary to include it, in line with Definition 2.1 we set
$\omega_{V}=\omega+dd^{c}\Psi_{V}$. This is a well-defined real $(1,1)$-form
on $\breve{N}$, but the first task will be to show that it can actually be
considered a metric, i.e. that it is a positive form. This may in fact not be
true for very large choices of vector field $V$, but the following argument
will make clear that so long as $V$ is small enough with respect to the
starting data, then $\omega_{V}$ is a genuine Kähler metric. As we are
interested eventually in a perturbation problem, only a small neighborhood of
$0$ in $\mathfrak{h}$ will be of great concern, so this will be sufficient for
the proof of Theorem 1.1.

######  Proposition 2.2.

There exists a neighborhood $\mathcal{W}\subset\mathfrak{h}$ about $0$ such
that for any $V\in\mathcal{W}$, $\omega_{V}$ is a symplectic form on
$\breve{N}$.

###### Proof.

Working on the punctured disk bundle is less convenient than that of the
corresponding upper-half-plane bundle, so we introduce the covering map

(2.3) $\upsilon:\mathbb{H}\times X\twoheadrightarrow\breve{N}\text{ via
}(\zeta,x)\mapsto(e^{i\zeta},x).$

This map pulls $\tau$ up to a new variable $w=\ln\mathfrak{Im}\zeta$, and then
the entire gnarled form $\omega_{V}$ for some unspecified $V$ can be written
out as

$\upsilon^{*}\omega_{V}=\sigma+dd^{c}\left(w+\psi_{wV}\right).$

What is more, since both bundles are trivial, the flow $F_{tV}$ can be
extended to $\breve{N}$ trivially and can be lifted up to $\mathbb{H}\times X$
as a $t$-dependent family of diffeomorphisms $\tilde{F}_{tV}$ according to

$\tilde{F}_{tV}(\zeta,x)=(e^{-t}\zeta,F_{tV}(x)).$

Note that for any given $t$, $\tilde{F}_{tV}$ is actually a biholomorphism.
With the goal of showing these form a family of isometries for $\omega_{V}$,
we must illustrate a handy property of the gnarl itself. Restricting our
attention back to $X$ again for a moment, observe that for a choice of
specific time $T$,

$F^{*}_{TV}dd^{c}\psi_{tV}=F^{*}_{(T+t)V}\sigma-F^{*}_{TV}\sigma=dd^{c}\left(\psi_{(T+t)V}-\psi_{TV}\right).$

Therefore there is a function $\kappa_{T}(t)$ that allows us to equate

$F^{*}_{TV}\psi_{tV}=\psi_{(T+t)V}-\psi_{TV}+\kappa_{T}(t).$

We would like to show this additional term is in fact 0. Initially we see that
for $t=0$, $\kappa_{T}(0)=0$. Additionally, taking the first derivative yields

$\kappa^{\prime}_{T}(t)=\frac{d}{dt}\psi_{(T+t)V}-F^{*}_{TV}\frac{d}{dt}\psi_{tV}=F^{*}_{(T+t)V}f_{V}-F^{*}_{TV}F^{*}_{tV}f_{V}=0.$

Hence $\kappa_{T}(t)=0$ for all $T$ and $t$, and we have a clear understanding
of how the flow along $V$ affects the associated gnarl, i.e. according to

(2.4) $F^{*}_{TV}\psi_{tV}=\psi_{(T+t)V}-\psi_{TV}.$

Returning to the cover, this formula arises when flowing the metric
$\upsilon^{*}\omega_{V}$:

$\tilde{F}^{*}_{TV}(\upsilon^{*}\omega_{V})=F^{*}_{TV}\sigma+dd^{c}w+dd^{c}\left(\psi_{(w+T-T)V}\right)-dd^{c}\left(\psi_{TV}\right).$

Note that since $\psi_{TV}$ is independent of the fiber variable, on the cover
we explicitly have

$\upsilon^{*}dd^{c}\psi_{TV}=\tilde{F}^{*}_{TV}(\upsilon^{*}\sigma)-\upsilon^{*}\sigma$

Therefore we arrive at some cancellations from equation 2.2:

$\tilde{F}^{*}_{TV}\upsilon^{*}(\omega+dd^{c}\psi_{\tau
V})=\sigma+dd^{c}(w+\psi_{wV})=\upsilon^{*}(\omega+dd^{c}\psi_{\tau V}).$

On any given strip, say $\mathcal{Q}=\\{1\leq\mathfrak{Im}\zeta\leq 2\\}\times
X$, we can certainly choose $V$ to be small enough so that the above form is
positive since the gnarl is independent of the angular coordinate $\theta$ and
thus the whole form only varies on a compact set. However, for any
$p\in\mathbb{H}\times X$, there exists a $T$ such that
$\tilde{F}_{TV}(p)\in\mathcal{Q}.$ Given that we now have chosen $V$ such that
$\upsilon^{*}(\omega+dd^{c}\psi_{\tau V})$ is positive in $\mathcal{Q}$ and
the form has been shown to be preserved by the map $\tilde{F}_{TV}$, it
follows $\upsilon^{*}(\omega+dd^{c}\psi_{\tau V})>0$ on $\mathbb{H}\times X$.
Since it is a local diffeomorphism, $\upsilon$ cannot pull a nonpositive form
back to a positive form, and we see that $\omega_{V}$ defines a Kähler metric
on $\breve{N}$. As this procedure can be done for any choice of direction
$V/\lvert\lvert V\rvert\rvert_{\mathcal{C}^{0}(X,\sigma)}$, we arrive at an
open neighborhood $\mathcal{W}$ of 0, as desired. ∎

A remark we make here that will be vital to the proof that follows arises from
noticing that we can study the scalar curvature of $\upsilon^{*}\omega_{V}$ by
looking at a single slice, say $\\{2i\\}\times X$. Here we do not mean the
scalar curvature of the restricted metric on the slice, but rather the
restriction of the scalar curvature as a function; the biholomorphisms
$\tilde{F}_{tV}$ can move this function to every other slice above this one as
$t$ varies, and in the case that this function happens to be a constant, the
value of $t$ is irrelevant. Certainly this remains true on $\breve{N}$ since
$\upsilon$ is a local diffeomorphism. Although we initially insisted $\sigma$
be cscK, this was not used in the proof of Proposition 2.2. The result would
remain true for any starting $\sigma+dd^{c}\varphi$ on $X$, and therefore the
procedure to prove Theorem 1.1 will be to search for a function $\varphi$ on
$X$ by looking at the restriction of the scalar curvature of
$\omega+dd^{c}\nu^{*}\varphi$ to $\\{e^{-2}\\}\times
X\hookrightarrow\breve{N}$, essentially converting the problem to one in which
we must only concern ourselves with functions on $X$.

More than simply showing $\omega_{V}$ is a metric, the covering space argument
can be used to get $\mathcal{C}^{k}$ bounds on $\omega_{V}$ which are derived
from estimating the size of the gnarl with respect to the PT metric $\omega$.

######  Proposition 2.3.

For every $k\geq 0$, there exists an $\varepsilon_{k}>0$ such that

$\lvert\lvert\Psi_{V}\rvert\rvert_{\mathcal{C}^{k}(\omega)}\lesssim
e^{\varepsilon_{k}\tau}.$

######  Corollary 2.4.

For every $k\geq 0$, there exists an $\tilde{\varepsilon}_{k}>0$ such that

$\lvert\lvert\omega_{V}\rvert\rvert_{\mathcal{C}^{k}(\omega)}\lesssim
e^{\tilde{\varepsilon}_{k}\tau}.$

###### Proof.

By using the above proposition,

$\lvert\lvert\nabla^{k}\omega_{V}\rvert\rvert_{\omega}\lesssim\lvert\lvert\nabla^{k}\omega\rvert\rvert_{\omega}+\lvert\lvert\nabla^{k+2}\Psi_{V}\rvert\rvert_{\omega}\lesssim
e^{\varepsilon_{k+2}\tau},$

since the first term is zero. Hence we merely set
$\tilde{\varepsilon}_{k}=\varepsilon_{k+2}$. ∎

The proposition itself requires a similar but more general and useful result
about flows of vector fields on compact manifolds.

######  Lemma 2.5.

Let $\beta$ be a smooth tensor on $(X,\sigma)$ and let $W$ be a smooth vector
field with associated time-$t$ flow $F_{tW}$. Then for each $k\geq 0$, there
exists $c_{k}>0$ depending on $W$ and the valence of $\beta$ such that

$\lvert\lvert F_{tW}^{*}\beta\rvert\rvert_{\mathcal{C}^{k}(X,\sigma)}\lesssim
e^{c_{k}t}\lvert\lvert\beta\rvert\rvert_{\mathcal{C}^{k}(X,\sigma)}.$

###### Proof.

We induct on $k$. Starting with $k=0$, we note that

$\lvert\lvert
F^{*}_{tW}\beta\rvert\rvert_{\sigma}=\lvert\lvert\beta\rvert\rvert_{F^{*}_{-tW}\sigma}\leq\lvert\lvert
F^{*}_{-tW}\sigma\rvert\rvert_{\sigma}\lvert\lvert\beta\rvert\rvert_{\sigma}\leq
e^{c_{0}t}\lvert\lvert\beta\rvert\rvert_{\sigma}.$

Here we find $c_{0}$ through the following ordinary differential inequality:

(2.5) $\partial_{t}\lvert\lvert
F^{*}_{tW}\sigma\rvert\rvert_{\sigma}\leq\lvert\lvert
F^{*}_{tW}\pounds_{W}\sigma\rvert\rvert_{\sigma}=\lvert\lvert
F^{*}_{tW}d(W\mathbin{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\lnot$}}}\sigma)\rvert\rvert_{\sigma}\leq
c_{0}\lvert\lvert F^{*}_{tW}\sigma\rvert\rvert_{\sigma},$

since $d(W\mathbin{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\lnot$}}}\sigma)$ is
just a differential form on $X$, so there exists a constant $c_{0}$ depending
on $W$ such that

$\lvert\lvert c_{0}\sigma\rvert\rvert_{L^{\infty}}\geq\lvert\lvert
d(W\mathbin{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\lnot$}}}\sigma)\rvert\rvert_{L^{\infty}}.$

For the case $k=1$, we define a 1-form $S_{t}$ with values in the endomorphism
bundle of whatever bundle contains $\beta$ according to
$S_{t}=\nabla-\nabla_{t}$ where $\nabla_{t}=F^{*}_{tW}(\nabla)$ is the Levi-
Civita connection of $F^{*}_{tW}\sigma.$ This will allow us to pass the flow
by the covariant derivative with a cost as seen by

(2.6)
$\nabla(F^{*}_{tW}\beta)=\nabla_{t}(F^{*}_{tW}\beta)+S_{t}F^{*}_{tW}\beta=F^{*}_{tW}(\nabla\beta)+S_{t}F^{*}_{tW}\beta.$

Taking the norm of the first covariant derivative then yields

$\lvert\lvert\nabla(F^{*}_{tW}\beta)\rvert\rvert\leq\lvert\lvert
F^{*}_{tW}(\nabla\beta)\rvert\rvert+\lvert\lvert
S_{t}F^{*}_{tW}\beta\rvert\rvert\leq
e^{\tilde{c}_{0}t}\lvert\lvert\nabla\beta\rvert\rvert+e^{c_{0}t}\lvert\lvert
S_{t}\rvert\rvert~{}\lvert\lvert\beta\rvert\rvert.$

Here we had to adjust the rate to a new $\tilde{c}_{0}$ (found the same way as
in the $k=0$ case) in the first term because the valence increased by 1 with
the derivative. Now the issue has been reduced to understanding the
derivatives of $S_{t}$.

Set $g$ and $g_{t}$ to be the Riemannian metrics associated to $\sigma$ and
$F^{*}_{tW}\sigma$. In general depending on the valence of $\beta$, $S_{t}$
might be several tensor products of the difference between the connections
acting on vector fields. Yet, this difference will control the product, so we
assume for a moment that $S_{t}$ is a section of $\Omega^{1}(TX\otimes
T^{*}X)$. Then we find the following ordinary differential inequality:

$\partial_{t}\lvert\lvert
S_{t}\rvert\rvert\leq\lvert\lvert\partial_{t}S_{t}\rvert\rvert\leq\lvert\lvert\dot{\nabla}_{t}\rvert\rvert=$
$\frac{1}{2}\lvert\lvert
g_{t}^{j\ell}(\nabla_{t,r}\pounds_{W}g_{t,u\ell}+\nabla_{t,u}\pounds_{W}g_{t,r\ell}-\nabla_{t,\ell}\pounds_{W}g_{t,ur})F^{*}_{tW}(\partial_{j}\otimes
dx^{u}\otimes dx^{r})\rvert\rvert.$

Since the flow commutes with the Lie derivative, we can simplify this
expression to

$\partial_{t}\lvert\lvert
S_{t}\rvert\rvert\leq\frac{1}{2}\left|\left|F^{*}_{tW}\left(g^{j\ell}(\nabla_{r}\pounds_{W}g_{u\ell}+\nabla_{u}\pounds_{W}g_{r\ell}-\nabla_{\ell}\pounds_{W}g_{ur})(\partial_{j}\otimes
dx^{u}\otimes dx^{r})\right)\right|\right|.$

From the identity
$\pounds_{W}g_{u\ell}=\nabla_{u}(W^{\flat}_{\ell})-\nabla_{\ell}(W^{\flat}_{u})$
we infer that

$\nabla_{r}\pounds_{W}g_{u\ell}+\nabla_{u}\pounds_{W}g_{r\ell}-\nabla_{\ell}\pounds_{W}g_{ur}=2\nabla^{2}W^{\flat}+2\text{Rm}\cdot
W^{\flat}.$

Sliding the flow out at the expense of another exponential term with rate
$\tilde{c}_{0}$ depending on $W$ (arising from measuring the flowed metric
tensor for the tangent bundle with respect to the unflowed metric as in
equation 2.5), we find

$\partial_{t}\lvert\lvert S_{t}\rvert\rvert\lesssim
e^{\tilde{c}_{0}t}\left(\lvert\lvert\nabla^{2}W^{\flat}\rvert\rvert+\lvert\lvert\text{Rm}\rvert\rvert~{}\lvert\lvert
W\rvert\rvert\right)\lesssim e^{\tilde{c}_{0}t}\lvert\lvert
W\rvert\rvert_{\mathcal{C}^{2}(X,\sigma)}.$

Recalling that $\tilde{c}_{0}$ depends on $W$, by integrating the ODI, we
arrive at $\lvert\lvert S_{t}\rvert\rvert\lesssim e^{\tilde{c}_{0}t}$ for the
case that $S_{t}$ operates on vector fields. However, in general, we may get a
multiple $m\tilde{c}_{0}$ in the exponent to account for the valence of
$\beta$. Note the above procedure unfolds similarly for $\nabla^{k}S_{t}$,
which then depends on $\lvert\lvert
W\rvert\rvert_{\mathcal{C}^{k+2}(X,\sigma)}.$ Taking the largest of the
exponents present gives the appropriate $c_{1}$ for the case at hand.

Now assume we have subexponential bounds with rate $c_{k}$ for
$\lvert\lvert\nabla^{k}F^{*}_{tW}\beta\rvert\rvert$, so we try to bound the
norm of the next derivative:

$\lvert\lvert\nabla^{k+1}F^{*}_{tW}\beta\rvert\rvert\leq\lvert\lvert\nabla^{k}\nabla_{t}F^{*}_{tW}\beta\rvert\rvert+\lvert\lvert\nabla^{k}S_{t}F^{*}_{tW}\beta\rvert\rvert.$

Since we have $\nabla_{t}F^{*}_{tW}\beta=F^{*}_{tW}(\nabla\beta)$, we can note
$\nabla\beta$ satisfies the hypothesis of having $k$-th derivative
subexponential bounds for some rate $\breve{c}_{k}$. Expanding out the other
term with the Leibniz Rule gives us

$\lvert\lvert\nabla^{k+1}F^{*}_{tW}\beta\rvert\rvert\lesssim
e^{\breve{c}_{k}t}\lvert\lvert\nabla^{k+1}\beta\rvert\rvert+\sum_{j=0}^{k}\binom{k}{j}\lvert\lvert\nabla^{k-j}S_{t}\rvert\rvert\lvert\lvert\nabla^{j}F^{*}_{tW}\beta\rvert\rvert.$

All of these terms have a subexponential growth rate as we have seen; merely
take the largest among the resulting addends to define $c_{k+1}$ to complete
the induction. ∎

###### Proof of Proposition 2.3.

We assume we are working about a level set of $\tau$. For the zeroth order
estimate, we note that

$|\Psi_{V}|\leq\left|\int_{0}^{\tau}F_{t}^{*}f_{V}~{}dt\right|\leq\tau\lvert\lvert
f_{V}\rvert\rvert_{L^{\infty}},$

which is certainly subexponential in $\tau$. For the other values of $k$, we
proceed as in the lemma, showing the first derivative explicitly for clarity.
First, we define a map from an bounded open subset of $\mathbb{H}\times X$ to
$\breve{N}$ as follows.

$\mathcal{A}=\\{(\zeta,x)|-1<\mathfrak{Re}\zeta<1,1<\mathfrak{Im}\zeta<3\\},$
$\upsilon_{T}:\mathcal{A}\rightarrow\breve{N}\text{ via
}(\zeta,x)\mapsto(e^{ie^{T}\zeta},x).$

Note that for a given $T$, the above map is certainly not surjective, but for
every $q\in\breve{N}$, there is always a value of $T$ for which $q$ is in the
image of $\upsilon_{T}$. Let us now consider $q$ in the level set of interest,
setting $T=\lfloor\tau\rfloor$ and taking $\hat{q}\in\upsilon^{-1}_{T}(q)$.
The utility of this map lies in the fact that
$\upsilon_{T}^{*}\omega=\upsilon^{*}\omega$ for the covering map defined
earlier in equation 2.3. In other words, this map still pulls $\omega$ back to
the standard Poincaré metric on $\mathcal{A}$, so we have the equality

$\lvert\lvert\Psi_{V}\rvert\rvert_{\mathcal{C}^{k}(\omega)}(q)=\lvert\lvert\upsilon^{*}_{T}\Psi_{V}\rvert\rvert_{\mathcal{C}^{k}(\upsilon^{*}\omega)}(\hat{q}).$

However, from one of the fundamental properties of the gnarl shown in 2.1,

$\upsilon^{*}_{T}\Psi_{V}=\psi_{(T+\ln\mathfrak{Im}\zeta)V}=\psi_{TV}+F^{*}_{TV}(\psi_{(\ln\mathfrak{Im}\zeta)V}).$

Here, the flow is just the trivial extension from $X$ to $\mathcal{A}$. From
this relation, we see that for the first derivative, which is just the
exterior derivative, we get

$\lvert\lvert d\Psi_{V}\rvert\rvert_{\omega}(q)=\lvert\lvert
d\upsilon^{*}_{T}\Psi_{V}\rvert\rvert_{\upsilon^{*}\omega}(\hat{q})\leq\lvert\lvert
d\psi_{TV}\rvert\rvert_{\upsilon^{*}\omega}(\hat{q})+\lvert\lvert
dF^{*}_{TV}(\psi_{(\ln\mathfrak{Im}\zeta)V})\rvert\rvert_{\upsilon^{*}\omega}(\hat{q}).$

For the first term, notice

$\partial_{T}\lvert\lvert d\psi_{TV}\rvert\rvert\leq\lvert\lvert
d(\partial_{T}\psi_{TV})\rvert\rvert=\lvert\lvert
dF^{*}_{TV}f_{V}\rvert\rvert\lesssim e^{c_{1}T}\lvert\lvert
f_{V}\rvert\rvert.$

The rate $c_{1}$ arises as in Lemma 2.5. Now we can integrate the ODI and find
$\lvert\lvert d\psi_{TV}\rvert\rvert$ is subexpoential. For the term with the
flow, observe that we can pull the flow out of the norm at the expense of an
exponential factor as in Lemma 2.5 which is then multiplied by the quantity

$\lvert\lvert
d\psi_{(\ln\mathfrak{Im}\zeta)V})\rvert\rvert\leq\sup_{\mathcal{A}}\lvert\lvert
d\psi_{(\ln\mathfrak{Im}\zeta)V})\rvert\rvert,$

which is independent of $T$, hence why we pulled back to a set whose closure
is compact. Together, these give us subexponential bounds for $k=1$.

For higher derivatives, setting $\tilde{\nabla}$ as the connection for
$\upsilon^{*}\omega$, we still have an inequality of the form

$\lvert\lvert\nabla^{k}\Psi_{V}\rvert\rvert(q)\leq\lvert\lvert\tilde{\nabla}^{k}\psi_{TV}\rvert\rvert(\hat{q})+\lvert\lvert\tilde{\nabla}^{k}F^{*}_{TV}(\psi_{(\ln\mathfrak{Im}\zeta)V})\rvert\rvert(\hat{q}).$

The only additional tool needed here is the endomorphism-valued 1-form
$\tilde{S}_{T}=\tilde{\nabla}-F^{*}_{TV}(\tilde{\nabla})$ which allows
commutation of $F^{*}_{TV}$ and $\tilde{\nabla}$. What is left in the end is a
polynomial expression in the $\tilde{\nabla}$ derivatives of
$\psi_{TV},\psi_{(\ln\mathfrak{Im}\zeta)V},$ and $\tilde{S}_{T}$, the last of
which was shown to be subexponential in $T\approx\tau$ in Lemma 2.5. Taking
the largest exponent among the resulting monomials gives the appropriate
$\varepsilon_{k}$ to complete the proof of the statement.

∎

Lastly in this section, we make the remark that the metric $\omega_{V}$ is
complete on $\breve{N}$. Again appealing to the covering space, we show that
$\upsilon^{*}\omega_{V}$ is complete by illustrating that the distance to the
boundary from any point is infinite. Starting with a point
$p=(\zeta_{0},x)\in\mathbb{H}\times X$, we can WLOG consider $\zeta_{0}$ to be
purely imaginary of sub-unit length and assume to the contrary of completeness
that there exists a geodesic
$\gamma:[0,a]\rightarrow\overline{\mathbb{H}\times X}$ which begins at $p$ and
terminates at the boundary. Then for $T=-\ln\frac{\zeta_{0}}{i}$, which is a
positive real number, we have the inequalities

$\text{length}(\gamma)\geq\text{dist}(p,b(\mathbb{H}\times
X))\geq\sum_{j<0}\text{dist}(\\{e^{jT}i\\}\times\mathbb{R}\times
X,\\{e^{(j-1)T}i\\}\times\mathbb{R}\times X),$

since indeed the geodesic must pass through each slice parameterized by the
values of $\mathfrak{Im}\zeta$ on the way to the boundary.

The quantity $\text{dist}(\\{e^{jT}i\\}\times\mathbb{R}\times
X,\\{e^{(j-1)T}i\\}\times\mathbb{R}\times X)$ is essentially the distance
between the boundary components of

$\\{(\zeta,x)|e^{(j-1)T}<\mathfrak{Im}\zeta<e^{jT}\\}.$

But as we have shown above, the map $\tilde{F}_{(j+1)TV}$ is an isometry for
$\omega_{V}$ that maps this set to
$\\{(\zeta,x)|e^{-2T}<\mathfrak{Im}\zeta<e^{-T}\\}$, which incidentally
contains $p$. This implies the distance between the slices is independent of
$T$, and since it is positive, each term in the earlier inequalities is
infinite, contradicting the assertion that $\gamma$ was of finite length.
Hence $\upsilon^{*}\omega_{V}$ and likewise $\omega_{V}$ are complete.

##  3\. Proof of Main Result

In order to make clear the hereditary property of extK and cscK PT metrics
proven by Auvray in [2], we want to explicitly state the corresponding
definition for a class of PT metrics in the global scenario wherein $X$ is a
hypersurface with trivial normal bundle in a compact Kähler manifold
$(M,\varpi_{0})$. The definitions regarding PT metrics are easily extended to
$M\setminus X$ by using a bump function $\chi$ around $X$ with the function
$\chi\tau$ replacing $\tau$ in all of the statements so that
$\varpi_{0}+dd^{c}(\chi\tau)$ becomes the baseline PT metric against which all
others are measured.

######  Definition 3.1.

For a smooth Kähler metric $\varpi_{0}$ on $M$, a PT metric $\varpi$ is said
to be in the class $[\varpi_{0}]$ if there exists a function $y=O(\chi\tau)$
on $M\setminus X$ such that $\varpi=\varpi_{0}+dd^{c}y$ and $\lvert\lvert
y\rvert\rvert_{\mathcal{C}^{k}(M\setminus X,\varpi_{0}+dd^{c}{(\chi\tau)})}$
is bounded for all $k>0$.

A natural desire for such a family of metrics would be that if the class
$[\varpi_{0}]$ contains an extremal PT metric, then for any closed
$(1,1)$-form $\eta$ small enough, $[\varpi_{0}+\eta]$ would also contain an
extremal PT metric just as in the case for compact Kähler manifolds. However,
an extK but not cscK PT metric would need to have an associated nontrivial
holomorphic vector field whose extension to $M$ fixes $X$, which does not
always exist, even if $X$ admits holomorphic vector fields of its own. In such
a case, one would have only a cscK PT metric, yet the perturbed class will
restrict to the Kähler class $[(\varpi_{0}+\eta)|_{X}]$ on $X$ which could
admit extK metrics but no cscK metrics. The following theorem of Auvray,
adapted for our present context, implies that in the above situation, no extK
PT metrics can exist in $[\varpi_{0}+\eta]$.

######  Theorem 3.2.

([2], Theorem 4) Let $\varpi_{0}$ be a Kähler metric on $M$ and assume there
exists an extK (resp. cscK) PT metric of class $[\varpi_{0}]$ on $M\setminus
X$. Then there exists an extK (resp. cscK) metric in the class
$[\varpi_{0}|_{X}]$ on $X$.

This result is proven by showing that locally any extremal PT metric splits
near the divisor as

$a~{}dd^{c}\tau+\nu^{*}\sigma+O(|\ln(|z|)|^{-r})$

for some $a,r>0$, where $\sigma$ is the resulting extK or cscK metric on $X$.
Therefore the obstruction to LeBrun-Simanca openness is local, and thus we
introduce the gnarl as a local construction in order to overcome this issue.

Our approach is to construct an appropriate scalar curvature operator to and
from functions on $X$ in accord with the remark at the end of Section 2. Then
through linearizing and applying a Banach space implicit function theorem
argument, we will arrive at a choice of gnarled PT metric in any nearby Kähler
class which is cscK near the divisor and hence has different asymptotic
properties than those delineated by Auvray for extK PT metrics. To this end,
we need the following lemma.

######  Lemma 3.3.

The linearization of the map $V\mapsto\text{Scal}(\omega_{V})$ at $\omega_{V}$
is given by

$W\mapsto 2\mathcal{D}_{V}^{\star_{V}}\mathcal{D}_{V}(\tau f_{W})+2\langle
d\text{Scal}(\omega_{V}),d(\tau f_{W})\rangle_{V},$

where $\mathcal{D}_{V}$ is the operator
$f\mapsto\bar{\partial}(\text{grad}_{V}^{1,0}f)$ and additionally $\star_{V}$
and $\langle,\rangle_{V}$ represent the duality and inner product provided by
$\omega_{V}$.

###### Proof.

We begin with the fact, as shown in [8], that the Fréchet derivative — defined
by convergence on compact subsets — of the map $\varphi\mapsto$
Scal$(\omega+dd^{c}\varphi)$ from distortion potentials to scalar curvature is
given by

$\phi\mapsto 2\mathcal{D}^{\star}\mathcal{D}\phi+2\langle
d\text{Scal}(\omega),d\phi\rangle,$

namely twice the Lichnerowicz operator for $\omega$ plus a correction
involving the derivative of the scalar curvature which of course vanishes when
based at a cscK metric. The new information here then arises from the
linearization of $V\mapsto\Psi_{V}$ at a gnarled metric, which we can derive
by differentiating the $X$-complex Hessian of the gnarl to find

$\displaystyle\left.\frac{d}{du}\right|_{u=0}d_{X}d^{c}_{X}\Psi_{V+uW}$
$\displaystyle=\left.\frac{d}{du}\right|_{u=0}d_{X}d^{c}_{X}\psi_{\tau(V+uW)}=\left.\frac{d}{du}\right|_{u=0}F^{*}_{\tau(V+uW)}\sigma-\sigma$
$\displaystyle=\left.\frac{d}{du}\right|_{u=0}F^{*}_{(\tau uF_{-\tau
V*}W)}F^{*}_{\tau V}\sigma=\pounds_{(\tau F_{-\tau V*}W)}F^{*}_{\tau V}\sigma$
$\displaystyle=d_{X}\left((\tau F_{-\tau
V*}W)\mathbin{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\lnot$}}}F^{*}_{\tau
V}\sigma\right)=d_{X}(\tau
W\mathbin{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\lnot$}}}\sigma)=d_{X}d^{c}_{X}(\tau
f_{W}).$

The complex Hessians can be removed as in equation 2.4, and then this can be
combined with the formula for the derivative with respect to general
distortion potentials to arrive at the desired result, proving the lemma. ∎

###### Proof of Theorem 1.1.

We will only perturb the metric $\omega$ by perturbing $\sigma$ and pulling
back, so in order to parameterize the directions in $H^{1,1}(X)$ we shall use
the $\sigma$-harmonic forms notated $\mathcal{H}^{1,1}(X)$. Moreover, because
taking the scalar curvature will require four derivatives of the distortion
potential, we will make of use the Hölder space $\mathcal{C}^{4,\alpha}(X)$ so
that scalar curvatures land in $\mathcal{C}^{0,\alpha}(X)$ after restriction
to the specific slice $\\{e^{-2}\\}\times X\cong X$.

Additionally, since we can define a gnarl for any holomorphic vector field in
$\mathfrak{h}$ and Kähler metric on $X$, given small enough $\eta$ and
$\varphi$, we will now set $\Psi_{V}^{\eta,\varphi}$ as the gnarl associated
to $V$ and $\sigma+\eta+dd^{c}\varphi$. This permits the definition of a new
gnarled metric:

$\omega_{V,\eta,\varphi}=\omega+\nu^{*}\eta+dd^{c}(\nu^{*}\varphi+\Psi_{V}^{\eta,\varphi}).$

The gnarling is so named because it simulates the effect of flowing in
directions tangent to $X$ more and more as one approaches the zero section,
wrapping around like a gnarl in a tree trunk, while still preserving the
holomorphic structure. The gnarl is required here instead of a literal flow
since although one can extend $V$ trivially to $N$, the vector field $\tau V$
on $\breve{N}$ along which one would flow is not holomorphic, so flowing the
original symplectic form would yield something no longer Kähler for the
original complex structure. However, the flow of $\tau V$ is still perfectly
well-defined here and will allow us to somewhat “ungnarl” the gnarl by
applying the inverse flow to the gnarled metric.

Define $\mathcal{F}$ to be the time-one flow of the vector field $-\tau V$ on
$\breve{N}$. Then for $J$ the almost complex structure on $\breve{N}$, we
notice that

$G(V,\eta,\varphi)=-\mathcal{F}^{*}J\mathbin{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\lnot$}}}\mathcal{F}^{*}(\omega_{V,\eta,\varphi})$

defines a Riemannian metric on the smooth manifold $\breve{N}$ through the
usual procedure of twisting the symplectic form by the almost complex
structure to get the symmetric tensor. Such a metric certainly has a scalar
curvature, and this is how we shall define our operator:

$\mathcal{S}:\mathfrak{h}\oplus\mathcal{H}^{1,1}(X)\oplus\mathcal{C}^{4,\alpha}(X)\rightarrow\mathcal{C}^{0,\alpha}(X)\text{
via
}(V,\eta,\varphi)\mapsto\text{Scal}(G(V,\eta,\varphi))|_{\\{e^{-2}\\}\times
X}$

The goal is to prove that there are $V_{\eta}$ and $\varphi_{\eta}$ such that
for any small $\eta$, the metric $G(V_{\eta},\eta,\varphi_{\eta})$ has
constant scalar curvature. This metric will indeed be a Kähler metric, but for
the new complex manifold $(\breve{N},\mathcal{F}^{*}J)$ and not our initial
$(\breve{N},J)$. The point is that once a constant scalar curvature metric,
the Kähler manifold
$(\breve{N},\mathcal{F}^{-1*}G(V_{\eta},\varphi_{\eta},\eta),J,\omega_{V_{\eta},\eta,\varphi_{\eta}})$
will also be cscK since the constancy of the scalar curvature will not be
altered under a diffeomorphism.

First we must establish smooth dependence on the variables, and then we will
invert the operator using its linearization and the implicit function theorem.
Moving outside in, restriction to a smooth submanifold is a smooth map, and
taking the scalar curvature of a Riemannian metric is analytic in the metric’s
components. Thus we now must only establish smoothness for the tensors
$\mathcal{F}^{*}J$ and $\mathcal{F}^{*}\omega_{V,\eta,\varphi}$.

The pullback of the complex structure is more straightforward to compute,
though it has both covariant and contravariant properties. The holomorphic
tangent bundle splits into

$T_{\breve{N}}=\mathcal{O}_{\breve{N}}\oplus\nu^{*}T_{X}$

where $\mathcal{O}_{\breve{N}}$ is just the trivial line bundle coming from
the tangent bundle of $\mathbb{D}^{*}$ pulled back up the product. Also note
that since $\mathcal{F}(z,x)=(z,F_{-\tau V}(x))$, we can split
$\mathcal{F}^{*}J$ up nicely along with $T_{\breve{N}}$. That is, for any
$(1,0)$ vector field $Y$ on $\breve{N}$,

$\mathcal{F}_{*}Y=\begin{bmatrix}1&0\cdots 0\\\ \partial_{z}F_{-\tau
V}&\partial_{X}F_{-\tau V}\end{bmatrix}Y,\text{ and
}\mathcal{F}_{*}\bar{Y}=\begin{bmatrix}1&0\cdots 0\\\
\partial_{\bar{z}}F_{-\tau V}&\bar{\partial}_{X}F_{-\tau
V}\end{bmatrix}\bar{Y}.$

We note here that by the chain rule, the vector field
$\partial_{z}=\frac{1}{z\ln(|z|)}\partial_{\tau}$, so that

$\partial_{z}F_{-\tau V}=-\frac{1}{z\ln(|z|)}V.$

We can now use the $J|_{X}$ equivariance of $d_{X}F_{-\tau V}$ and the fact
that

$\mathcal{F}^{*}J=\mathcal{F}_{*}\circ J\circ\mathcal{F}^{-1}_{*}$

to see that for any vector field $Z$ lifted from $X$,

$\displaystyle(\mathcal{F}^{*}J)Z=d_{X}F_{-\tau V}\circ J\circ d_{X}F_{\tau
V}(Z)=d_{X}F_{-\tau V}\circ d_{X}F_{\tau V}(JZ)=JZ.$

For vertical vectors, we calculate on the frame for
$\mathcal{O}_{\breve{N}}\oplus\overline{\mathcal{O}_{\breve{N}}}$ given by
$\\{\partial_{z},\partial_{\bar{z}}\\}$ to see

$\displaystyle(\mathcal{F}^{*}J)\partial_{z}$
$\displaystyle=\mathcal{F}_{*}\left(J\left(\partial_{z}+\frac{1}{z\ln(|z|)}V\right)\right)=\mathcal{F}_{*}\left(i\partial_{z}+\frac{1}{z\ln(|z|)}JV\right)$
$\displaystyle=i\partial_{z}-\frac{i}{z\ln(|z|)}V+\frac{1}{z\ln(|z|)}JV=i\partial_{z}-\frac{2i}{z\ln(|z|)}V^{0,1}.$

Here we write $V^{0,1}$ for the $(0,1)$ part of $V$, defined so that
$V=V^{1,0}+V^{0,1}$. A similar calculation for $\partial_{\bar{z}}$ shows that
as a tensor, we get the following smooth linear expression in terms of $V$:

$(\mathcal{F}^{*}J)=J+4\mathfrak{Re}\left(\frac{i}{\bar{z}\ln(|z|)}V^{1,0}\otimes
d\bar{z}\right)=J+4\mathfrak{Re}(iV^{1,0}\otimes\bar{\partial}\tau)$
$=J-e^{-\tau}V\otimes d\theta+JV\otimes d\tau=J-V\otimes d^{c}\tau+JV\otimes
d\tau.$

For the symplectic form, we take note that around the slice we are interested
in,

$dd^{c}\Psi_{V}^{\eta,\varphi}=dd^{c}(\Psi_{V}^{\eta}+F^{*}_{\tau
V}\nu^{*}\varphi-\nu^{*}\varphi).$

Recalling that $\tau$ is unchanged by $\mathcal{F}$, pulling back the form
altogether leaves us with

$\mathcal{F}^{*}\varpi_{V,\eta,\varphi}=dd^{c}\tau+\mathcal{F}^{*}\sigma+\mathcal{F}^{*}\eta+\mathcal{F}^{*}dd^{c}\Psi_{V}^{\eta}+\mathcal{F}^{*}\left(dd^{c}F^{*}_{\tau
V}\nu^{*}\varphi\right).$

The map $(V,\eta)\mapsto\mathcal{F}^{*}(\sigma+\eta)$ is smooth — clearly in
$\eta$ — with linearization in $V$ arising from Cartan’s magic formula as the
map

$W\mapsto-\tau\mathcal{F}^{*}d(W\mathbin{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\lnot$}}}(\sigma+\eta)).$

For the gnarl $\Psi_{V}^{\eta}$ which was already shown to be smooth in $V$ in
Lemma 3.3, we notice that if we use an appropriately normalized Green’s
operator $\mathfrak{G}_{\sigma}$ for the Laplacian $\Delta_{\sigma}$ of
$\sigma$, we can write the gnarl as an affine linear expression in $\eta$:

$\Psi_{V}^{\eta}=\nu^{*}\left(\mathfrak{G}_{\sigma}\Lambda_{\sigma}(F^{*}_{\tau
V}(\sigma+\eta)-\sigma-\eta)\right).$

Lastly then, we can see that the operation applied to $\varphi$ can be
simplified to $\varphi\mapsto d(\mathcal{F}^{*}J)d\nu^{*}\varphi$, the complex
Hessian in terms of the pulled back complex structure. Yet we already showed
this flowed complex structure is smooth in $V$ with smooth components, so
composing with its linear endomorphism action on differential forms will be
smooth in $V$ and $\varphi$. Hence, $\mathcal{F}^{*}\varpi_{V,\eta,\varphi}$
is a smooth expression in the variables, and the Riemannian metric
$-\mathcal{F}^{*}J\mathbin{\raisebox{0.0pt}{\scalebox{1.0}[-1.0]{$\lnot$}}}\mathcal{F}^{*}(\varpi_{V,\eta,\varphi})$
is likewise a smooth expression, and we can finally conclude the operator
$\mathcal{S}$ is at least $\mathcal{C}^{1}$.

To finish the proof, we must show that the differential of $\mathcal{S}$ in
the variables $V$ and $\varphi$ at $\eta=0$ is invertible. This will be
sufficient for the the local submersion theorem for Banach spaces as stated in
_A3_ of Donaldson-Kronheimer [5] to define a cscK gnarled metric for each
nonzero $\eta$ in a small neighborhood $U$ of the origin in
$\mathcal{H}^{1,1}(X)$, as specified in the theorem statement.

Since the complex Hessian operator is linear in itself, we can replace
$\Psi_{V}$ by its $V$-linearization when differentiating $\mathcal{S}$, as the
final pullback by $\mathcal{F}$ introduces only quadratic or higher-order
expressions in $V$ which will vanish in the derivative.

As in Lemma 3.3, we set $\mathcal{D}^{\star}\mathcal{D}$ as the Lichnerowicz
operator for $\omega$, and additionally we consider $\mathcal{D}_{\sigma}$ and
$\star_{\sigma}$ as the analogous operators for $\sigma$ on $X$.
Differentiating in these two variables then gives

$\left.\frac{d}{du}\right|_{u=0}\mathcal{S}(uV,0,u\varphi)=2\mathcal{D}^{\star}\mathcal{D}(\nu^{*}(\varphi+\tau
f_{V}))|_{\\{e^{-2}\\}\times X}.$

Here we refer to the proof of Lemma 4.7 in Sektnan [11] where the problem of
blowing up manifolds with PT metrics is considered, particularly with respect
to how to handle holomorphic vector fields on $X$. When applied to a function
$h$ which is independent of the angle $\theta$, the Lichnerowicz operator
expands to

$\mathcal{D}^{\star}\mathcal{D}h=(\partial_{\tau}^{2}-\partial_{\tau})^{2}h-(\partial_{\tau}^{2}-\partial_{\tau})\Delta_{\sigma}h-(\partial_{\tau}^{2}-\partial_{\tau})h+\mathcal{D}_{\sigma}^{\star}\mathcal{D}_{\sigma}h.$

Recall also that
$f_{V}\in\text{ker}\mathcal{D}_{\sigma}^{\star}\mathcal{D}_{\sigma}$ since it
is a holomorphy potential for $V$. Certainly both $\varphi$ and $\tau f_{V}$
are independent of $\theta$, so

$\left.\frac{d}{du}\right|_{u=0}\mathcal{S}(uV,0,u\varphi)=2\mathcal{D}^{\star}_{\sigma}\mathcal{D}_{\sigma}\varphi+2f_{V}+2\Delta_{\sigma}f_{V}.$

Additionally, we can show that this map surjects to
$\mathcal{C}^{0,\alpha}(X)$ through an integration by parts due to the self-
adjointness of the Lichnerowicz operator on $X$, and additionally noting that
for any holomorphy potential $f$, the expression $f+\Delta_{\sigma}f$ is
orthogonal to im$\mathcal{D}_{\sigma}^{\star}\mathcal{D}_{\sigma}$, forming a
complementary basis. Thus every function in $\mathcal{C}^{0,\alpha}(X)$ can be
written as
$\mathcal{D}^{\star}_{\sigma}\mathcal{D}_{\sigma}\varphi+f_{V}+\Delta_{\sigma}f_{V}+\text{
const.}$ for some $\varphi$ and $V$, just as we need.

This surjectivity allows invocation of the implicit function theorem so that
there exists a $U\subset\mathcal{H}^{1,1}(X)$ such that if $\eta\in U$, we can
find $V_{\eta}\in\mathfrak{h}$ and
$\varphi_{\eta}\in\mathcal{C}^{4,\alpha}(X)$ such that
$\varpi_{V_{\eta}}=\omega+\eta+dd^{c}\Psi_{V_{\eta}}^{\eta,\varphi_{\eta}}$ is
both cscK and complete on $\breve{N}$. Since we now know
$\Delta_{\varpi_{V_{\eta}}}\left(\ln\varpi_{V_{\eta}}^{n+1}\right)$ is a
constant and therefore smooth, by elliptic regularity for the two operators
$f\mapsto\Delta_{\varpi_{V_{\eta}}}f$ and
$\varphi\mapsto\ln(\varpi_{V_{\eta}}+dd^{c}\varphi)^{n+1}$, one can bootstrap
$\varphi_{\eta}$ to be $\mathcal{C}^{\infty}(X)$. Finally, set
$z_{0}\in\mathbb{D}^{*}$ and define $s(x)=(z_{0},x)$ to be a fiberwise
constant section. Then lastly we clearly have the pullback

$s^{*}(\varpi_{V_{\eta}})=\sigma+\eta+d_{X}d_{X}^{c}(\Psi_{V_{\eta}}^{\eta,\varphi_{\eta}}|_{\\{z_{0}\\}\times
X})\in[\sigma+\eta].$

∎

With the existence of these metrics on a product model, a natural extension
would be to define their extensions on the complement of a complex
hypersurface $X$ in a compact manifold $M$ as is done for PT metrics, or in
even greater generality to the complement of a normal crossing divisor in $M$.
This introduces various technical complications mainly stemming from the lack
of an obvious choice for the extension of a vector field on $X$ to a tubular
neighborhood. When there is a holomorphic tubular neighborhood, one can
consider a holomorphic lift, but more generally a horizontal lift that
preserves the log-polar distance may be preferred. However, in the case that
$X$ has a holomorphically trivial normal bundle, the gnarl $\Psi_{V}$ can
still be introduced as-is to existing PT metrics to construct new gnarled
metrics on $M\setminus X$ which are asymptotically cscK without necessarily
splitting as a product of cscK metrics near the divisor.

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