# Asymptotic behavior of unstable perturbations of the Fubini–Study metric in
Ricci flow

David Garfinkle  Oakland University<EMAIL_ADDRESS>,  James Isenberg
University of Oregon<EMAIL_ADDRESS>http://www.uoregon.edu/$\sim$isenberg/ ,  Dan Knopf  University of Texas at
Austin<EMAIL_ADDRESS>http://www.ma.utexas.edu/users/danknopf and
Haotian Wu  The University of Sydney<EMAIL_ADDRESS>
###### Abstract.

Kröncke has shown that the Fubini–Study metric is an unstable generalized
stationary solution of Ricci flow [Kro20]. In this paper, we carry out
numerical simulations which indicate that Ricci flow solutions originating at
unstable perturbations of the Fubini–Study metric develop local singularities
modeled by the blowdown soliton discovered in [FIK03].

DG thanks the NSF for support in PHY-1806219 and PHY-2102914. JI thanks the
NSF for support in PHY-1707427. DK thanks the Simons Foundation for support in
Award 635293. HW thanks the Australian Research Council for support in
DE180101348.

###### Contents

  1. 1 Introduction
  2. 2 An unstable conformal perturbation
  3. 3 Cohomogeneity-one metrics
  4. 4 Initial data
  5. 5 Ricci–DeTurck flow
  6. 6 A system better suited to simulation
  7. 7 The blowdown soliton
  8. 8 Results of simulations
  9. 9 Directions for future work

##  1\. Introduction

In studying dynamical systems, it is always important to identify and classify
fixed points. For Ricci flow, regarded as a dynamical system on the infinite-
dimensional space of Riemannian metrics, one must consider generalized fixed
points. These are the _Ricci solitons_ , specified by the data
$\big{(}\mathcal{M}^{n},g,\lambda,X\big{)}$ for which

$-2\operatorname{Rc}[g]=2\lambda g+\mathcal{L}_{X}g,$

where $\operatorname{Rc}[g]$ is the Ricci curvature of the metric $g$,
$\lambda\in\mathbb{R}$, and $\mathcal{L}_{X}g$ is the Lie derivative of $g$
with respect to the vector field $X$.

A prominent example of a Ricci soliton is the Fubini–Study metric on complex
projective space, an Einstein metric corresponding to
$\big{(}\mathbb{CP}^{N},G_{\mathrm{FS}},2(N+1),0\big{)}$.111 $N$ is the
complex dimension, so that the real dimension is $n=2N$. In §10 of [Ham95],
Hamilton conjectures that it is a stable fixed point. We refer the reader to
Section 4 for an explicit description of the Fubini–Study metric. In Example
2.3 of [CHI04], Cao, Hamilton, and Ilmanen note that the Fubini–Study metric
is neutrally linearly stable with respect to the second variation of
Perelman’s shrinker entropy, crediting this observation to unpublished work of
Goldschmidt. However, in his dissertation, Kröncke computes the third
variation of Perelman’s entropy to prove the surprising result that
Fubini–Study is in fact unstable. This calculation has appeared as [Kro20] and
has been independently verified in [KS19]. The instability is conformal and
hence not Kähler.

The instability of the Fubini–Study raises an interesting question. What
happens to Ricci flow solutions that start at arbitrarily small but unstable
perturbations of $G_{\mathrm{FS}}$? According to the conjectural hierarchy of
$4$-dimensional solutions given in [CHI04], one expects the Ricci flow of
unstable perturbations of $\big{(}\mathbb{CP}^{2},G_{\mathrm{FS}}\big{)}$ to
develop local singularities modeled by the blowdown gradient shrinking soliton
$\mathcal{L}^{2}_{-1}$ discovered in [FIK03]. These solutions would start
arbitrarily close to (but _not_ in) the subset of Kähler metrics and, in
finite time, develop local singularities that are asymptotically Kähler (but
with the complex structure reversed — see below). A closely related type of
singularity formation, an asymptotic approach to the space of Kähler metrics
in the blowup limit, has been partially investigated in [IKS19]. It should be
noted that, while the blowdown soliton has been shown to occur as a model of
singularity formation on compact manifolds by Máximo [Max14], the solutions
considered in that work are Kähler, whereas the unstable perturbations studied
here all lie outside the space of Kähler initial data.

In this paper, we describe numerical simulations that provide evidence in
favor of the conjecture that there exists an unstable Ricci flow “orbit”
starting arbitrarily close to $\big{(}\mathbb{CP}^{2},G_{\mathrm{FS}}\big{)}$
and ending at the blowdown soliton $\mathcal{L}^{2}_{-1}$ in real dimension
$n=4$. This is not a true orbit, because convergence to the blowdown soliton
happens only locally and after parabolic dilation of the developing
singularity. Nonetheless, it provides evidence in favor of the hierarchy
outlined in [CHI04]. Furthermore, while $\mathcal{L}^{2}_{-1}$ is known to be
unique among Kähler solitons, our results suggest that there is no non-Kähler
gradient shrinking soliton with the topology of $\mathbb{C}^{2}$ blown up at a
point, at least none with a density $\Theta$ above that of $\mathbb{CP}^{2}$;
_cf_. [CHI04].

Because unperturbed $\big{(}\mathbb{CP}^{2},G_{\mathrm{FS}}\big{)}$ is a
positively-curved Einstein manifold, under Ricci flow it vanishes in a point.
In particular, the area of the distinguished $\mathbb{CP}^{1}$ goes to zero at
the same time that the diameter of the manifold goes to zero. In each of our
numerical simulations, we begin with initial data that are small non-Kähler
conformal perturbations $\big{(}\mathbb{CP}^{2},\tilde{G}\big{)}$. The
conformal factor lies in the nullspace of the second variation of Perelman’s
shrinker entropy $\nu$ and is constructed in Section 2. Our numerics indicate
that evolution of this initial data by Ricci–DeTurck flow crushes the
$\mathbb{CP}^{1}$ fiber before the remainder of the manifold vanishes, that
is, before the diameter goes to zero. Performing a parabolic rescaling of the
solution fixes the area of the $\mathbb{CP}^{1}$ fiber. Our numerics indicate
that the developing singularity is Type-I; that is, it develops at the natural
parabolic rate. Because of this, one expects by [EMT11] to see local
convergence of the parabolic blowups to a nonflat gradient shrinking soliton.
How do we recognize the soliton numerically? We do so in two ways.

First, using a measure of local closeness to Kähler derived in Section 7
below, we find that the blowups are becoming asymptotically Kähler near the
distinguished fiber, but with the opposite orientation of the complex
structure on unperturbed $\big{(}\mathbb{CP}^{2},G_{\mathrm{FS}}\big{)}$. This
is significant because we know by [FIK03] that the blowdown soliton is the
unique gradient shrinking Kähler soliton with the topology of $\mathbb{C}^{2}$
blown up at the origin, which corresponds in the original solution to the
distinguished $\mathbb{CP}^{1}$.

Second, we use the fact that the blowdown soliton is asymptotically conical.
By [KW15], there is at most one shrinking gradient soliton asymptotic to any
cone. So, as further verification, we determine that the parabolic blowup is
converging numerically to the correct cone near the $\mathbb{CP}^{1}$, in a
neighborhood suitable to the rescaling. Thus, while numerical simulations
cannot provide proof, we regard these two observations as compelling evidence
in favor of the formation of the [FIK03] soliton in solutions emerging from
unstable perturbations of the Fubini–Study metric.

##  2\. An unstable conformal perturbation

We seek a conformal factor $\psi$ so that $(1+\delta\psi)G_{\mathrm{FS}}$ is
an unstable perturbation of the Fubini–Study metric for small $\delta>0$.

As shown in [KS19], the function $\psi$ must satisfy three conditions:

$\Big{(}\Delta+\frac{1}{\tau}\Big{)}\psi=0,\qquad\int_{\mathbb{CP}^{2}}\psi\,\mathrm{d}V=0,\qquad\int_{\mathbb{CP}^{2}}\psi^{3}\,\mathrm{d}V>0.$

These are, respectively: the condition that $\psi$ lie in the nullspace of the
second variation of Perelman’s entropy $\nu$, a normalization necessary for
that functional, and a sufficient condition for the perturbation generated by
$\psi$ to be unstable. Here, $1/(2\tau)$ is the Einstein constant of the
metric, which in our case is $2(N+1)=6$.

Because $\operatorname{Rc}(G_{\mathrm{FS}})=6\,G_{\mathrm{FS}}$ and the
Laplacian of a rotationally-symmetric function $\psi$ is

$\Delta\psi=\psi_{\theta\theta}+\big{\\{}3\cot(\theta)-\tan(\theta)\big{\\}}\psi_{\theta},$

to specify rotationally symmetric unstable perturbations of the Fubini–Study
metric, we seek solutions $\psi$ of the second-order linear ode

$\displaystyle 0$ $\displaystyle=(\Delta+12)\psi(\theta)$
$\displaystyle=\psi^{\prime\prime}(\theta)+\big{\\{}3\cot(\theta)-\tan(\theta)\big{\\}}\psi^{\prime}(\theta)+12\psi(\theta).$

Only one fundamental solution of this ode is smooth for $\theta\in[0,\pi/2]$.
Up to an arbitrary multiplicative constant, it is

$\psi=\csc^{2}(\theta)\big{\\{}2-8\cos(2\theta)+6\cos(4\theta)\big{\\}}=1+3\cos(2\theta).$

Using $\det(G_{\mathrm{FS}})=\sin^{6}(\theta)\cos^{2}(\theta)$, one readily
verifies that this solution satisfies

$\displaystyle\int_{\mathbb{CP}^{2}}\psi\,\mathrm{d}V=\int_{0}^{\pi/2}\big{\\{}1+3\cos(2\theta)\big{\\}}\sin^{3}(\theta)\cos(\theta)\,\mathrm{d}\theta=0,$
$\displaystyle\int_{\mathbb{CP}^{2}}\psi^{3}\,\mathrm{d}V=\int_{0}^{\pi/2}\big{\\{}1+3\cos(2\theta)\big{\\}}^{3}\sin^{3}(\theta)\cos(\theta)\,\mathrm{d}\theta=\frac{2}{5}.$

Hence, given $\delta>0$, we choose as initial data

$\tilde{G}=\frac{1+\delta\psi}{1+\delta}G_{\mathrm{FS}}=\Big{(}1+3\frac{\delta}{1+\delta}\cos(2\theta)\Big{)}G_{\mathrm{FS}}=h\,G_{\mathrm{FS}},$

where $h$ is defined in (7) below, with $\varepsilon=\frac{\delta}{1+\delta}$.

##  3\. Cohomogeneity-one metrics

On $[0,\pi/2]\times\mathrm{SU}(2)$, we consider cohomogeneity-one metrics of
the form

(1)
$G=\mathrm{\rho}^{2}\,d\theta^{2}+\sum_{i=1}^{3}f_{i}^{2}\,\omega^{i}\otimes\omega^{i},$

where $\omega^{i}\otimes\omega^{i}$ is a Milnor coframe for $\mathrm{SU}(2)$.
The cohomogeneity-one condition means that the fiber over a generic $\theta$
is diffeomorphic to $\mathbb{S}^{3}$, except for $\theta\in\\{0,\pi/2\\}$. We
are interested in initial data that are unstable conformal perturbations of
the Fubini–Study metric on $\mathbb{CP}^{2}$. Accordingly, we impose an
_Ansatz_ of $\mathrm{U}(2)$ symmetry in the form

(2) $f_{1}=f\qquad\mbox{ and }\qquad f_{2}=f_{3}=g.$

From the viewpoint of analysis, it is convenient to fix a gauge and let
$s(\theta,t)$ represent arclength with respect to $G$. Then $s$ and $\theta$
satisfy the infinitesimal spatial relation

$\mathrm{d}s=\rho\,\mathrm{d}\theta.$

In this gauge, the metric takes the form

$G=ds^{2}+\sum_{i=1}^{3}f_{i}^{2}\,\omega^{i}\otimes\omega^{i}.$

Then the sectional curvatures of $G$ are convex linear combinations of

(3a) $\displaystyle\kappa_{12}=\kappa_{31}$
$\displaystyle=\frac{f^{2}}{g^{4}}-\frac{f_{s}g_{s}}{fg},$ (3b)
$\displaystyle\kappa_{23}$
$\displaystyle=\frac{4g^{2}-3f^{2}}{g^{4}}-\frac{g_{s}^{2}}{g^{2}},$ (3c)
$\displaystyle\kappa_{01}$ $\displaystyle=-\frac{f_{ss}}{f},$ (3d)
$\displaystyle\kappa_{02}=\kappa_{03}$ $\displaystyle=-\frac{g_{ss}}{g},$

where $e_{0}=\frac{\mathrm{d}}{\mathrm{d}s}$ and $(e_{1},e_{2},e_{3})$ is a
Milnor frame for $\mathrm{SU}(2)$.

The variables $(s,t)$ do not commute, but this approach yields a strictly
parabolic system for Ricci flow on these geometries,

(4a) $\displaystyle f_{t}$
$\displaystyle=f_{ss}+2\frac{g_{s}}{g}f_{s}-2\frac{f^{3}}{g^{4}},$ (4b)
$\displaystyle g_{t}$
$\displaystyle=g_{ss}+\left(\frac{f_{s}}{f}+\frac{g_{s}}{g}\right)g_{s}+2\frac{f^{2}-2g^{2}}{g^{3}},$

where all time derivatives on the lhs are taken at fixed $\theta$. This system
has been studied in [IKS19]. One can recover the behavior of $\rho$ under
Ricci flow using its evolution equation

$\rho_{t}=\left(\frac{f_{ss}}{f}+2\frac{g_{ss}}{g}\right)\rho.$

From the viewpoint of numerical simulation, on the other hand, it is
preferable to use fixed commuting variables. Using the fact that any smooth
$\zeta(s)$ must satisfy

$\zeta=\frac{\zeta_{\theta}}{\rho}\qquad\mbox{ and
}\qquad\zeta_{ss}=\frac{\zeta_{\theta\theta}}{\rho^{2}}-\frac{\rho_{\theta}\zeta_{\theta}}{\rho^{3}},$

we rewrite system (4) in terms of the fixed commuting variables $(\theta,t)$
as

(5a) $\displaystyle\rho_{t}$ $\displaystyle=\frac{f_{\theta\theta}}{\rho
f}-\frac{\rho_{\theta}f_{\theta}}{\rho^{2}f}+2\frac{g_{\theta\theta}}{\rho
g}-2\frac{\rho_{\theta}g_{\theta}}{\rho^{2}g},$ (5b) $\displaystyle f_{t}$
$\displaystyle=\frac{f_{\theta\theta}}{\rho^{2}}-\frac{\rho_{\theta}f_{\theta}}{\rho^{3}}+2\frac{f_{\theta}g_{\theta}}{\rho^{2}g}-2\frac{f^{3}}{g^{4}},$
(5c) $\displaystyle g_{t}$
$\displaystyle=\frac{g_{\theta\theta}}{\rho^{2}}-\frac{\rho_{\theta}g_{\theta}}{\rho^{3}}+\left(\frac{f_{\theta}}{f}+\frac{g_{\theta}}{g}\right)\frac{g_{\theta}}{\rho^{2}}+2\frac{f^{2}-2g^{2}}{g^{3}}.$

The evolution equation (5a) for $\rho$ prevents this system from being
parabolic. To remedy this, we implement DeTurck’s trick below.

##  4\. Initial data

Our reference (unperturbed) Fubini–Study metric $G_{\mathrm{FS}}$ is
determined by three functions
$\rho_{\mathrm{FS}},f_{\mathrm{FS}},g_{\mathrm{FS}}\colon[0,\pi/2]\rightarrow\mathbb{R}$,
given by

$\rho_{\mathrm{FS}}(\theta)=1,\qquad
f_{\mathrm{FS}}(\theta)=\frac{1}{2}\sin(2\theta)=\sin(\theta)\cos(\theta),\quad\mbox{
and }\quad g_{\mathrm{FS}}(\theta)=\sin(\theta).$

That is, using (1), we have

$G_{\mathrm{FS}}=d\theta^{2}+\sin^{2}(\theta)\cos^{2}(\theta)\,\omega^{1}\otimes\omega^{1}+\sin^{2}(\theta)\,\omega^{2}\otimes\omega^{2}+\sin^{2}(\theta)\,\omega^{3}\otimes\omega^{3}.$

It is readily verified that these choices have the correct boundary behavior
to induce a smooth metric on $\mathbb{CP}^{2}$, with the geometry of an
asymptotically round $4$-ball near $\theta=0$ and with a distinguished
$\mathbb{CP}^{1}$ fiber at $\theta=\pi/2$. Furthermore, it follows easily from
(3) that $\operatorname{Rc}[G_{\mathrm{FS}}]=6\,G_{\mathrm{FS}}$.

For use in specifying a Ricci-DeTurck flow below, which shadows the Ricci flow
and is parabolic, unlike the Ricci flow itself, we note that the Fubini–Study
connection is determined by

(6)
$(\Gamma_{\mathrm{FS}})_{00}^{0}=0,\quad(\Gamma_{\mathrm{FS}})_{11}^{0}=-\frac{1}{4}\sin(4\theta),\quad(\Gamma_{\mathrm{FS}})_{22}^{0}=(\Gamma_{\mathrm{FS}})_{33}^{0}=-\frac{1}{2}\sin(2\theta).$

Given any $\varepsilon>0$, we recall that the conformal factor determining an
unstable perturbation of the Fubini–Study metric is given by

(7) $h(\theta):=\big{(}1+3\varepsilon\cos(2\theta)\big{)},$

as shown in Section 2. As initial data for our perturbed Ricci flow, we take
$G(0)=\tilde{G}=h\,G_{\mathrm{FS}}$, for which
$(\rho,f,g)=(h\rho_{\mathrm{FS}},hf_{\mathrm{FS}},hg_{\mathrm{FS}})$, namely

$\displaystyle\rho(\theta)$
$\displaystyle=\Big{(}1+3\varepsilon\cos(2\theta)\Big{)},$ $\displaystyle
f(\theta)$
$\displaystyle=\frac{1}{4}\Big{(}2\sin(2\theta)+3\varepsilon\sin(4\theta)\Big{)},$
$\displaystyle g(\theta)$
$\displaystyle=\sin(\theta)\Big{(}1+3\varepsilon\cos(2\theta)\Big{)}$

at time $t=0$. The Levi–Civita connection of $G(0)$ is determined by

$\displaystyle\Gamma_{00}^{0}$
$\displaystyle=-\frac{6\varepsilon\sin(2\theta)}{1+3\varepsilon\cos(2\theta)},$
$\displaystyle\Gamma_{11}^{0}$
$\displaystyle=-\frac{\sin(2\theta)\Big{(}\cos(2\theta)+3\varepsilon\cos(4\theta)\Big{)}}{2\Big{(}1+3\varepsilon\cos(2\theta)\Big{)}},$
$\displaystyle\Gamma_{22}^{0}=\Gamma_{33}^{0}$
$\displaystyle=-\frac{\sin(\theta)\cos(\theta)\Big{(}1-6\varepsilon+9\varepsilon\cos(2\theta)\Big{)}}{1+3\varepsilon\cos(2\theta)},$

with the unlisted Levi-Civita coefficients vanishing.

##  5\. Ricci–DeTurck flow

The formulas below that suppress time apply at all $t\geq 0$ such that a
solution exists.

We consider the pde system

(8) $\partial_{t}G=-2\operatorname{Rc}[G]+\mathcal{L}_{V}G$

with initial data $G(0)=h\,G_{\mathrm{FS}}$, where our DeTurck vector field
$V$ is defined by

(9a)
$V^{\beta}=G^{\alpha\alpha}\Big{(}\Gamma_{\alpha\alpha}^{\beta}-(\Gamma_{\mathrm{FS}})_{\alpha\alpha}^{\beta}\Big{)},$
in which the implicit summation ranges over $\alpha\in\\{0,1,2,3\\}$. For the
metrics under consideration, this reduces to
$V=V^{0}\,\frac{\mathrm{d}}{\mathrm{d}\theta}$, where (9b)
$V^{0}=G^{\alpha\alpha}\Big{(}\Gamma_{\alpha\alpha}^{0}-(\Gamma_{\mathrm{FS}})_{\alpha\alpha}^{0}\Big{)}.$

######  Remark.

We note that at $t=0$, one has

$V^{0}(\theta,0)=\frac{12\varepsilon\sin(2\theta)}{\Big{(}1+3\varepsilon\cos(2\theta)\Big{)}^{3}},$

which is smooth for all sufficiently small values of $\varepsilon\geq 0$.

In working with the DeTurck vector field
$V=V^{0}\,\frac{\mathrm{d}}{\mathrm{d}\theta}$ and the 1-form
$V^{\flat}=V_{0}\,\mathrm{d}\theta$, where $V_{0}=\rho^{2}V^{0}$, it is
convenient to relabel $V^{0}$ as the function $v(\theta,t)$ defined by

(10)
$v=\frac{\rho_{\theta}}{\rho^{3}}+\frac{\tfrac{1}{4}\rho^{2}\sin(4\theta)-ff_{\theta}}{\rho^{2}f^{2}}+\frac{\rho^{2}\sin(2\theta)-2gg_{\theta}}{\rho^{2}g^{2}}.$

One finds that $(\mathcal{L}_{V}G)_{\alpha\alpha}=2\nabla_{\alpha}V_{\alpha}$,
where the only nonzero components are

$\displaystyle\nabla_{0}V_{0}$
$\displaystyle=\rho^{2}v_{\theta}+v\,\rho\rho_{\theta}=\rho(\rho v)_{\theta},$
$\displaystyle\nabla_{1}V_{1}$ $\displaystyle=v\,ff_{\theta},$
$\displaystyle\nabla_{2}V_{2}$ $\displaystyle=\nabla_{3}V_{3}=v\,gg_{\theta}.$

Thus from system (5) above, we obtain a manifestly parabolic system
corresponding to the Ricci-DeTurck flow,

(11a) $\displaystyle\rho_{t}$
$\displaystyle=\frac{\rho_{\theta\theta}}{\rho^{2}}-3\frac{\rho_{\theta}^{2}}{\rho^{3}}+\Big{(}\frac{f_{\theta}}{\rho^{2}f}+2\frac{g_{\theta}}{\rho^{2}g}+v\Big{)}\rho_{\theta}$
$\displaystyle\quad+\left(\Big{[}\frac{\sin(4\theta)}{4f^{2}}\Big{]}_{\theta}+\Big{[}\frac{\sin(2\theta)}{g^{2}}\Big{]}_{\theta}\right)\rho+\frac{f_{\theta}^{2}}{\rho
f^{2}}+2\frac{g_{\theta}^{2}}{\rho g^{2}},$ (11b) $\displaystyle f_{t}$
$\displaystyle=\frac{f_{\theta\theta}}{\rho^{2}}+\Big{(}v-\frac{\rho_{\theta}}{\rho^{3}}\Big{)}f_{\theta}+2\frac{f_{\theta}g_{\theta}}{\rho^{2}g}-2\frac{f^{3}}{g^{4}},$
(11c) $\displaystyle g_{t}$
$\displaystyle=\frac{g_{\theta\theta}}{\rho^{2}}+\Big{(}v-\frac{\rho_{\theta}}{\rho^{3}}\Big{)}g_{\theta}+\left(\frac{f_{\theta}}{f}+\frac{g_{\theta}}{g}\right)\frac{g_{\theta}}{\rho^{2}}+2\frac{f^{2}-2g^{2}}{g^{3}}.$

Below, we convert this into an equivalent form for which suitable boundary
behaviors as $\theta\searrow 0$ and $\theta\nearrow\pi/2$ can be robustly
enforced in numerical simulations.

##  6\. A system better suited to simulation

Here, we convert system (11) into an equivalent one appropriate for numerical
simulation of the evolution. The issue we must address is this: for the
evolving functions $(\rho,f,g)$ to induce smooth metrics on $\mathbb{CP}^{2}$
at each time requires that we enforce the identities

$f=0,\quad f_{s}=1,\quad\mbox{ and }\quad g=0,\quad g_{s}=1\qquad\mbox{ at
}\qquad\theta=0,$

and the identities

$f=0,\quad f_{s}=-1,\quad\mbox{ and }\quad g>0,\quad g_{s}=0\qquad\mbox{ at
}\qquad\theta=\frac{\pi}{2}.$

However, we may impose only one Dirichlet or Neumann condition at each
boundary for the functions that we evolve numerically. We proceed to resolve
this issue in two steps, as follows:

We begin by defining

$\tilde{\rho}:=\log\rho,\qquad\tilde{f}:=\log f,\qquad\tilde{g}:=\log g,$

and

$Q:=\rho^{2}\sin(\theta)\cos(\theta)\left(\frac{\cos(2\theta)}{f^{2}}+\frac{2}{g^{2}}\right).$

Then using (11) and diligently doing some algebra, we obtain the system:

(12a) $\displaystyle\rho^{2}\,\tilde{\rho}_{t}$
$\displaystyle=\tilde{\rho}_{\theta\theta}-Q\tilde{\rho}_{\theta}-\tilde{\rho}_{\theta}^{2}+\tilde{f}_{\theta}^{2}+2\tilde{g}_{\theta}^{2}+Q_{\theta},$
(12b) $\displaystyle\rho^{2}\,\tilde{f}_{t}$
$\displaystyle=\tilde{f}_{\theta\theta}+Q\tilde{f}_{\theta}-2\frac{\rho^{2}}{g^{4}}f^{2},$
(12c) $\displaystyle\rho^{2}\,\tilde{g}_{t}$
$\displaystyle=\tilde{g}_{\theta\theta}+Q\tilde{g}_{\theta}+\frac{\rho^{2}}{g^{4}}\big{(}2f^{2}-4g^{2}\big{)}.$

Next we define $A,Y,Z$ via the relations

$\rho=:e^{A+Y+Z},\qquad f=:e^{Y}\cos(\theta)\,g,\qquad g=:e^{A}\sin(\theta),$

noting that

$Q=2e^{2Z}\left(\cot(2\theta)+e^{2Y}\cot(\theta)\right).$

Then we obtain this system:

(13a) $\displaystyle\rho^{2}\,A_{t}$
$\displaystyle=A_{\theta\theta}+QA_{\theta}+\csc^{2}(\theta)\left(2e^{4Y+2Z}\cos^{2}(\theta)-4e^{2Y+2Z}-1\right)$
$\displaystyle\quad+2e^{2Z}\cot(\theta)\left(\cot(2\theta)+e^{2Y}\cot(\theta)\right),$
(13b) $\displaystyle\rho^{2}\,Y_{t}$
$\displaystyle=Y_{\theta\theta}+QY_{\theta}-2e^{2Z}\tan(\theta)\cot(2\theta)-2e^{2Y+2Z}-\sec^{2}(\theta)$
$\displaystyle\quad+4e^{2Y+2Z}\csc^{2}(\theta)\left(1-e^{2Y}\cos^{2}(\theta)\right),$
(13c) $\displaystyle\rho^{2}\,Z_{t}$
$\displaystyle=Z_{\theta\theta}-Q(2A_{\theta}+2Y_{\theta}+Z_{\theta})+Q_{\theta}-(A_{\theta}+Y_{\theta}+Z_{\theta})^{2}+3\big{(}A_{\theta}+\cot(\theta)\big{)}^{2}$
$\displaystyle+\big{(}Y_{\theta}-\tan(\theta)\big{)}^{2}+2\big{(}A_{\theta}+\cot(\theta)\big{)}\big{(}Y_{\theta}-\tan(\theta)\big{)}+2e^{2Y+2Z}$
$\displaystyle+2e^{2Y+2Z}\cot^{2}(\theta)(e^{2Y}-1)+\sec^{2}(\theta)+\csc^{2}(\theta)+2e^{2Z}\cot(2\theta)\big{(}\tan(\theta)-\cot(\theta)\big{)}.$

Now consider what boundary conditions we need to impose smoothness. We need
${A_{\theta}}=0$ at $\theta=0$ and $\theta=\pi/2$. We need $Y=0$ and
${Y_{\theta}}=0$ at $\theta=0$ and ${Y_{\theta}}=0$ at $\theta=\pi/2$. We need
$Z=0$ and ${Z_{\theta}}=0$ at both $\theta=0$ and $\theta=\pi/2$. Thus, $A$ is
the sort of variable we need, but $Y$ and $Z$ are not. However, defining $B$
and $C$ by

$Y=:B{\sin^{2}}(\theta),\qquad Z=:C{\sin^{2}}(2\theta),$

we see that the conditions needed for smoothness can be imposed as long as
${B_{\theta}}={C_{\theta}}=0$ at $\theta=\pi/2$.

After some tedious algebra we find that equation (13b) yields

(14) $\displaystyle{\partial_{t}}B$
$\displaystyle={\rho^{-2}}\biggl{[}{B_{\theta\theta}}+\left(Q+4\cot(\theta)\right){B_{\theta}}+B\left({{2+2{e^{2Z}}-4{e^{2Y+2Z}}}\over{{\sin^{2}}(\theta)}}\right)$
$\displaystyle-4B(1+{e^{2Z}}+{e^{2Y+2Z}})+{e^{2Z}}\left({{-2-2{e^{2Y}}+4{e^{4Y}}}\over{{\sin^{2}}(\theta)}}\right)$
$\displaystyle+4\left({{e^{2Z}-1}\over{{\sin^{2}}(2\theta)}}\right)+4{e^{2Y+2Z}}\left({{1+2Y-{e^{2Y}}}\over{{\sin^{4}}(\theta)}}\right)\biggr{]}.$

Similarly some tedious algebra applied to equation (13c) yields the following:

(15) $\displaystyle{\partial_{t}}C$
$\displaystyle={\rho^{-2}}\biggl{[}{C_{\theta\theta}}+\left(Q+8\cot(2\theta)\right){C_{\theta}}-8C(2+{e^{2Z}}+{e^{2Y+2Z}})$
$\displaystyle+\left({{1-{e^{2Y+2Z}}}\over{{\sin^{2}}(\theta)}}\right)\left({{2{A_{\theta}}}\over{\sin(2\theta)}}-4C\right)$
$\displaystyle+\left({{1-{e^{2Z}}}\over{{\sin^{2}}(2\theta)}}\right)\left(-8C+4\cot(2\theta){A_{\theta}}+2\cos(2\theta)\left(\tan(\theta){B_{\theta}}+2B\right)-6\right)$
$\displaystyle+{e^{2Y+2Z}}\left({{\cosh(2Y)-1}\over{{\sin^{4}}(\theta)}}\right)+8\left(1+{\cos^{2}}(\theta)\right)\left({{1+2Z-{e^{2Z}}}\over{{\sin^{4}}(2\theta)}}\right)+2{{\left({{A_{\theta}}\over{\sin(2\theta)}}\right)}^{2}}$
$\displaystyle-\left(\sin(2\theta){C_{\theta}}+4\cos(2\theta)C\right)\left(\sin(2\theta){C_{\theta}}+4\cos(2\theta)C+\tan(\theta){B_{\theta}}+2B+{{2{A_{\theta}}}\over{\sin(2\theta)}}\right)\biggr{]}.$

In summary, we evolve the variables $A$, $B$ and $C$, and their evolution
equations are equations (13a), (14), and (15), respectively. The Neumann
boundary conditions are that ${A_{\theta}},\;{B_{\theta}}$ and $C_{\theta}$
vanish at $\theta=0$ and $\theta=\pi/2$.

##  7\. The blowdown soliton

We first derive a necessary and sufficient condition for a metric of the form
(1) to be Kähler, under the symmetry assumptions $f_{1}=f$ and
$f_{2}=f_{3}=g$. Calabi observed [Cal82] that any $\mathrm{U}(2)$-invariant
Kähler metric on $\mathbb{C}^{2}\backslash(0,0)$ may be written in complex
form as

(16)
$G_{\mathbb{C}}=\Big{\\{}e^{-r}\varphi\,\delta_{\alpha\beta}+e^{-2r}(\varphi_{r}-\varphi)\,\bar{z}_{\alpha}z_{\beta}\Big{\\}}\,\mathrm{d}z^{\alpha}\otimes\mathrm{d}\bar{z}^{\beta},$

with respect to his coordinate $r:=\log(|z_{1}|^{2}+|z_{2}|^{2})$. Written in
real coordinates, the same metric becomes

(17)
$G_{\mathbb{R}}=\varphi_{r}\Big{(}\frac{1}{4}\mathrm{d}r\otimes\mathrm{d}r+\omega^{1}\otimes\omega^{1}\Big{)}+\varphi\Big{(}\omega^{2}\otimes\omega^{2}+\omega^{3}\otimes\omega^{3}\Big{)},$

which shows that $\varphi>0$ and $\varphi_{r}>0$ are necessary conditions for
this to be a Kähler metric.

A comparison of equations (1) and (17) shows that a coordinate transformation
is needed to write a Kähler metric with respect to arclength $s$, where
$\mathrm{d}s=\rho\,\mathrm{d}\theta$. To see this, we observe that if $s$ and
$r$ are related by the ode

$\frac{\mathrm{d}r}{\mathrm{d}s}=\frac{2}{f},$

then equation (1) takes the form

(18)
$G=f^{2}\Big{(}\frac{1}{4}\mathrm{d}r\otimes\mathrm{d}r+\omega^{1}\otimes\omega^{1}\Big{)}+g^{2}\Big{(}\omega^{2}\otimes\omega^{2}+\omega^{3}\otimes\omega^{3}\Big{)},$

which matches (17) if and only if $f^{2}=\varphi_{r}$ and $g^{2}=\varphi$ are
related by

(19) $f=gg_{s}.$

It follows that a cohomogeneity-one metric with the $\mathrm{U}(2)$ symmetries
we have imposed is Kähler if and only if condition (19) holds. So, we may use
the ratio $f/(gg_{s})$ to provide a local measurement of the closeness of a
metric to the Kähler subspace.

Now, it follows from Lemma 6.1 and equation (27) of [FIK03] with $\lambda=-1$,
$\mu=\sqrt{2}$, and $\nu=0$, that the metric on the blowdown soliton
$\mathcal{L}^{2}_{-1}$ is determined by a function $\phi(r)$ that solves the
separable first-order ode

(20)
$\phi_{r}=\frac{1}{\sqrt{2}}\phi-(\sqrt{2}-1)-\left(1-\frac{1}{\sqrt{2}}\right)\phi^{-1}.$

One can solve (20) implicitly up to an arbitrary constant $\eta$, obtaining

(21) $e^{r+\eta}=\frac{\phi-1}{\big{(}\phi+\sqrt{2}-1\big{)}^{\sqrt{2}-1}}.$

As shown in [FIK03], the $\mathcal{L}^{2}_{-1}$ soliton is complete and exists
for all $r\in\mathbb{R}$. Using this fact and the positivity of $\phi$ and
$\phi_{r}$, it is not difficult to see from (20) and (21) that
$\phi\nearrow\infty$ as $r\nearrow\infty$ and that the metric is
asymptotically conical in the precise sense that

(22) $\frac{\phi_{r}}{\phi}\rightarrow\frac{1}{\sqrt{2}}\qquad\mbox{ as
}\qquad\phi\nearrow\infty.$

Specifically, if we define $\gamma:=2^{-1/4}$, then the asymptotic cone of the
blowdown soliton corresponds to

(23) $f=\gamma^{2}s\quad\mbox{ and }\quad g=\gamma
s\qquad\Rightarrow\qquad\frac{f^{2}}{g^{2}}=\gamma^{2}=\frac{1}{\sqrt{2}}.$

All sectional curvatures of the cone vanish except
$\kappa_{23}=\frac{4(\sqrt{2}-1)}{s^{2}}$.

##  8\. Results of simulations

The simulations are performed using standard numerical methods for parabolic
equations: centered differences for spatial derivatives and Euler’s method for
time evolution. We describe the method here in more detail.

Any function $F(t,\theta)$ is represented by the values $F^{k}_{i}$ that the
function takes at points $\theta_{i}$ equally spaced with spacing
$\Delta\theta$ and with times $t_{k}$ equally spaced with spacing $\Delta t$.
We use standard centered finite differences, so that $F_{\theta}$ and
$F_{\theta\theta}$ are approximated by

(24) ${F_{\theta}}={\frac{{F^{k}_{i+1}}-{F^{k}_{i-1}}}{2\Delta\theta}}$

and

(25)
${F_{\theta\theta}}={\frac{{F^{k}_{i+1}}+{F^{k}_{i-1}}-2{F^{k}_{i}}}{{(\Delta\theta)}^{2}}},$

respectively.

Time evolution is carried out using the Euler method, so that

(26) ${F^{k+1}_{i}}={F^{k}_{i}}+\Delta t{\partial_{t}}F,$

where ${\partial_{t}}F$ is the finite difference version of the right hand
side of equation (13a), (14) or (15) with the spatial derivatives evaluated
using equations (24) and (25). The standard von Neumann stability analysis of
equation (26) reveals that the time step must satisfy the Courant condition

(27) $\Delta t<{\textstyle{\frac{1}{2}}}{{({\rho_{\rm
min}}\Delta\theta)}^{2}}.$

where $\rho_{\rm min}$ is the minimum value of $\rho$.

We now show the results of a simulation with $\epsilon=0.1$. To get an idea of
how the evolution proceeds we plot $1/{\kappa_{23}}$ at $\theta=\pi/2$ as a
function of time. This curve goes to zero (i.e. ${\kappa_{23}}\to\infty$) at
the final time. Note that the curve is linear near the final time. Thus
$1/{\kappa_{23}}$ is a good proxy (up to some overall scale) for time
remaining until the singularity.

Figure 1. $1/{\kappa_{23}}$ plotted vs. time

We now want to examine the extent to which the metric becomes Kähler as the
singularity is approached. Recall that a Kähler metric has $f=g{g_{s}}$. We
define the quantity $K$ by $K=g{g_{s}}/f$. Then a metric is Kähler if $K=1$,
but it is also Kähler (but with the opposite orientation of $s$) if $K=-1$. In
Figure 2, we plot $K$ vs. $\theta$ at a time near the final time. Note that
near $\theta=\pi/2$ the quantity $K$ is approaching $-1$, thus indicating that
the metric in this region is becoming Kähler.

Figure 2. $K=g{g_{s}}/f$ plotted vs. $\theta$ near the final time

We would now like to know the form of the Kähler metric that is being
approached. Here we take a small region near $\theta=\pi/2$ and examine the
behavior of the cone angle $\gamma^{2}={f^{2}}/{g^{2}}$. In Figure 3, we plot
$\gamma^{2}={f^{2}}/{g^{2}}$ vs. the length from $\pi/2$ for three times near
the final time. We see that $\gamma^{2}$ approaches $0.707\approx 1/\sqrt{2}$
numerically, indicating local convergence of the metric cone angle
$\gamma^{2}$ to that of the blowdown soliton; _cf_. equation (23). What is
perhaps surprising is how close one needs to get to the singularity to obtain
this asymptotic behavior. In figure (3) the bottom curve corresponds to
$1/{\kappa_{23}}=4.7\times{{10}^{-8}}$ and has an asymptotic value for
${f^{2}}/{g^{2}}$ of 0.666. The middle curve has
$1/{\kappa_{23}}=9.4\times{{10}^{-9}}$ and has an asymptotic value for
${f^{2}}/{g^{2}}$ of 0.700. The top curve has
$1/{\kappa_{23}}=1.9\times{{10}^{-9}}$ and has an asymptotic value for
${f^{2}}/{g^{2}}$ of 0.707.

Figure 3. $\gamma^{2}=f^{2}/g^{2}$ plotted vs. length near the final time

##  9\. Directions for future work

Our results here provide evidence in favor of the conjecture that unstable
perturbations of the Fubini–Study Einstein metric — perturbations which are
conformal and not Kähler — develop finite-time local singularities modeled by
the Kähler blowdown soliton $\mathcal{L}_{-1}^{2}$ discovered in [FIK03], but
with the opposite complex structure.

Short of proving the full conjecture, a useful next step would be to derive
formal matched asymptotics that describe how parabolic dilations at these
singularities approach $\mathcal{L}_{-1}^{2}$, analogous to the matched
asymptotics formally derived in [AIK11] for solutions that approach the Bryant
soliton. However, because $\mathcal{L}_{-1}^{2}$ is only known in the implicit
form (21), a somewhat more approachable next step would be to develop formal
matched asymptotics for parabolic dilations that approach its asymptotic cone
$(f,g)=(\gamma^{2}s,\gamma s)$, where $\gamma=2^{-1/4}$, as described in (23).

## References