Positive Einstein metrics with S4m3as principal orbit Hanci Chi Department of Foundational Mathematics Xian Jiaotong-Liverpool University

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Positive Einstein metrics with S4m+3 as principal orbit

Hanci Chi

Department of Foundational Mathematics, Xi’an Jiaotong-Liverpool University

hanci.chi@xjtlu.edu.cn

April 10, 2024

Abstract

We prove that there exists at least one positive Einstein metric on HPm+1♯HPm+1 for m≥2.

Based on the existence of the ﬁrst Einstein metric, we give a criterion to check the existence of

a second Einstein metric on HPm+1♯HPm+1. We also investigate the existence of cohomogeneity

one positive Einstein metrics on S4m+4 and prove the existence of a non-standard Einstein metric

on S8.

1 Introduction

A Riemannian manifold (M, g) is Einstein if its Ricci curvature is a constant multiple of g:

Ric(g) = Λg.

The metric gis then called an Einstein metric and Λ is the Einstein constant. Depending on the

sign of Λ, we call ga positive Einstein (Λ >0) metric, a negative Einstein (Λ <0) metric or a

Ricci-ﬂat (Λ = 0) metric. A positive Einstein manifold is compact by Myers’ Theorem [Mye41].

In this article, we investigate the existence of positive Einstein metrics of cohomogeneity one.

A Riemannian manifold (M, g) is of cohomogeneity one if a Lie Group Gacts isometrically on M

such that the principal orbit G/Kis of codimension one. The ﬁrst example of an inhomogeneous

positive Einstein metric was constructed in [Pag78]. The metric is deﬁned on CP2♯CP2and

is of cohomogeneity one. The result was later generalized in [BB82], [KS86], [Sak86], [PP87],

[KS88] and [WW98]. A common feature shared by positive Einstein metrics constructed in this

series of works is the principal orbits being principal U(1)-bundles over either a Fano manifold

or a product of Fano manifolds. From this perspective, one can view the Einstein metric on

HP2♯HP2in [B¨oh98] as another type of generalization to the Page’s metric, whose principal

orbit is a principal Sp(1)-bundle over HP1.

A natural question arises whether there exists a positive Einstein metric of cohomogeneity one

on HPm+1♯HPm+1 with m≥2, where the principal orbit is the total space of the quaternionic

Hopf ﬁbration formed by the following group triple:

(K,H,G) = (Sp(m)∆Sp(1), Sp(m)Sp(1)Sp(1), Sp(m+ 1)Sp(1)).(1.1)

The condition of being G-invariant reduces the Einstein equations to an ODE system deﬁned on

the 1-dimensional orbit space. The solution takes the form of g=dt2+g(t),where g(t) is a G-

invariant metric on S4m+3 for each t. One looks for a g(t) that is deﬁned on a closed interval [0, T ]

with an initial condition and a terminal condition. If g(t) collapses to the quaternionic-K¨ahler

metric on the singular orbit HPmat t= 0 and t=T, then gdeﬁnes a positive Einstein metric

on the connected sum HPm+1♯HPm+1, or equivalently, an S4-bundle over HPm. It has been

arXiv:2210.13216v5 [math.DG] 9 Apr 2024

conjectured that such an Einstein metric exists on HPm+1♯HPm+1 for all m≥2, as indicated

by numerical evidence provided in [PP86], [B¨oh98] and [DHW13].

Some well-known Einstein metrics are realized as integral curves to the cohomogeneity one

Einstein equation. For example, the standard sphere metric, the sine cone over Jensen’s sphere,

and the quaternionic-K¨ahler metric on HPm+1 are represented by integral curves to the cohomo-

geneity one system. Furthermore, the cone solution is an attractor to the system. It was realized

in [B¨oh98] that the winding of integral curves around the cone solution plays an important role

in the existence problem described above. To investigate the winding, one studies a quantity

(denoted as ♯Cw(¯

h) in [B¨oh98]) that is assigned to each local solution that does not globally

deﬁne a complete Einstein metric on HPm+1♯HPm+1. From the point of view of geometry, the

quantity records the number of times that the principal orbit becomes isoparametric while its

mean curvature remains positive. In general, an estimate for ♯Cw(¯

h) can be obtained from the

linearization along the cone solution. For m= 1, the estimate is good enough to prove the global

existence. This is not the case, however, if m≥2. For higher dimensional cases, it is from the

global analysis of the system that we obtain a further estimate for ♯Cw(¯

h) and we prove the

following existence theorem.

Theorem 1.1. On each HPm+1♯HPm+1 with m≥2, there exists at least one positive Einstein

metric with G/K=S4m+3 as its principal orbit.

Numerical studies in [B¨oh98] and [DHW13] indicate that there exists another Einstein metric

on HPm+1♯HPm+1 with m≥2. Based on Theorem 1.1, an estimate for ♯Cw(¯

h) in a limiting

subsystem (essentially obtained from the linearization along the cone solution) helps us propose

a criterion to check the existence of the second Einstein metric. Let nbe the dimension of G/K.

Such a criterion only depends on n(or m).

Theorem 1.2. Let θΨbe the solution to the following initial value problem:

dθ

dη =n−1

2ntanh η

nsin(2θ) + 2

ns(2m+ 1)(2m+ 2)(2m+ 3)

(2m+ 3)2+ 2m, θ(0) = 0.(1.2)

Let Ω = lim

η→∞ θΨ. For m≥2, there exist at least two positive Einstein metrics on HPm+1♯HPm+1

if Ω<3π

The upper bound for Ω in Theorem 1.2 is not sharp. Although it is diﬃcult to solve the initial

value problem (1.2) explicitly, one can use the Runge–Kutta 4-th order algorithm to approximate

Ω. Since the RHS of (1.2) does not vanish at η= 0, the initial Runge–Kutta step is well-deﬁned.

Our numerical study shows that Ω <3π

4for integers m∈[2,100].

We also look into the case where G/Kcompletely collapses at two ends of a compact manifold.

In that case, the cohomogeneity one space is S4m+4. No new Einstein metric is found on S4m+4

for m≥2. For m= 1, however, we obtained a non-standard positive Einstein metric on S8.

Such a metric is inhomogeneous by the classiﬁcation in [Zil82].

Theorem 1.3. There exists a non-standard Sp(2)Sp(1)-invariant positive Einstein metric ˆgS8

on S8.

It is worth mentioning that all new solutions found are symmetric. Metrics that are repre-

sented by these solutions all have a totally geodesic principal orbit.

This article is structured as follows. In Section 2, we present the dynamical system for

positive Einstein metrics of cohomogeneity one with G/Kas the principal orbit. Then we apply

a coordinate change that makes the cohomogeneity one Ricci-ﬂat system serve as a limiting

subsystem. Initial conditions and terminal conditions are transformed into critical points of

the new system. The new system admits Z2-symmetry. By a sign change, one can transform

initial conditions into terminal conditions. Hence the problem of ﬁnding globally deﬁned positive

Einstein metrics boils down to ﬁnding heteroclines that join two diﬀerent critical points.

In Section 3, we compute linearizations of the critical points mentioned above and obtain

two 1-parameter families of locally deﬁned positive Einstein metrics. One family is deﬁned on

a tubular neighborhood around HPm, represented by a 1-parameter family of integral curves

γs1. The other family is deﬁned on a neighborhood of a point in S4m+4, represented by another

1-parameter family of integral curves ζs2.

In Section 4, we make a little modiﬁcation on the quantity ♯Cw(¯

h) in [B¨oh98] and it is assigned

to both γs1and ζs2(hence denoted as ♯C(γs1) and ♯C(ζs2)). We construct a compact set to

obtain an estimate for ♯C(γs1) of some local solutions. Then we apply Lemma 4.4 in [B¨oh98]

and prove Theorem 1.1.

In Section 5, we apply another coordinate change that allows us to obtain more information

on ♯C(γs1) and ♯C(ζs2), which is encoded in the initial value problem (1.2) in Theorem 1.2. We

also prove Theorem 1.3.

Visual summaries of Theorem 1.1-1.3 are presented at the end of this article.

Acknowledgments The research is funded by NSFC (No. 12071489), the Foundation for

Young Scholars of Jiangsu Province, China (BK-20220282), and XJTLU Research Development

Funding (RDF-21-02-083). The author is grateful to McKenzie Wang for his constant support

and encouragement. The author would like to thank Christoph B¨ohm for his helpful suggestions

and remarks on this project. The author also thanks Cheng Yang and Wei Yuan for many

inspiring discussions.

2 Cohomogeneity one system

Consider the group triple (K,H,G) in (1.1). The isotropy representation g/kconsists of two

inequivalent irreducible summands p1=h/kand p2=g/h. Let the standard sphere metric gS4m+3

on G/K=S4m+3 be the background metric. As any G-invariant metric on G/Kis determined by

its restriction to one tangent space g/k, the metric has the form of

1gS4m+3 |p1+f2

2gS4m+3 |p2.

Let f1and f2be functions that are deﬁned on the 1-dimensional orbit space. We consider

Einstein equations for the cohomogeneity one metric

g:= dt2+f2

1gS4m+3 |p1+f2

2gS4m+3 |p2.

By [EW00], the metric gis an Einstein metric on (t∗−ϵ, t∗+ϵ)×G/Kif (f1, f2) is a solution to

f1− ˙

f1!2

=− 3˙

+ 4m˙

f2!˙

+ 2 1

+ 4mf2

2−Λ,

f2− ˙

f2!2

=− 3˙

+ 4m˙

f2!˙

+ (4m+ 8) 1

2−6f2

2−Λ,

(2.1)

with a conservation law

3 ˙

f1!2

+4m ˙

f2!2

− d1

+d2

f2!2

+6 1

+4m(4m+8) 1

2−12mf2

2−(n−1)Λ = 0.(2.2)

To ﬁx homothety, we set Λ = nin this article. We leave Λ in the equations for readers to trace

the Einstein constant.

Remark 2.1. If we replace the principal orbit G/Kby S4m+3 = [Sp(m+1)U(1)]/[Sp(m)∆U(1)],

then the isotropy representation g/kconsists of three inequivalent irreducible summands. The

principal orbit can collapse either as HPmor CP2m+1, depending on the choice of intermediate

group. For such a principal orbit, the dynamical system of cohomogeneity one Einstein metrics

involves three functions and has (2.1) as its subsystem. A numerical solution in [HYI03] indicates

the existence of a positive Einstein metric where G/Kcollapse to HPmon one end and CP2m+1

on the other end.

We consider (2.1) and (2.2) with the following two initial conditions. By [EW00], for the

metric gto extend smoothly to the singular orbit HPm, we have

lim

t→0f1, f2,˙

f1,˙

f2= (0, f, 1,0) (2.3)

for some f > 0. On the other hand, for gto extend smoothly to a point where G/Kfully

collapses, one considers

lim

t→0f1, f2,˙

f1,˙

f2= (0,0,1,1) .(2.4)

By Myers’ theorem, any solution obtained from (2.1) that represents an Einstein metric on

HPm+1♯HPm+1 must be deﬁned on [0, T ] for some ﬁnite T > 0. Speciﬁcally, one looks for

solutions with the initial condition (2.3) and the terminal condition

lim

t→Tf1, f2,˙

f1,˙

f2=0,¯

f, −1,0(2.5)

for some ¯

f > 0. Similarly, to construct an Einstein metric on S4m+4, one looks for solutions with

the initial condition (2.4) and the terminal condition

lim

t→Tf1, f2,˙

f1,˙

f2= (0,0,−1,−1) .(2.6)

Remark 2.2. In [Koi81], one takes a non-collapsed principal orbit G/Kas the initial data.

Speciﬁcally, consider

f1, f2,˙

f1,˙

f2=¯

f1,¯

f2,¯

h1,¯

h2

for some positive ¯

fi’s. To construct a positive Einstein metric, one looks for a solution that

extends backward and forward smoothly to either HPmor a point on S4m+4 in ﬁnite time.

Inspired by a personal communication with Wei Yuan, we introduce a coordinate change that

transforms (2.1) to a polynomial ODE system. Let Lbe the shape operator of principal orbit.

Deﬁne

X1:=

p(trL)2+nΛ, X2:=

p(trL)2+nΛ, Y :=

p(trL)2+nΛ, Z :=

p(trL)2+nΛ.

Also, deﬁne

H:= 3X1+ 4mX2, G := 3X2

1+ 4mX2

R1:= 2Y2+ 4mZ2, R2:= (4m+ 8)Y Z −6Z2.

Consider dη =ptr(L)2+nΛdt. Let ′denote taking the derivative with respect to η. Then (2.1)

becomes













′

=V(X1, X2, Y, Z) = 





X1HG+1

n(1 −H2)−1+R1−1

n(1 −H2)

X2HG+1

n(1 −H2)−1+R2−1

n(1 −H2)

YHG+1

n(1 −H2)−X1

ZHG+1

n(1 −H2)+X1−2X2







.(2.7)

The conservation law (2.2) becomes

CΛ≥0:G+1

n(1 −H2)+6Y2+ 4m(4m+ 8)Y Z −12mZ2= 1.(2.8)

Or equivalently,

CΛ≥0:12m

n(X1−X2)2+ 6Y2+ 4m(4m+ 8)Y Z −12mZ2= 1 −1

n.(2.9)

We can retrieve the original system by

t=Zη

η∗r1−H2

nΛd˜η, f1=1

Yr1−H2

nΛ, f2=1

√Y Z r1−H2

nΛ.(2.10)

It is clear that H2≤1 by the deﬁnition of Hand Xi’s. However, such a piece of information

can be obtained from the new system alone without (2.1) and (2.2). Note that

H′=⟨∇H, V ⟩

=H2G+1

n(1 −H2)−1+ 6Y2+ 4m(4m+ 8)Y Z −12mZ2−(1 −H2)

=H2G+1

n(1 −H2)−1+ 1 −G−1

n(1 −H2)−(1 −H2) by (2.8)

= (H2−1) G+1

n(1 −H2)= (H2−1) 1

n+12m

n(X1−X2)2.

(2.11)

Therefore, the following algebraic surface in R4with boundary

E:= CΛ≥0∩ {Y, Z ≥0}∩{H2≤1}

is invariant. Moreover, E ∩ {H=±1}are two invariant sets of lower dimension. The Z2-

symmetry on the sign of (X1, X2) gives a one to one correspondence between integral curves on

E ∩ {H= 1}and those on E ∩ {H=−1}.

Remark 2.3. The restricted system of (2.7) on E ∩ {H= 1}is in fact (2.1) with Λ = 0

under the coordinate change dη = (trL)dt. The dynamical system is essentially the same as

the one that appears in [Win17]. An integral curve on the subsystem is known for representing

a complete Ricci-ﬂat metric deﬁned on the non-compact manifold HPm+1\{∗} [B¨oh98]. The

Ricci-ﬂat metric on HPm+1\{∗} is the limit cone for locally deﬁned positive Einstein metrics on

the tubular neighborhood around HPm.

Remark 2.4. If an integral curve to (2.7) enters E ∩ {H < 1}and is deﬁned on R, then from

(2.11) it must cross E ∩ {H= 0}transversally. The crossing point corresponds to the turning

point in [B¨oh98]. For any integral curve to (2.7) that has a turning point, we choose the η∗in

(2.10) so that t∗:= R0

η∗q1−H2

nΛd˜ηis the value at which trLvanishes. By our choice of η∗, the

integral curve crosses E ∩{H= 0}at η= 0. There are cohomogeneity one Einstein systems with

additional geometric structure, e.g. the one considered in [FH17], where every trajectory has a

turning point.

Remark 2.5. From (2.9), the inequality 6Y2+4m(4m+8)Y Z −12mZ2≤1−1

nis always valid.

Therefore, the set E∩{Z−ρY ≤0}is compact for any ﬁxed ρ∈0,m(4m+8)+√m2(4m+8)2+18m

6m.

If the maximal interval of existence of an integral curve to (2.7) is (−∞,¯η) for some ¯η∈R, it

must escape E ∩ {Z−ρY ≤0}. The crossing point corresponds to the W-intersection point in

[B¨oh98]. In Proposition 3.3 and Deﬁnition 4.9, we introduce an invariant set Wand a modiﬁed

deﬁnition for the W-intersection point, which ﬁxes ρ= 1 in the original deﬁnition in [B¨oh98].

3 Linearization at critical points

The local existence of positive Einstein metrics around the singular orbit HPmis well-established

in [B¨oh98]. We interpret the result using the new coordinate. For m≥2, the vector ﬁeld Vhas

in total 10 critical points (12 critical points for m= 1) on E. As indicated by their superscripts,

these critical points lie on either E ∩ {H= 1}or E ∩ {H=−1}.

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PositiveEinsteinmetricswithS4m+3asprincipalorbitHanciChiDepartmentofFoundationalMathematics,Xi’anJiaotong-LiverpoolUniversityhanci.chi@xjtlu.edu.cnApril10,2024AbstractWeprovethatthereexistsatleastonepositiveEinsteinmetriconHPm+1♯HPm+1form≥2.BasedontheexistenceofthefirstEinsteinmetric,wegiveacriterio...

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Positive Einstein metrics with S4m3as principal orbit Hanci Chi Department of Foundational Mathematics Xian Jiaotong-Liverpool University

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