ricci flow in homogeneous manifolds - ime.unicamp.brrmiranda/talks/palestra-geometria.pdf · oops,...

Ricci flow in homogeneousmanifolds:geometry meets dynamical systems.

Ricardo M. [email protected]://www.ime.unicamp.br/~rmiranda

SEMINARIOS DE GEOMETRIA DIFERENCIAL

IMECC/UNICAMP - 11/05/2018

http://www.ime.unicamp.br/~rmiranda

Oops, last slide: references

This talk is based on the following papers:

# L. Grama, R. M. Martins, The Ricci flow of left invariant metrics on full flag

manifold SU(3)/T from a dynamical systems point of view. Bull. Sci. math. 133(5) (2009) 463 – 469.

# L. Grama, R. M. Martins, Global behavior of the Ricci flow on homogeneous

manifolds with two isotropy summands. Indagationes Mathematicae 23 (2012)95–104.

# L. Grama, R. M. Martins, A brief survey on the Ricci flow in homogeneous

manifolds. Sao Paulo Journal of Mathematical Sciences 9 (2015) 37–52.

# L. Grama, R. M. Martins, A numerical treatment to the problem of the quantity

of Einstein metrics on flag manifolds SU(5)/T 4 and SU(6)/T 5, preprint.

Ricci flow

Reference: N. Sheridan, Hamilton’s Ricci Flow. PhD thesis, 2006.

Let M be a Riemannian manifold with metric g0.

The Ricci flow equation (Hamilton’82) is given by∂g(t)

∂t= −2Ric(g(t)),

g(0) = g0

where Ric(g(t)) is the Ricci curvature of the metric g(t)

Hamilton, 1982

Let M be a closed 3-dimensional which admits a Riemannian metricwith strictly positive Ricci curvature. Then M also admits a metric ofconstant positive curvature.

Ricci flow

Corollary

Any simply-connected closed 3-dimensional manifold which admitsa metric of strictly positive Ricci curvature is diffeomorphic to the3-sphere.

Poincare’s Conjecture

Every simply connected closed 3-manifold is homeomorphic to the3-sphere.

Ricci flow

The Ricci flow equation∂g(t)

∂t= −2Ric(g(t)),

g(0) = g0

in general is a system of PDEs. In some special cases we can solve these

equations.

Sn

Let M be the sphere of radius r > 0. The metric is given by g = r2g1,where g1 is the metric on the unit sphere. The curvatures are given

by 1/r2 so Ric(v , v) =n − 1

r2and thus Ric = (n − 1)g1.

Ricci flow

Sn

The Ricci flow equation in this case is given by

∂r2(t)

∂t= −2(n − 1),

g(0) = g0

The solution is r(t) =√

R20 − 2(n − 1)t, where R0 is the initial radius

(note that M → ? when t → R20/(2n − 2).

Hn

If M = Hn then r(t) =√R20 + 2(n − 1)t so r(t)→∞ with r →∞.

Ricci flow on homogeneous manifolds

Very interesting reference: Jorge Lauret, Ricci flow of homogeneous

manifolds. Math. Z. 274 (2013). (Lie Groups)

Theorem

The Ricci flow equation for a homogeneous manifold is a system of

ODEs.

So the Ricci flow is the flow of a system of autonomous differential

equations (a dynamical system) on the finite dimensional space of the

metrics.

If g(t) is the flow with inicial condition g(0) = g0, what happens if

t →∞ (or even t → t0 for some t0 > 0)? Or: how to classify/calculate

the ω-limits of the Ricci flow?

First step: how to obtain the ODE?

New context: left-invariant metrics on flag manifolds.

Bonus: the space of invariant metrics can be parametrized by a

low-dimensional manifold.

The Ricci tensor Ric(g) of (M, g) is given by

Ric(g)(X ,Y ) = −1

2

∑i |[X ,Xi ]p|2 +

1

2(X ,X )

+1

4

∑i,j〈[Xi ,Xj ]p,X 〉2 − 〈[Z ,X ]p,X 〉,

where (·, ·) is the Killing form and etc etc (Besse’s book “Einstein

Manifolds”, 1987).

First case: SO(2n + 1)/(U(m)× SO(2k + 1))

If (λ1, λ2) is a Riemannian invariant metric on

SO(2n + 1)/(U(m)× SO(2k + 1)) then the components of the Ricci

tensor are

r1 =

2(m − 1)

2n − 1+

1 + 2k

2(2n − 1)

λ21λ22,

r2 =n + k

2n − 1+

m − 1

2(2n − 1)

(λ22 − (λ1 − λ2)2

)λ1λ2

,

Second case: SU(3)/T

Let Λ = (λ12, λ23, λ13) be a Riemannian invariant metric on

M6 = SU(3)/T 2.

The components of the Ricci tensor are given by

r12 =1

2λ12+

1

12

(λ12

λ13λ23− λ13λ12λ23

− λ23λ12λ13

)r13 =

1

2λ13+

1

12

(λ13

λ12λ23− λ12λ13λ23

− λ23λ12λ13

)r23 =

1

2λ23+

1

12

(λ23

λ12λ13− λ13λ23λ12

− λ12λ23λ13

)

Third case: SU(n + m + p)/S(U(n)× U(m)× U(p))

For SU(n + m + p)/S(U(n)× U(m)× U(p)), the components of the

Ricci tensor are given by

r1 =p

4(m + n + p)

(λ12

λ13λ23− λ23λ12λ13

− λ13λ12λ23

)+

1

2λ12

r2 =n

4(m + n + p)

(− λ12λ13λ23

− λ23λ12λ13

+λ13

λ12λ23

)+

1

2λ13

r3 =m

4(m + n + p)

(− λ12λ13λ23

+λ23

λ12λ13− λ13λ12λ23

)+

1

2λ23

Turning into polynomials (SO(2n+1)/(U(m)×SO(2k+1)))

For λ1, λ2 > 0 systems

r1 =

2(m − 1)

2n − 1+

1 + 2k

2(2n − 1)

λ21λ22,

r2 =n + k

2n − 1+

m − 1

2(2n − 1)

(λ22 − (λ1 − λ2)2

)λ1λ2

,

and

x = −

(2 + 4k

8n − 4

)x2 −

(2m − 2

2n − 1

)y2,

y = −(2− 2m

8n − 4

)xy −

(n + k +m − 1

2n − 1

)y2,

are equivalent in the first quadrant (x = λ1, y = λ2). The same is true

for the other systems: they are equivalent to a polynomial system.

Qualitative Theory of Differential Equations

x = −

(2 + 4k

8n − 4

)x2 −

(2m − 2

2n − 1

)y2,

y = −(2− 2m

8n − 4

)xy −

(n + k +m − 1

2n − 1

)y2,

Take initial conditions (x0, y0) and consider the flow φt(x0, y0) (the

solution of the above differential equation with initial conditions

x(0) = x0, y(0) = y0).

Note that {φt(x0, y0)}t∈R is a curve with limt→−∞

φt(x0, y0) = 0 and

limt→∞

||φt(x0, y0)|| =∞.

So it is not possible to distinguish the ω-limit sets!

End of the talk?

Poincare compactification I

The Poincare compactification allows us to take a vector field defined on

a non-compact manifold to the sphere (compact) so we can better

understand its behavior at infinity (Velasco, TAMS ’69; Poincare’1881).

Consider the polynomial differential system{x1 = P1(x1, x2),

x2 = P2(x1, x2),

with the associated vector field X = (P1,P2). The degree of X is defined

as d = max{deg(P1), deg(P2)}.

Poincare compactification II

Let

S2 = {y = (y1, y2, y3) ∈ R3; ||y || = 1}

be the unit sphere with northern hemisphere S2+ = {y ∈ S2; y3 > 0},

southern hemisphere S2− = {y ∈ S2; y3 < 0} and equator

S20 = {y ∈ S2; y3 = 0}.

Consider the central projections f+ : R2 → S2+ and f− : R2 → S2

− given by

f+(x) =1

∆(x)(x1, x2, 1)

and

f−(x) = − 1

∆(x)(x1, x2, 1),

where ∆(x) =√

1 + x21 + x22 . We shall use coordinates y = (y1, y2, y3)

for a point y ∈ S2.

Poincare compactification III

As f+ and f− are homeomorphisms, we can identify R2 with both S2+ and

S2−.

The maps f+ and f− define two copies of X , Df+(x)X (x) in the northern

hemisphere, based on f+(x), and Df−(x)X (x) in the southern

hemisphere, based on f−(x).

Note that, for x ∈ R2, when ||x || → ∞, f+(x), f−(x) converge to points

on the equation S20 . This allow us to identify S2

0 with the infinity of R2.

Denote by X the vector field on S2 \ S20 = S2

+ ∪ S2−.

Poincare compactification IV

To extend X (y) to the sphere S2, we define the Poincare

compactification of X as

p(X )(y) = yd−13 X (y).

Theorem (Poincare)

The vector field p(X ) extends X analytically to the whole sphere, and

in such a way that the equator is invariant.

If we know the behavior of p(X ) around the equator S20 ≡ S1, then we

know the behavior of X in the neighborhood of infinity.

Poincare compactification V

The natural projection π of S2 on y3 = 0 is called the Poincare disc and

it is denoted by D2. If we understand the dynamics of p(X ) on D2, then

we completely understand the dynamics of X , even at infinity.

We remark that Theorem 2 works just for polynomial vector fields.

Poincare compactification: charts I

Consider the coordinate neighborhoods Ui = {y ∈ S2, yi > 0} and

Vi = {y ∈ S2, yi < 0}, for i = 1, 2, 3, and the corresponding coordinate

maps φi : Ui → R2 and ψi : Vi → R2, given by

φi (y1, y2, y3) =

(yjyi,ykyi

), ψi (y1, y2, y3) =

(yjyi,ykyi

),

for i , j , k ∈ {1, 2, 3} with j < k . Denote by z = (z1, z2) the value of

φi (y) or ψi (y), according to the local chart that is being used.

Poincare compactification: charts II

Then we have the following expressions for p(X ) written in local charts:

U1 :zd2

(∆(z))d−1

(−z1P1

(1

z2,z1z2

)+ P2

(1

z2,z1z2

),−z2P1

(1

z2,z1z2

));

U2 :zd2

(∆(z))d−1

(−z1P2

(z1z2,

1

z2

)+ P1

(z1z2,

1

z2

),−z2P2

(z1z2,

1

z2

));

U3 :1

(∆(z))d−1(P1(z1, z2),P2(z1, z2)) .

For the charts V1, V2 and V3, we obtain the same expressions (18), (18)

and (18), now multiplied by (−1)d−1.

Poincare compactification: charts III

In this way, the expression of p(X ) is polynomial in each local chart.

Note that the singularities at infinity have z2 = 0.

Poincare comp: SO(2n + 1)/(U(m)× SO(2k + 1)) I

• Chart U1:z1 =

1

2

(m + 2 k) z12 n − 1

− 1

2

(2 k + 2 n + 2m − 2) z12

2 n − 1− 1

2

(4− 4m) z13

2 n − 1,

z2 =1

2

z2 (1 + 2 k)

2 n − 1+

1

2

(4m − 4) z12z2

2 n − 1;

(1)

The singularities at infinity (that is, with z2 = 0) with their local behavior

are:

p1 =

(1

2

m + 2k

m − 1, 0

): stable node; p2 =

(1

2, 0

): saddle; p3 = (0, 0):

stable node.

Poincare comp: SO(2n + 1)/(U(m)× SO(2k + 1)) II

• Chart U2:z1 = −

1

2

(−m − 2 k) z1

2 n − 1−

1

2

(2 k + 2 n + 2m − 2) z12

2 n − 1−

1

2

(−4m + 4) z13

2 n − 1,

z2 =1

2

z2 (1 + 2 k)

2 n − 1+

1

2

(4m − 4) z12z2

2 n − 1.

The singularities at infinity with their local behavior are:

q1 = (2, 0) (≡ p2): saddle; q2 =

(2(m − 1)

m + 2k, 0

)(≡ p1): stable node.

Poincare comp: SO(2n + 1)/(U(m)× SO(2k + 1)) III

Using the coordinates maps φi , ψi , i = 1, 2, 3, see that these singularities

correspond to the following points in S2:

p1 =

(2(m − 1)

√5m2 − 8m + 4mk + 4k2 + 4

,(m + 2k)

√5m2 − 8m + 4mk + 4k2 + 4

, 0

)(stable

node);

p2 =

(2

5

√5,

1

5

√5, 0

)(saddle);

p3 = (1, 0, 0) (stable node).

Main result for SO(2n + 1)/(U(m)× SO(2k + 1))

Theorem

The invariant lines f+ ◦ γ1, f+ ◦ γ2 and f+ ◦ γ3 satisfy the following:

(i) f+(γ1(0)) = f+(γ2(0)) = f+(γ3(0)) = 0;

(ii) limt→−∞ f+(γ1(t)) = p1;

(iii) limt→−∞ f+(γ2(t)) = p2;

(iv) limt→−∞ f+(γ3(t)) = p3.

Main result for SO(2n + 1)/(U(m)× SO(2k + 1))

Main result for SO(2n + 1)/(U(m)× SO(2k + 1)) I

The following result is the main theorem for the flag manifold

SO(2n + 1)/(U(m)× SO(2k + 1)).

Theorem

Let R1,R2 and R3 be the distinguished open regions in the last figure,

and let ϕt be the flow of the projection of p(X ) to D2.

(a) If (x0, y0) ∈ R1 then limt→−∞ ϕt(x0, y0) = q;

(b) If (x0, y0) ∈ R2 ∪ γ2 ∪ R3 then limt→−∞ ϕt(x0, y0) = p2;

(c) If (x0, y0) ∈ γ1 then limt→−∞ ϕt(x0, y0) = p1.

Complete classification of the flow!

Bonus: Einstein metrics for full flag manifolds SU(n)/T n−1

Since the Ricci flow equation for a homogeneous manifold is a system of

ODEs, the singularities of this system are something important to

calculate.

A metric g is called an Einstein metric if Ric(g) = λg . The constant λ is

called Einstein constant.

So the Einstein metrics are the singularities of the system of ODEs

produced by the Ricci flow.

Or: the Einstein metrics are the solutions of a system of polynomials

equations in several variables.


In general:

Theorem (Wang-Ziller)

G/T full flag manifold. For each α ∈ R+ the Ricci component rαcorresponding to the isotropy summand uα is given by

rα =1

2λα +

1

8

∑β,γ∈R+

λαλβλγ

[α

βγ

]− 1

4

∑β,γ∈R+

λγλαλβ

[γ

αβ

].


Definition of

[i

j k

]: Let G/H be a compact homogeneous space of a

compact semissimple Lie group G whose the isotropy representation m

decomposes into k pairwise inequivalent irreducible Ad(H)-submodules

mi as m = m1 ⊕ . . .⊕mk . We choose a Q-orthonormal basis {ep}adapted to m =

⊕ki=1 mi . Let Ar

pq = Q([ep, eq], er ) so that

[ep, eq]m =∑

Arpqer , and set[i

j k

]=∑

(Arpq)2 =

∑(Q([ep, eq], er ))2,

where the sum is taken over all indices p, q, r with ep ∈ mi , eq ∈ mj and

er ∈ mk .


For SU(n)/T :

Theorem (Sakane)

The components of the Ricci tensor of an invariant metric on SU(n +

1)/T are given by

rij = rαij =1

2λij+

1

4(n + 1)+∑k 6=i,j

(λij

λikλkj− λikλijλkj

− λjkλijλik

).


Example (SU(3)/T 2, Arvanitoyergos 1993)

In this case the Einstein equations are given by

r12 =1

x12+

1

12

(x12

x13x23− x13

x12x23− x23

x12x13

)= k

r13 =1

x13+

1

12

(x13

x12x23− x12

x13x23− x23

x12x13

)= k

r23 =1

x23+

1

12

(x23

x12x13− x13

x12x23− x12

x23x13

)= k.

Solutions: the SU(3)-invariant Einstein metrics:

(1, 1, 1), (2, 1, 1), (1, 2, 1), (1, 1, 2).


Arvanitoyergos, 1993: SU(n + 1)/T n admits at least(n + 1)!

2+ n + 1

invariant Einstein metrics.

Recently, Wang and Li proved that there are exactly 29 SU(4)-invariant

Einstein metrics on SU(4)/T 3.

Theorem (Grama-R.)

SU(4)/T 3 admits at least 29 invariant Einstein metrics.


Theorem (Grama-R.)

SU(5)/T 4 admits at least 396 invariant Einstein metrics, classified into

12 classes up to isometries and homoteties.

Conjecture

SU(5)/T 4 admits exactly 396 invariant Einstein metrics, classifiedinto 12 classes up to isometries and homoteties.

Theorem (Grama-R.)

SU(6)/T 5 admits at least 3941 invariant Einstein metrics, classified into

at least 35 classes up to isometries and homoteties.


Proofs.

# Almost exhaustive search

# Probabilistic algorithm

# SU(5)/T : 9 (homogeneous) polynomial equations in 9 variables,108 initial conditions (target in 105).

# SU(6)/T : 14 (homogeneous) polynomial equations in 14variables, 106 initial conditions (not stable).

# Both cases: the estimate of Arvanitoyergos is improved.

References I

A. Arvanitoyergos, I. Chrysikos, Invariant Einstein metrics on flag

manifolds with four isotropy summands, Ann. Glob. Anal. Geom. 37(2010), 185–219.

A. Arvanitoyergos, I. Chrysikos, Y. Sakane, Homogeneous Einstein

metrics on G2/T , Proc. AMS 141 (2013), 2485–2499.

A. Arvanitoyergos, New invariant einstein metrics on generalized

flag manifolds. Trans. Amer. Math. Soc. 337, 981-995 (1993).

C. Bohm, B. Wilking, Nonnegatively curved manifolds with finite

fundamental groups admit metrics with positive Ricci curvature,

GAFA Geometric and Functional Analysis 17, 665-681 (2007).

J. Lauret, Ricci flow of homogeneous manifolds. MathematischeZeitschrift 274, 373–403 (2013)

References II

Y. Sakane, Homogeneous Einstein metrics on flag manifolds,Lobachevskii J. Math. 4 (1999), 71–87.

Y. Wang, T. Li, Invariant Einstein metrics on SU(4)/T . Adv. Math.(China) 43 (2014), no. 5, 781 – 788.

M.Wang, W.ZillerThe existence and nonexistence of

homogeneous Einstein metric. Invent. Math 84, 177–194 (1986).

Thank you!

ricci flow in homogeneous manifolds - ime.unicamp.brrmiranda/talks/palestra-geometria.pdf · oops,...

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