Homogeneous Ricci curvature and the beta operator

(1)

Homogeneous Ricci curvature and the beta operator

Jorge Lauret

Universidad Nacional de C´ordoba - CONICET

Sao Paulo, July 23th, 2018

J. Lauret Homogeneous Ricci Sao Paulo, July 23th, 2018 1 / 17

(2)

CURVATURE TOPOLOGY

(3)

CURVATURE TOPOLOGY .%

SYMMETRY Lie groups

(4)

CURVATURE TOPOLOGY

&- .%

SYMMETRY Lie groups

(5)

CURVATURE

Ricci TOPOLOGY

&- .%

SYMMETRY Lie groups

(6)

CURVATURE Ricci

Einstein, Ricci solitons, ...

Ricci pinching Ricci <0

TOPOLOGY

&- .%

SYMMETRY Lie groups

Structure Uniqueness Classification Compact quotients

(7)

Ricci pinching functional

C_homog :={n−dim non-flat homogeneous Riemannian manifolds}.

F :C_homog −→R, F(g) = scal²_g

|Ric_g |².

F ≤n and equality holds atEinstein metrics (i.e. Ricg =cg). n−1<F(g) ⇒Ric_g definite.

Among a particular subclass C ⊂ C_homog, any g ∈ C can be at most qsupF|C

n -Ricci pinched.

|Rm_g| ≤C_n|Ric_g |, for all g ∈ C_homog,[B¨ohm-Lafuente-Simon, 16]. C_G_/K :={G−invariant metrics onG/K} infF|_C_G/K, supF|_C_G/K, nice invariants of G/K, up to equivariant diffeomorphism (canonical metrics).

(8)

Ricci pinching functional

|Ric_g |².

F ≤n and equality holds atEinstein metrics (i.e. Ricg =cg). n−1<F(g) ⇒Ric_g definite.

n -Ricci pinched.

(9)

Ricci pinching functional

|Ric_g |².

F ≤n and equality holds atEinstein metrics (i.e. Ricg =cg).

n−1<F(g) ⇒Ric_g definite.

n -Ricci pinched.

(10)

Ricci pinching functional

|Ric_g |².

n−1<F(g)⇒ Ric_g definite.

n -Ricci pinched.

(11)

Ricci pinching functional

|Ric_g |².

n -Ricci pinched.

(12)

Ricci pinching functional

|Ric_g |².

n -Ricci pinched.

Very useful combined with

(13)

Ricci pinching functional

|Ric_g |².

n -Ricci pinched.

|Rm_g| ≤C_n|Ric_g |, for all g ∈ C_homog,[B¨ohm-Lafuente-Simon, 16].

C_G_/K :={G−invariant metrics onG/K} infF|_C_G/K, supF|_C_G/K, nice invariants of G/K, up to equivariant diffeomorphism (canonical metrics).

(14)

Ricci pinching functional

|Ric_g |².

n -Ricci pinched.

C_G_/K :={G −invariant metrics onG/K} infF|_C_G/K, supF|_C_G/K, nice invariants of G/K, up to equivariant diffeomorphism

(canonical metrics).

(15)

Ricci pinching functional

|Ric_g |².

n -Ricci pinched.

C_G_/K :={G −invariant metrics onG/K} infF|_C_G/K, supF|_C_G/K, nice invariants of G/K, up to equivariant diffeomorphism (canonical metrics).

(16)

G/K homogeneous space,

C_G_/K :={G−invariant metrics onG/K}.

Under the absence of Einstein metrics on G/K (e.g.G unimodular), What kind of metrics can be globalmaxima of F|_C

G/K?

Strong candidates for such a distinction are of courseRicci solitons, i.e. Ric_g =cg +L_Xg, X ∈X(G/K)

⇔ g(t) =c(t)ψ(t)^∗g, ψ(t)∈Diff(G/K), Ricci flow. (G/K,g) semi-algebraicRicci soliton: ψ(t)∈Aut(G/K),

⇔ Ricg =cI+D|_p+ (D|_p)^t, D∈Der(g), g=k⊕p. (G/K,g) algebraicRicci soliton: in addition g(t) diagonal,

⇔ Ricg =cI +D|_p, D ∈Der(g), g=k⊕p.

(17)

Under the absence of Einstein metrics on G/K (e.g.G unimodular),

What kind of metrics can be globalmaxima of F|_C

G/K?

(18)

G/K?

(19)

G/K?

Strong candidates for such a distinction are of courseRicci solitons, i.e.

Ric_g =cg +L_Xg, X ∈X(G/K)

⇔ g(t) =c(t)ψ(t)^∗g, ψ(t)∈Diff(G/K), Ricci flow.

(G/K,g) semi-algebraicRicci soliton: ψ(t)∈Aut(G/K),

(20)

G/K?

(21)

G/K?

⇔ Ricg =cI+D|_p+ (D|_p)^t, D∈Der(g), g=k⊕p.

(G/K,g) algebraicRicci soliton: in addition g(t) diagonal,

(22)

G/K?

(23)

G/K?

(24)

The beta operator

(N,g) nilpotent Lie group endowed with a left-invariant metric, dimN =m.

(N,g) β :n−→n, β^t =β, called the beta operator of (N,g)[L 07].

Spec(β) ={b₁, . . . ,bm} depends only onN (only finitely many possibilities in a given dimension).

trβ=−1, |β|² > _m¹, β₊ :=β+|β|²I >0 [L 07].

F(g)≤ _|β|¹₂ , where equality holds⇔ (N,g) is a Riccisoliton. [L 10] F increases along the Ricci flow. Ricci solitons are therefore the only (local) global maxima of F on the set of all left-invariant metrics on a nilpotent Lie group.

(25)

The beta operator

(N,g) β :n−→n, β^t =β, called the beta operatorof (N,g)[L 07].

trβ=−1, |β|² > _m¹, β₊ :=β+|β|²I >0 [L 07].

(26)

The beta operator

Spec(β) ={b₁, . . . ,bm} depends only onN

(only finitely many possibilities in a given dimension).

trβ=−1, |β|² > _m¹, β₊ :=β+|β|²I >0 [L 07].

(27)

The beta operator

trβ=−1, |β|² > _m¹, β₊ :=β+|β|²I >0 [L 07].

(28)

The beta operator

trβ=−1,

|β|² > _m¹, β₊ :=β+|β|²I >0 [L 07].

(29)

The beta operator

trβ=−1, |β|² > _m¹,

β₊ :=β+|β|²I >0 [L 07].

(30)

The beta operator

trβ=−1, |β|² > _m¹, β₊ :=β+|β|²I >0 [L 07].

(31)

The beta operator

trβ=−1, |β|² > _m¹, β₊ :=β+|β|²I >0 [L 07].

(N,g) is a Riccisoliton⇔ Ric_g =|scal_g |β

(32)

The beta operator

trβ=−1, |β|² > _m¹, β₊ :=β+|β|²I >0 [L 07].

(N,g) is a Riccisoliton⇔ Ric_g =|scal_g |β (β₊∈Der(n)).

(33)

The beta operator

trβ=−1, |β|² > _m¹, β₊ :=β+|β|²I >0 [L 07].

(34)

The beta operator

trβ=−1, |β|² > _m¹, β₊ :=β+|β|²I >0 [L 07].

F(g)≤ _|β|¹₂ , where equality holds⇔ (N,g) is a Riccisoliton.

[L 10] F increases along the Ricci flow. Ricci solitons are therefore the only (local) global maxima of F on the set of all left-invariant metrics on a nilpotent Lie group.

(35)

The beta operator

trβ=−1, |β|² > _m¹, β₊ :=β+|β|²I >0 [L 07].

[L 10] F increases along the Ricci flow.

Ricci solitons are therefore the only (local) global maxima of F on the set of all left-invariant metrics on a nilpotent Lie group.

(36)

The beta operator

trβ=−1, |β|² > _m¹, β₊ :=β+|β|²I >0 [L 07].

[L 10] F increases along the Ricci flow. Ricci solitons are therefore the only (local) global maxima of F on the set of all left-invariant metrics on a nilpotent Lie group.

(37)

(G/K,g) homogeneous space endowed with a G-invariant metric, dimG/K =n,G non-semisimple.

g=k⊕p, B(k,p) = 0 ⇒ n⊂p,nthe nilradical ofg, dimn=m. (G/K,g) β_p:p−→p, β_p:=h

−|β|²I β

i

, p=h⊕n, called the beta operator of (G/K,g) [L 07].

Spec(β) =

−P

b²_i,b1, . . . ,bm depends only onG/K (orn). Only finitely many possibilities in a given dimension.

[Lafuente-L 12](G/K,g) is a semi-algebraicsoliton ⇔

D+:= h₀

0 β+

i

∈Der(g), β+=β+|β|²I >0 (⇔ [h,h]⊂k⊕h, [β,adh|_n] = 0, β₊∈Der(n)).

(G/K,g) is Einstein⇔ in addition S(ad_pH) =|scal_N|D₊|_p .

(38)

g=k⊕p, B(k,p) = 0

⇒ n⊂p,nthe nilradical ofg, dimn=m. (G/K,g) β_p:p−→p, β_p:=h

−|β|²I β

i

Spec(β) =

−P

D+:= h₀

0 β+

i

(39)

g=k⊕p, B(k,p) = 0 ⇒ n⊂p,nthe nilradical ofg, dimn=m.

(G/K,g) β_p:p−→p, β_p:=h

−|β|²I β

i

Spec(β) =

−P

D+:= h₀

0 β+

i

(40)

−|β|²I β

i

, p=h⊕n, called the beta operatorof (G/K,g) [L 07].

Spec(β) =

−P

D+:= h₀

0 β+

i

(41)

−|β|²I β

i

Spec(β) =

−P

b²_i,b1, . . . ,bm depends only onG/K (orn).

Only finitely many possibilities in a given dimension.

D+:= h₀

0 β+

i

(42)

−|β|²I β

i

Spec(β) =

−P

D+:= h₀

0 β+

i

(43)

−|β|²I β

i

Spec(β) =

−P

[Lafuente-L 12](G/K,g) is a semi-algebraicsoliton⇔ Ric^∗_g =|scal_N|β_p

(discovered in [B¨ohm-Lafuente 17]); in that case, Ricg = scal_N|β|²I−scal_ND+|_p−S(ad_pH),

D+:= h₀

0 β+

i

(44)

−|β|²I β

i

Spec(β) =

−P

[Lafuente-L 12](G/K,g) is a semi-algebraicsoliton⇔ Ric^∗_g =|scal_N|β_p (discovered in [B¨ohm-Lafuente 17]);

in that case, Ricg = scal_N|β|²I−scal_ND+|_p−S(ad_pH),

D+:= h₀

0 β+

i

(45)

−|β|²I β

i

Spec(β) =

−P

[Lafuente-L 12](G/K,g) is a semi-algebraicsoliton⇔

D+:= h₀

0 β+

i

(46)

−|β|²I β

i

Spec(β) =

−P

D+:=

h₀

0 β+

i

∈Der(g), β+ =β+|β|²I >0

(⇔ [h,h]⊂k⊕h, [β,adh|_n] = 0, β₊∈Der(n)).

(47)

−|β|²I β

i

Spec(β) =

−P

D+:=

h₀

0 β+

i

∈Der(g), β+ =β+|β|²I >0 (⇔ [h,h]⊂k⊕h, [β,adh|_n] = 0, β₊∈Der(n)).

(48)

−|β|²I β

i

Spec(β) =

−P

D+:=

h₀

0 β+

i

∈Der(g), β+ =β+|β|²I >0 (⇔ [h,h]⊂k⊕h, [β,adh|_n] = 0, β₊∈Der(n)).

(49)

(G/K,g), dimG/K =n,G non-semisimple, g=k⊕p,n⊂p.

(G/K,g) β_p :p−→p, β_p :=h_−|β|₂

I β

i

, p=h⊕n. [B¨ohm-Lafuente 17] |Ric^∗| ≥ −scal^∗

n−m+P¹ b²_i

−¹₂

, where equality holds⇔ (G/K,g) is a semi-algebraicsoliton

(Ric^∗_g =|scalN|βp).This estimate was used to prove the convergence of any immortal homogeneous Ricci flow to a homogeneous

expanding Ricci soliton.

For G unimodular, scal≤0, the above estimate can be rewritten as F(g)≤n−m+ 1

Pb_i² <n.

For G unimodular and non-semisimple, semi-algebraicsolitons are therefore global maximaof F on the set of all G-invariant metrics with scal≤0 on G/K. Are there other global or local maxima ??

(50)

(G/K,g) β_p :p−→p, β_p :=h

−|β|²I β

i

, p=h⊕n.

[B¨ohm-Lafuente 17] |Ric^∗| ≥ −scal^∗

n−m+P¹ b²_i

−¹₂

Pb_i² <n.

(51)

(G/K,g) β_p :p−→p, β_p :=h

−|β|²I β

i

, p=h⊕n.

n−m+^P¹_b2 i

−¹₂

(Ric^∗_g =|scalN|βp).

This estimate was used to prove the convergence of any immortal homogeneous Ricci flow to a homogeneous

Pb_i² <n.

(52)

(G/K,g) β_p :p−→p, β_p :=h

−|β|²I β

i

, p=h⊕n.

n−m+^P¹_b2 i

−¹₂

Pb_i² <n.

(53)

(G/K,g) β_p :p−→p, β_p :=h

−|β|²I β

i

, p=h⊕n.

n−m+^P¹_b2 i

−¹₂

Pb_i² <n.

(54)

(G/K,g) β_p :p−→p, β_p :=h

−|β|²I β

i

, p=h⊕n.

n−m+^P¹_b2 i

−¹₂

Pb_i² <n.

For G unimodular and non-semisimple, semi-algebraicsolitons are therefore global maximaof F on the set of all G-invariant metrics with scal≤0 on G/K.

Are there other global or local maxima ??

(55)

(G/K,g) β_p :p−→p, β_p :=h

−|β|²I β

i

, p=h⊕n.

n−m+^P¹_b2 i

−¹₂

Pb_i² <n.

(56)

Behavior of F beyond unimodular

Joint work with Cynthia Will.

S almost-abelian Lie group (i.e. shas a codimension one abelian ideal), dimS =n.

S ←→ A_S ∈gl_n−1(R), matrix. (up to conjugation and nonzero scaling).

S unimodular ⇔ trA_S = 0. S nilpotent ⇔A_S nilpotent.

S admits an Einstein metric ⇔the real semisimple part of A_S is cI. [Arroyo 12] S admits asolvsolitononS ⇔AS is either semisimple or nilpotent.

[Jablonski 14] S admits a Ricci soliton which is not a solvsoliton⇔ Spec(A_S)⊂iRandA_S is neither semisimple nor nilpotent.

Which are the local and global maxima of F :C_S −→R??

(57)

Behavior of F beyond unimodular

S ←→ A_S ∈gl_n−1(R), matrix. (up to conjugation and nonzero scaling).

S unimodular ⇔ trA_S = 0. S nilpotent ⇔A_S nilpotent.

(58)

Behavior of F beyond unimodular

S ←→ A_S ∈gl_n−1(R), matrix.

(up to conjugation and nonzero scaling). S unimodular ⇔ trA_S = 0.

S nilpotent ⇔A_S nilpotent.