text: K-ENERGY ON POLARIZED COMPACTIFICATIONS OF LIE GROUPS 19 (cid:90) (cid:90) (cid:90) (cid:88) ≤ (cid:104) ∇ (cid:105) L (cid:90) − 2 (cid:104) (cid:105) 0 4 ρ, π u dy = (u) + S¯ uπ dy u y, ν π dσ λ A 0 2P+ 2P+ A A FA ≤ L (cid:104) (cid:105) (4.12) (u) + C u y, ν π dσ . 0 ∂(2P+) Moreover, (cid:90) (cid:12) (cid:12) (cid:90) (cid:12) (cid:12) (cid:12) (cid:12) ≤ (4.13) uij uπ dy C uπ dy, 0,ij 1 (cid:90) (cid:12) 2P(cid:12)+ (cid:90) 2P+ (cid:12) (cid:12) (cid:12) (cid:12) ≤ (cid:104) ∇ (cid:90)(cid:105) uij π u dy C ρ, π u dy 0,j ,i 2 2P+ 2P+ ≤ L (cid:48) (cid:104) (cid:105) (4.14) C (u) + C u y, ν π dσ , 2 2 0 ∂(2P+) (cid:48) since uij , uij are smooth up to the boundary, where C , C , C > 0 are con- 0,ij 0,j 1 2 2 stants independent of u. Hence, substituting (4.12), (4.13) and (4.14) into (4.11), (cid:3) we obtain (4.7). Since Q is singular and π vanishes along each W , we shall give an explicit α estimate for the singular order of Q. In the following, we will divide 2P into two (cid:48)∪ ≥ parts 2P = 2P U, where U is a union of small neighborhoods of faces of codim 2 ∪ (cid:48) (cid:48) ∩ ∗ (cid:48) which are contained in ∈ W , and 2P = 2P a¯ , where 2P is a W-invariant α Φ+ α + + polytope whose boundary intersects the Weyl walls orthogonally. By Proposition 4.4, to ﬁnish the proof of Proposition 4.1, it suﬃces to prove (cid:12) (cid:12) Propos(cid:12)i(cid:90)tion 4.5. T(cid:12)here ar(cid:90)e constants C , C > 0(cid:90)independent of u such that (cid:12) (cid:12) I II (cid:12) (cid:12) (cid:12) (cid:12) ≤ (cid:104) ∇ (cid:105) ∀ ∈ C Quπ dy C ρ, π u dy + C uπ dy, u ˆ∞ . I II ,W 2P+ 2P+ 2P+ It is easy to see th(cid:90)at Qπ is uniforml(cid:90)y bounded in 2P . Then + ≤ ∀ ∈ C (4.15) Quπ dy C u dy, u ˆ∞ . ,W (cid:48) (cid:48) 2P 2P + + In this subsection, we further prove ##LTLine##

text: K-ENERGY

text: ON

text: POLARIZED

text: COMPACTIFICATIONS

text: OF

text: LIE

text: GROUPS

text: L

text: ρ,

text: π

text: u

text: dy

text: (u)

text: S¯

text: uπ

text: dy

text: u

text: y,

text: ν

text: π

text: dσ

text: λ

text: A

text: A

text: A

text: FA

text: L

text: (u)

text: C

text: u

text: y,

text: ν

text: π

text: dσ

text: Moreover,

text: uij

text: uπ

text: dy

text: C

text: uπ

text: dy,

text: uij

text: π

text: u

text: dy

text: C

text: ρ,

text: π

text: u

text: dy

text: ,i

text: L

text: C

text: (u)

text: C

text: u

text: y,

text: ν

text: π

text: dσ

text: since

text: uij

text: uij

text: are

text: smooth

text: up

text: to

text: the

text: boundary,

text: where

text: C

text: C

text: C

text: are

text: con-

text: stants

text: independent

text: of

text: u.

text: Hence,

text: substituting

text: and

text: into

text: we

text: obtain

text: Since

text: Q

text: is

text: singular

text: and

text: π

text: vanishes

text: along

text: each

text: W

text: we

text: shall

text: give

text: an

text: explicit

text: α

text: estimate

text: for

text: the

text: singular

text: order

text: of

text: Q.

text: In

text: the

text: following,

text: we

text: will

text: divide

text: into

text: two

text: parts

text: U,

text: where

text: U

text: is

text: a

text: union

text: of

text: small

text: neighborhoods

text: of

text: faces

text: of

text: codim

text: which

text: are

text: contained

text: in

text: W

text: and

text: a¯

text: where

text: is

text: a

text: W-invariant

text: α

text: Φ+

text: α

text: polytope

text: whose

text: boundary

text: intersects

text: the

text: Weyl

text: walls

text: orthogonally.

text: By

text: Proposition

text: to

text: ﬁnish

text: the

text: proof

text: of

text: Proposition

text: it

text: suﬃces

text: to

text: prove

text: constants

text: C

text: C

text: of

text: u

text: such

text: that

text: I

text: II

text: C

text: Quπ

text: dy

text: C

text: ρ,

text: π

text: u

text: dy

text: C

text: uπ

text: dy,

text: u

text: ˆ∞

text: I

text: II

text: ,W

text: It

text: is

text: easy

text: to

text: see

text: Qπ

text: is

text: bounded

text: in

text: Then

text: C

text: Quπ

text: dy

text: C

text: u

text: dy,

text: u

text: ˆ∞

text: ,W

text: In

text: this

text: subsection,

text: we

text: further

text: prove

text: ##LTLine##


text: K-ENERGY ON POLARIZED COMPACTIFICATIONS OF LIE GROUPS 23 (cid:18) (cid:19) (cid:88) (cid:104) ∇ (cid:105) 1 2α(u )α(y) (4.22) α, u = α(u ) log 1 + | | a 0 2 a α 2l (y) a a and (cid:32) (cid:33) (cid:32) (cid:33) (cid:88) (cid:88) 1 ui uj ui uj 1 uiuj (4.23) u = a a + a,α a,α + b b . 0,ij 2 l (y) l (y) 2 l (y) a a,α b a b ∈  (cid:32) (cid:33) Lemma 4.8. Let y W . Then 0 (cid:88)α − 1  (α(ua))2 |α|4 + O(α(y)), if α(y) → 0, ∀ a, ui0jαiαj = a la(y) la(y) O(α(y)), otherwise. Proof. Since l (y) >(cid:34)l (y) > 0, we have (cid:35) a,α (cid:88)a (cid:0) (cid:1) (cid:88) 1 (cid:34) 1 uiuj (cid:35) 0 < M1 := 2 (cid:88)l (y) (cid:0)uiauja + uia,αuja,α (cid:1)+ (cid:88)l b(yb) a,α b a b ≤ ≤ 1 1 uiuj (u ) ui uj + ui uj + b b =: M . 0,ij 2 l (y) a a a,α a,α l (y) 2 a b a b It is easy to see (cid:32) (cid:33) (cid:32) (cid:33) (cid:88) (cid:88) (α(u ))2 (α(u ))2 (4.24) M α = | | a α, M α = | | a α. 1 α 2l (y) 2 α 2l (y) a,α a a a Thus (cid:32) (cid:33) (cid:88) − 1 − (α(ua))2 = αT|M|2 1α  l (y) α 4(cid:88) a a   − − 1 ≤ uijα α ≤ αT M 1α (α(u ))2 0| |i j | |1 = a . α 4 α 4 l (y) + 2α|(u|a)α(y) a a α 2 → Moreover, if α(y) 0 for all a, by Lemma 7.1 in Appendix, la(y) ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: K-ENERGY

label: paragraph

text: ON

text: POLARIZED

text: COMPACTIFICATIONS

text: OF

text: LIE

text: GROUPS

label: equation

text: )α(y)

text: α,

text: u

text: α(u

text: log

text: a

text: a

text: α

text: (y)

text: a

text: a

text: and

text: ui

text: uj

text: ui

text: uj

text: uiuj

text: u

text: a

text: a

text: a,α

text: a,α

text: b

text: b

text: l

text: (y)

text: l

text: (y)

text: l

text: (y)

text: a

text: a,α

text: b

text: a

text: b

text: Lemma

text: Let

text: y

text: W

text: Then

text: 0

text: O(α(y)),

text: if

text: α(y)

text: a,

text: a

text: la(y)

text: la(y)

text: O(α(y)),

text: otherwise.

text: Proof.

text: Since

text: l

text: (y)

text: (y)

text: we

text: have

text: a,α

text: uiuj

text: (y)

text: uia,αuja,α

text: b(yb)

text: a,α

text: b

text: a

text: b

text: uiuj

text: (u

text: ui

text: uj

text: ui

text: uj

text: b

text: b

text: M

text: l

text: (y)

text: a

text: a

text: a,α

text: a,α

text: l

text: (y)

text: a

text: b

text: a

text: b

text: It

text: is

text: easy

text: to

text: see

text: (α(u

text: (α(u

text: M

text: α

text: a

text: α,

text: M

text: α

text: a

text: α.

text: α

text: (y)

text: α

text: (y)

text: a,α

text: a

text: a

text: a

text: Thus

text: 1α

text: l

text: (y)

text: α

text: a

text: a

text: uijα

text: α

text: αT

text: M

text: 1α

text: (α(u

text: |i

text: j

text: a

text: α

text: α

text: l

text: (y)

text: a

text: a

text: α

text: Moreover,

text: if

text: α(y)

text: for

text: all

text: a,

text: by

text: Lemma

text: in

text: Appendix,

text: la(y)

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: ∗ ∗∗ 24 YAN LI BIN ZHOU XIAOHUA ZHU   (cid:32) (cid:33) (cid:88) (cid:88)   −1 −1 ≤ (α(u ))2 − (α(u ))2 0 a a a la(y) + 2α|(αu|a2)α(y) a la(y) (4.25) = O(α(y)). (cid:32) (cid:33) (cid:88) This implies − 1 uijα α ≤ (α(u ))2 0| |i j a + O(α(y)). α 4 l (y) a a The ﬁrst case is proved. ∈ (cid:54) ≥ In the second case, there exists an F F such that α(u ) = 0 and α(y) (cid:15) a0 α a0 la0(y) 0 for some (cid:15)0 > 0. Then (cid:32) (cid:33) (cid:88) − 1 (α(u ))2 (4.26) a = O(l (y)) l (y) a0 a a and   (cid:88)   − 1 (α(u ))2 (4.27) a = O(α(y)). l (y) + 2α|(u|a)α(y) a a α 2 Thus uijα α ≤ 0| |i j O(l (y)) + O(α(y)) = O(α(y)). α 4 a0 (cid:3) The lemma is proved. ∈ Lemma 4.9. Let y W . Suppose that y also lies on another Weyl wall W . → 0 α 0 β Then as y y , it holds (cid:16) (cid:17) 0 − − (4.28) αT (uij) M 1 α =O(α(y)), 0 2 (cid:16) (cid:17) − − (4.29) βT (uij) M 1 β = O(α(y) + β(y)). 0 2 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: YAN

text: LI

text: BIN

text: ZHOU

text: XIAOHUA

text: ZHU

text: −1

text: −1

text: (α(u

text: (α(u

text: a

text: a

text: a

text: la(y)

text: a

text: la(y)

text: O(α(y)).

text: This

text: implies

text: uijα

text: α

text: (α(u

text: |i

text: j

text: a

text: O(α(y)).

text: α

text: l

text: (y)

text: a

text: a

text: The

text: ﬁrst

text: case

text: is

text: proved.

font: default

text: In

text: the

text: second

text: case,

text: there

text: exists

text: an

text: F

text: F

text: such

text: that

text: α(u

text: and

text: α(y)

text: α

text: for

text: some

text: Then

text: (α(u

text: a

text: O(l

text: (y))

text: l

text: (y)

text: a

text: a

text: and

text: (α(u

text: a

text: O(α(y)).

text: l

text: (y)

text: a

text: a

text: α

text: Thus

text: uijα

text: α

text: |i

text: j

text: O(l

text: (y))

text: O(α(y))

text: O(α(y)).

text: α

text: The

text: lemma

text: is

text: proved.

text: Lemma

text: Let

text: y

text: W

text: Suppose

text: that

text: y

text: also

text: lies

text: on

text: another

text: Weyl

text: wall

text: W

text: α

text: β

text: Then

text: as

text: y

text: y

text: it

text: holds

text: αT

text: (uij)

text: M

text: α

text: =O(α(y)),

text: βT

text: (uij)

text: M

text: β

text: O(α(y)

text: β(y)).

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: K-ENERGY ON POLARIZED COMPACTIFICATIONS OF LIE GROUPS 25 Proof. (4.28) follows from the estimate in Lemma 4.8 immediately. It remains to ⊂ prove (4.29). Let S W be the group generated by the reﬂections s and α,β α s . We want to relabel faces of 2P according to this S -action. In each orbit {β } { } − α,β S F , where F is a (r 1)-dimensional face, we take a face F such that α,β A˜ ≥ A˜ c α(u ), β(u ) 0. Let c c { } { } S F = F , ..., F , { ∈ (cid:32) (cid:80)| α,β c (cid:33)−c,s1 c,s}p(c)(cid:32) (cid:80) (cid:33) where F = y(cid:88)∂(2P) l (y) = λ ui y = 0(cid:88). Set c,s c,s c,s c,s i 1 p(c) ui uj 1 p(c) ui uj Mˆ = q=1{ c,sq c,sq} , Mˆ = q=1{ c,sq c,sq} . 1 2 max (l ) (y) 2 2 min (l ) (y) s c,s s c,s c (cid:88) (cid:88) c Then we rewrite u as 0 1 p(c) ui uj u = c,sq c,sq . 0 2 l (y) c q=1 c,sq Thus it is easy to see ≤ ≤ ≤ 0 < Mˆ (u ) M Mˆ . 1 0,ij 2 2 { } Note that S F is invariant under the reﬂection associated to W . Then as α,β c { } ∈ β in the proof of Lemma 4.8, we relabel S F : faces F F ; faces F with ∈ α,β c a β a,β (cid:88) F F ; and Faces F which is orthogonal to W . Thus similar to (4.24), we get a β b β p(c) ui uj β = λ β , c,sq c,sq j c i q=1 (cid:88) ≥ where λ 0 is a constant with at least one λ > 0 since Mˆ > 0. As a consequence, c c 1 1 λ Mˆ β = { c }β 1 2 (cid:88) maxs lc,s(y) c and 1 λ Mˆ β = { c }β. 2 2 min l (y) s c,s c This means that β is an eigenvector of both Mˆ and Mˆ . 1 2 On the other hand, there are constants µc, µc such that 1 2 l (y) = l (y) + µcα(y) + µcβ(y). c,s c 1 2 In particular, { } { } max l (y) = min l (y) + µ˜cα(y) + µ˜cβ(y). s c,(cid:16)s s c(cid:17),s 1 2 Then as in the estimate (4.25) (also see (4.26), (4.27)), we get − − − βT (cid:16)Mˆ 1 Mˆ 1(cid:17) β = O(α(y) + β(y)). 1 2 It follows − − βT (uij) M 1 β = O(α(y) + β(y)). 0 2 The lemma is proved. (cid:3) ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: K-ENERGY

text: ON

text: POLARIZED

text: COMPACTIFICATIONS

text: OF

text: LIE

text: GROUPS

text: Proof.

text: follows

text: from

text: the

text: estimate

text: in

text: Lemma

text: immediately.

text: It

text: remains

text: to

text: prove

text: Let

text: S

text: W

text: be

text: the

text: group

text: generated

text: by

text: the

text: reﬂections

text: s

text: and

text: α,β

text: α

text: s

text: We

text: want

text: to

text: relabel

text: faces

text: of

text: according

text: to

text: this

text: S

text: -action.

text: In

text: each

text: orbit

text: {β

text: α,β

text: S

text: F

text: where

text: F

text: is

text: a

text: (r

text: face,

text: we

text: take

text: a

text: face

text: F

text: such

text: that

text: α,β

text: A˜

text: A˜

text: c

text: α(u

text: β(u

text: Let

text: c

text: c

text: S

text: F

text: F

text: F

text: α,β

text: c

text: where

text: F

text: l

text: (y)

text: λ

text: ui

text: y

text: Set

text: c,s

text: c,s

text: c,s

text: c,s

text: i

text: p(c)

text: ui

text: uj

text: p(c)

text: ui

text: uj

text: Mˆ

text: c,sq

text: c,sq}

text: Mˆ

text: c,sq

text: c,sq}

text: max

text: (l

text: (y)

text: min

text: (l

text: (y)

text: s

text: c,s

text: s

text: c,s

text: c

text: c

text: Then

text: we

text: rewrite

text: u

text: as

text: p(c)

text: ui

text: uj

text: u

text: c,sq

text: c,sq

text: l

text: (y)

text: c

text: c,sq

text: Thus

text: it

text: is

text: easy

text: to

text: see

text: Mˆ

text: (u

text: M

text: Mˆ

text: Note

text: that

text: S

text: F

text: is

text: invariant

text: under

text: the

text: reﬂection

text: associated

text: to

text: W

text: Then

text: as

text: α,β

text: c

text: β

text: in

text: the

text: proof

text: of

text: Lemma

text: we

text: relabel

text: S

text: F

text: faces

text: F

text: F

text: faces

text: F

text: with

text: α,β

text: c

text: a

text: β

text: a,β

text: F

text: F

text: and

text: Faces

text: F

text: which

text: is

text: orthogonal

text: to

text: W

text: Thus

text: similar

text: to

text: we

text: get

text: a

text: β

text: b

text: β

text: p(c)

text: ui

text: uj

text: β

text: λ

text: β

text: c,sq

text: c,sq

text: j

text: c

text: i

text: where

text: λ

text: is

text: a

text: constant

text: with

text: at

text: least

text: one

text: λ

text: since

text: Mˆ

text: As

text: a

text: consequence,

text: c

text: c

text: λ

text: Mˆ

text: β

text: c

text: }β

text: maxs

text: lc,s(y)

text: c

text: and

text: λ

text: Mˆ

text: β

text: c

text: }β.

text: min

text: l

text: (y)

text: s

text: c,s

text: c

text: This

text: means

text: that

text: β

text: is

text: an

text: eigenvector

text: of

text: both

text: Mˆ

text: and

text: Mˆ

text: On

text: the

text: other

text: hand,

text: there

text: are

text: constants

text: µc,

text: µc

text: such

text: that

text: l

text: (y)

text: l

text: (y)

text: µcα(y)

text: µcβ(y).

text: c,s

text: c

text: In

text: particular,

text: max

text: l

text: (y)

text: min

text: l

text: (y)

text: µ˜cα(y)

text: µ˜cβ(y).

text: s

text: s

text: Then

text: as

text: in

text: the

text: estimate

text: (also

text: see

text: we

text: get

text: βT

text: Mˆ

text: β

text: O(α(y)

text: β(y)).

text: It

text: follows

text: βT

text: (uij)

text: M

text: β

text: O(α(y)

text: β(y)).

text: The

text: lemma

text: is

text: proved.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: ∗ ∗∗ 26 YAN LI BIN ZHOU XIAOHUA ZHU (cid:32) (cid:33) (cid:32) (cid:33) (cid:88) (cid:88) Remark 4.10. Since (4.23) can be rewritten as 1 ui uj ui uj 1 uiuj u = a a + a,β a,β + b b 0,ij 2 l (y) l (y) 2 l (y) a a,β b a b for any Weyl wall W , as in the proof of Lemma 4.8 for the second case, one can ∈ β prove: if y / W , then 0 β → (4.30) uijβ β = O(1), as y y . 0 i j 0 Similar to (4.9), (cid:16) (cid:17) − − ∈ \ (4.31) βT (uij) M 1 β = O(1), if y W W . 0 2 0 α β From (4.(cid:88)8), a(cid:20)direct computation shows (cid:21) | | (cid:104) ∇ (cid:105) Q = (cid:88)4 α(cid:20) 2 coth α, u0 − 2 u0,i(cid:104)jαi∇αj (cid:105) − 2uij αiαj (cid:21) α∈Φ+ α(y) (cid:104) (cid:105)sinh2 α, u0 (cid:104) 0 ((cid:105)α(y))2 (cid:104) ∇ (cid:105) α, β (cid:104) ∇ (cid:105) α, β − α β (4.32) + 2 coth α, u + coth β, u 2uij i j . (cid:54) ∈ 0 β(y) 0 α(y) 0 α(y)β(y) α=β Φ+ For simplicity, we denote each term in these two sums by I (y) and I (y), α α,β respectively. We need to estimate them in the following key lemma. ∈ Lemma 4.11. Let y W . Then there exist C , C > 0, such that 0 α α,y0 α,β,y0 | | ≤ C (4.33) I (y) α,y0 α α(y) and (cid:18) (cid:19) | | ≤ 1 1 (4.34) I (y) C + α,β α,β,y0 α(y) β(y) → as y y . 0 → Proof. We consider the following three cases as y y : 0 ≤ ∀ ∈ (i) α(y) (cid:15) << 1, F F . la(y) 0 a α ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: YAN

text: LI

text: BIN

text: ZHOU

text: XIAOHUA

text: ZHU

text: Remark

text: Since

text: can

text: be

text: rewritten

text: as

text: ui

text: uj

text: ui

text: uj

text: uiuj

text: u

text: a

text: a

text: a,β

text: a,β

text: b

text: b

text: l

text: (y)

text: l

text: (y)

text: l

text: (y)

text: a

text: a,β

text: b

text: a

text: b

text: for

text: any

text: Weyl

text: wall

text: W

text: as

text: in

text: the

text: proof

text: of

text: Lemma

text: for

text: the

text: second

text: case,

text: one

text: can

text: β

text: prove:

text: if

text: y

text: W

text: then

text: β

text: uijβ

text: β

text: as

text: y

text: y

text: i

text: j

text: Similar

text: to

text: βT

text: (uij)

text: M

text: β

text: if

text: y

text: W

text: W

text: α

text: β

text: From

text: computation

text: shows

text: Q

text: coth

text: α,

text: αiαj

text: α∈Φ+

text: α(y)

text: α,

text: α,

text: β

text: α,

text: β

text: α

text: β

text: coth

text: α,

text: u

text: coth

text: β,

text: u

text: i

text: j

text: β(y)

text: α(y)

text: α(y)β(y)

text: α=β

text: Φ+

text: For

text: simplicity,

text: we

text: denote

text: each

text: term

text: in

text: these

text: two

text: sums

text: by

text: I

text: (y)

text: and

text: I

text: (y),

text: α

text: α,β

text: respectively.

text: We

text: need

text: to

text: estimate

text: them

text: in

text: the

text: following

text: key

text: lemma.

text: Lemma

text: Let

text: y

text: W

text: Then

text: there

text: exist

text: C

text: C

text: such

text: that

text: α

text: C

text: I

text: (y)

text: α

text: α(y)

text: and

text: I

text: (y)

text: C

text: α,β

text: α(y)

text: β(y)

text: as

text: y

text: y

text: Proof.

text: We

text: consider

text: the

text: following

text: three

text: cases

text: as

text: y

text: y

text: (i)

text: α(y)

text: F

text: F

text: la(y)

text: a

text: α

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: ∗ ∗∗ 34 YAN LI BIN ZHOU XIAOHUA ZHU ∈ (5.2) bar 4ρ + Ξ. X · H Then µX ( ) is proper on × (ω ) modulo Z(G). ω0 K K 0 Since the properness of the modiﬁed K-energy implies the existence of K¨ahler- Ricci solitons [28], Theorem 5.1 gives a proof for the existence of K¨ahler-Ricci solitons under the condition (5.2). As in the proof of Proposition 3.4, one can also show that (5.2) is a necessary condition by using the computation as for toric manifolds [30]. The following is a generalization of Proposition 3.1 in [30]. ∈ H Proposition 5.2. Let φ × (ω ) and u be the Legendre function of ψ = ψ +φ. K K 0 0 Then 1 K µX (φ) = X(u) + const., ω0 V K N L where X(u) = X(u) + X(u), and (cid:90) L (cid:104) − ∇ (cid:105) X(u) = y 4ρ, u eθX(y)π dy, . 2P+ (cid:90) N − − ∇ − (cid:104) ∇ (cid:105) (5.3) X(u) = (log det (uij) χ ( u) 4 ρ, u ) eθX(y)π dy. 2P+ Proof. By (2.2) and (3.12), we reduce ωωφnn e(cid:81)φ−h0 to a function on a+ by 0 (cid:104) ∇ (cid:105) ωn − MAR(ψ) ∈ α, ψ 2 φ eφ h0 = α Φ+ eψ. ωn J(x) 0 Then ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: YAN

text: LI

text: BIN

text: ZHOU

text: XIAOHUA

text: ZHU

text: bar

text: 4ρ

text: Ξ.

text: X

text: H

text: Then

text: µX

text: is

text: proper

text: on

text: (ω

text: modulo

text: Z(G).

text: ω0

text: K

text: K

text: Since

text: the

text: properness

text: of

text: the

text: modiﬁed

text: K-energy

text: implies

text: the

text: existence

text: of

text: K¨ahler-

text: Ricci

text: solitons

text: Theorem

text: gives

text: a

text: proof

text: for

text: the

text: existence

text: of

text: K¨ahler-Ricci

text: solitons

text: under

text: the

text: condition

text: As

text: in

text: the

text: proof

text: of

text: Proposition

text: one

text: can

text: also

text: show

text: that

text: is

text: a

text: necessary

text: condition

text: by

text: using

text: the

text: computation

text: as

text: for

text: toric

text: manifolds

text: The

text: following

text: is

text: a

text: generalization

text: of

text: Proposition

text: in

text: H

text: Proposition

text: Let

text: φ

text: (ω

text: and

text: u

text: be

text: the

text: Legendre

text: function

text: of

text: ψ

text: ψ

text: +φ.

text: K

text: K

text: Then

text: K

text: µX

text: (φ)

text: X(u)

text: const.,

text: ω0

text: V

text: K

text: N

text: L

text: where

text: X(u)

text: X(u)

text: X(u),

text: and

text: L

text: X(u)

text: y

text: u

text: eθX(y)π

text: dy,

text: N

text: X(u)

text: (log

text: det

text: (uij)

text: χ

text: u)

text: ρ,

text: u

text: eθX(y)π

text: dy.

text: Proof.

text: By

text: and

text: we

text: reduce

text: ωωφnn

text: to

text: a

text: function

text: on

text: a+

text: by

text: ωn

text: MAR(ψ)

text: α,

text: ψ

text: φ

text: eφ

text: α

text: Φ+

text: eψ.

text: ωn

text: J(x)

text: Then

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: ∗ ∗∗ 40 YAN LI BIN ZHOU XIAOHUA ZHU (cid:90) (cid:0) (cid:1) (cid:90) log det(∂2u˜ ) − χ(∂u˜(cid:90) ) − 4ρ(cid:16)(∂u˜ ) π dy (cid:17) h h h 2Pδ + ≤ −(cid:90) (cid:104) Q˜iν u˜ π dσ + uijν u˜ (cid:105)π − uij ν u˜ π dσ i h 0 0 j h,i 0,i j h 0 ∂(2Pδ) ∂(2Pδ) + + (cid:104) ∇ (cid:105) − (6.6) + uij π + 2uij π + 4 ρ, π Qπ u˜ dy. 0,ij 0,j ,i h 2Pδ (cid:12) + (cid:12) (cid:12) Here (cid:12) ∂χ uijπ Q˜i = + 4ρ + 0 ,j , ∂xi ∇ i π x= u0 | | and Q is given by (4.8). Let ξi = πuijν . We see that ξ = O((α(y))2) near W . 0 j α By the convexity of u , h | | ≤ {| − | | − − |} ξiu˜ (y) max u˜ (y + ξ) u˜ (y) , u˜ (y ξ) u˜ (y) . h,i h h h h S(cid:90)ince π = 0 on Weyl walls, we h(cid:90)ave (cid:16) (cid:17) − uijν u˜ π dσ = 0, uijν u˜ π uij ν u˜ π dσ = 0. ∩ 0 j h,i 0 ∩ 0 j h,i 0,i j h 0 ∂(2P+δ) (cid:90)Wα (cid:16) ∂(2P+δ) (cid:17)Wα (cid:88) (cid:90) → → By taking h 0 and then δ 0 with Lemma 2.3, we get(cid:101) − → 2 (cid:104) (cid:105) uijν u˜ π uij ν u˜ π dσ u y, ν π dσ . 0 j h,i 0,i j h 0 λ A 0 ∂(2P+δ) A FA A The last term in (6.6) can be settled by (4.12)-(4.14) and Proposition 4.5. It remains to deal with the ﬁrst term involving Q˜. In fact, by using the similar argument as in the proof of Lemma 4.11 (checking the Cases (i)-(iii) there), we can | | ≤ get Q˜iν C for some uniform C depending only on P and u . Now by Lemma i →Q˜ → Q˜ 0 (cid:90) 6.3, taking h 0 and then(cid:88)δ 0 in (6.6), we get a uniform constant C such that −N+(u˜) ≤ (1(cid:90)+ C(cid:104)(cid:48) ) 2 u˜(cid:104)y, ν (cid:105)π dσ (cid:105) Q˜ λ A 0 A FA A (cid:32) (cid:104) ∇ (cid:105) − (cid:33) + (cid:90) uij π + 2uij π + (cid:90)4 ρ, π Qπ u˜dy + C(u ) 0,ij 0,j ,i 0 2P+ ≤ (cid:104) (cid:105) (cid:104) ∇ (cid:105) C u˜ y, ν π dσ + ( ρ, π + π)u˜ dy + C(u ). 0 0 ∂(2P+) 2P+ Replacing u by (cid:15)u, we obtain (6.2). ∈ C · · → For a general u ˜∗, we consider ut( ) = u(t ) for 0 < t < 1. Then ∂ut → → − ∈ ∂u and ∂2ut ∂2u almost everywhere when t 1 . Since ut C(2P), (6.2) holds for all ut. Note that the constants in (6.2) are independent of t. Thus the (cid:3) proposition is proved. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: YAN

text: LI

text: BIN

text: ZHOU

text: XIAOHUA

text: ZHU

text: log

text: π

text: dy

text: h

text: h

text: h

text: Q˜iν

text: u˜

text: π

text: dσ

text: uijν

text: u˜

text: uij

text: ν

text: u˜

text: π

text: dσ

text: i

text: h

text: j

text: h,i

text: j

text: h

text: uij

text: π

text: π

text: ρ,

text: π

text: Qπ

text: u˜

text: dy.

text: ,i

text: h

text: Here

text: ∂χ

text: uijπ

text: Q˜i

text: 4ρ

text: ,j

text: ∂xi

text: i

text: π

text: x=

text: and

text: Q

text: is

text: given

text: by

text: Let

text: ξi

text: πuijν

text: We

text: see

text: that

text: ξ

text: near

text: W

text: j

text: α

text: By

text: the

text: convexity

text: of

text: u

text: h

text: ξiu˜

text: (y)

text: max

text: u˜

text: (y

text: ξ)

text: u˜

text: (y)

text: u˜

text: (y

text: ξ)

text: u˜

text: (y)

text: h,i

text: h

text: h

text: h

text: h

text: π

text: on

text: Weyl

text: walls,

text: we

text: uijν

text: u˜

text: π

text: dσ

text: uijν

text: u˜

text: π

text: uij

text: ν

text: u˜

text: π

text: dσ

text: j

text: h,i

text: j

text: h,i

text: j

text: h

text: By

text: taking

text: h

text: and

text: then

text: δ

text: with

text: Lemma

text: we

text: uijν

text: u˜

text: π

text: uij

text: ν

text: u˜

text: π

text: dσ

text: u

text: y,

text: ν

text: π

text: dσ

text: j

text: h,i

text: j

text: h

text: λ

text: A

text: A

text: FA

text: A

text: The

text: last

text: term

text: in

text: can

text: be

text: settled

text: by

text: and

text: Proposition

text: It

text: remains

text: to

text: deal

text: with

text: the

text: ﬁrst

text: term

text: involving

text: Q˜.

text: In

text: fact,

text: by

text: using

text: the

text: similar

text: argument

text: as

text: in

text: the

text: proof

text: of

text: Lemma

text: (checking

text: the

text: Cases

text: (i)-(iii)

text: there),

text: we

text: can

text: get

text: Q˜iν

text: C

text: for

text: some

text: uniform

text: C

text: depending

text: only

text: on

text: P

text: and

text: u

text: Now

text: by

text: Lemma

text: i

text: →Q˜

text: Q˜

text: taking

text: h

text: and

text: in

text: we

text: get

text: a

text: uniform

text: constant

text: C

text: such

text: that

text: −N+(u˜)

text: ν

text: dσ

text: Q˜

text: λ

text: A

text: A

text: FA

text: A

text: uij

text: π

text: π

text: ρ,

text: π

text: Qπ

text: u˜dy

text: C(u

text: ,i

text: C

text: u˜

text: y,

text: ν

text: π

text: dσ

text: ρ,

text: π

text: π)u˜

text: dy

text: C(u

text: Replacing

text: u

text: by

text: we

text: obtain

text: C

text: For

text: a

text: general

text: u

text: we

text: consider

text: ut(

text: u(t

text: for

text: t

text: Then

text: ∂ut

text: ∂u

text: and

text: almost

text: everywhere

text: when

text: t

text: Since

text: ut

text: holds

text: for

text: all

text: ut.

text: Note

text: that

text: the

text: constants

text: in

text: are

text: independent

text: of

text: t.

text: Thus

text: the

text: proposition

text: is

text: proved.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: K-ENERGY ON POLARIZED COMPACTIFICATIONS OF LIE GROUPS 7 → → − Note that uijν 0 as y F , where (uij) = (u ) 1 and ν = (ν , ..., ν ) is 0 i A˜ { | 0 } 0,ij − A˜ → 1 (cid:104) r (cid:105) the unit normal vector of face F = y l (y) = 0 . Similarly uij ν 2 y, ν , A˜ A˜ 0,j i λA˜ A˜ where uij = ∂ui0j . Thus we get 0,k ∂yk ∈ C → Lemma 2.3. If u ∞ , then for any A˜, as y F , ,W A˜ → → 2 (cid:104) (cid:105) (2.5) uijν 0 and uijν y, ν , i ,j i λ A˜ A˜ − where (uij) = (u ) 1 and uij = ∂uij . ,ij ,k ∂yk × We compute the Ricci curvature of ω . Clearly it is also K K-invariant. As φ in Lemma 2.2, in the local coordinates in Sect. 2.2, Ric(ω ) can be expressed as φ   − HessC(log det(∂∂¯Ψ))(exp(x))     1HessR(ψ˜)(x) 0 0  4   0 M˜ (x) 0   α(1)  = −  0 0 ... ...  ... ... ... 0 0 0 M˜α(n−2r)(x) ∈ for any x a+, where (cid:88) ∇ ∇ ψ˜ = log det( 2ψ) + 2 (cid:88) log α( ψ) + χ(x), ∈ α Φ+ − − χ(x) = (cid:68)log J(x)(cid:69)=(cid:18) 2 log sinh α(x)(cid:19), ∈ α Φ+ √ − 1 ∇ coth√α(x) 1 M˜ (x) = α, ψ˜ − − . α 2 1 coth α(x) Then the scalar curvature (cid:16) (cid:17) (cid:88) (cid:104) ∇ (cid:105) | ∇ − ∇ α, ψ˜ (2.6) S(ω ) = tr 2ψ) 1 2ψ˜ + (cid:104) ∇ (cid:105). φ exp(x) ∈ α, ψ α Φ+ By using the Legendre function u, we get ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: K-ENERGY

text: ON

text: POLARIZED

text: COMPACTIFICATIONS

text: OF

text: LIE

text: GROUPS

text: Note

text: that

text: uijν

text: as

text: y

text: F

text: where

text: (uij)

text: (u

text: and

text: ν

text: (ν

text: ν

text: is

text: i

text: A˜

text: A˜

text: r

text: the

text: unit

text: normal

text: vector

text: of

text: face

text: F

text: y

text: l

text: (y)

text: Similarly

text: uij

text: ν

text: y,

text: ν

text: A˜

text: A˜

text: i

text: λA˜

text: A˜

text: where

text: uij

text: Thus

text: we

text: get

text: ∂yk

text: C

text: Lemma

text: If

text: u

text: then

text: for

text: any

text: A˜,

text: as

text: y

text: F

text: ,W

text: A˜

text: uijν

text: and

text: uijν

text: y,

text: ν

text: i

text: ,j

text: i

text: λ

text: A˜

text: A˜

text: where

text: (uij)

text: (u

text: and

text: uij

text: ∂uij

text: ,ij

text: ,k

text: ∂yk

text: We

text: compute

text: the

text: Ricci

text: curvature

text: of

text: ω

text: Clearly

text: it

text: is

text: also

text: K

text: K-invariant.

text: As

text: φ

text: in

text: Lemma

text: in

text: the

text: local

text: coordinates

text: in

text: Sect.

text: Ric(ω

text: can

text: be

text: expressed

text: as

text: φ

text: HessC(log

text: det(∂∂¯Ψ))(exp(x))

text: 4

text: M˜

text: (x)

text: for

text: any

text: x

text: a+,

text: where

text: ψ˜

text: log

text: det(

text: log

text: α(

text: ψ)

text: χ(x),

text: α

text: Φ+

text: χ(x)

text: log

text: sinh

text: α

text: Φ+

text: coth√α(x)

text: M˜

text: (x)

text: α,

text: ψ˜

text: α

text: coth

text: α(x)

text: Then

text: the

text: scalar

text: curvature

text: α,

text: ψ˜

text: S(ω

text: tr

text: 2ψ˜

text: φ

text: exp(x)

text: α,

text: ψ

text: α

text: Φ+

text: By

text: using

text: the

text: Legendre

text: function

text: u,

text: we

text: get

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: INCOMPRESSIBLE FILLINGS OF MANIFOLDS 11 The map s∗(cid:102)is an (cid:101)isomo(cid:102)rphism by Proposition 10, so δ∗ is also an isomor- phism. The composition i∗δ∗ is injective because it is induced by a section → × → of the bundle M ∂ M ∂. Since δ∗ is an isomorphism, the map i∗ is Γ also injective. → → Suppose there is a homotopy section c : M ∂ of ∂ (cid:44) M. Then i∗ is (cid:55)→ surjective (because it has a section of the form (x, y) (x, cy)) and then all Z homology groups in the dia(cid:101)gram(cid:101)above a(cid:101)re iso(cid:101)morphic to . So, the map × → × ∂ ∂ ∂ ∂, Γ Γ (cid:55)→ (x, y) (x, cy), (cid:55)→ is an H − -isomorphism, and hence the ﬂipped map (x, y(cid:101)) ((cid:101)cx, y) is, n 1 (cid:55)→ (cid:55)→ (cid:55)→ as well16. So, the composite x (x, x) (x, cx) (cx, cx) is an H − - → → × n 1 isomorphism. But, this composite map factors as ∂ (cid:44) M ∂ ∂, so it is Γ zero on H − because ∂ bounds M. This gives a contradiction, so there is n 1 (cid:3) no homotopy section c. Proposition 11 is a restatement of Theorem 3, so we are now done with the proof of that theorem. (cid:101) Note that the same argument proves the following variant of Theorem 3: If ∂ is connected, (M, ∂) is a ﬁlling and the π M-cover(cid:101) ∂ of ∂ has a 1 small singular model, then (M, ∂) is incompressible. The point here is that → → ∂ (cid:44) M doesn’t need to be π -injective as long as the cover ∂ ∂ that has 1 a small singular model comes from the ﬁlling M. 3. The curve complex is small (Proof of Proposition 5) We need to verify inequality (2) for the curve complex C. Let St0 (σ) be Γ the pointwise stabilizer of a simplex σ in C. It is a ﬁnite index subgroup of ∩ the stabilizer St (σ). Therefore, St0 (σ) St0 (τ) is a ﬁnite index subgroup ∩ Γ Γ Γ of St (σ) St (τ). Since the group Γ is torsion-free, these two groups have Γ Γ the same homological dimension. So, we can use poinwise stabilizers when verifying (2). In the case of the curve complex the inequality for pointwise stabilizers we need to prove has the following form. Lemma 12. Let A be a collection of disjoint simple closed curves and B another collection of disjoint simple closed curves, such that no two curves ∪ in A B are homotopic. Then ∪ | | − | | − − (26) hdim(Stab (A B)) + A 1 + B 1 < 6g 7. Γ (cid:55)→ 16Because (x, y) (y, x) is an isomorphism. ##LTLine##

text: INCOMPRESSIBLE

text: FILLINGS

text: OF

text: MANIFOLDS

text: The

text: map

text: an

text: by

text: Proposition

text: so

text: δ∗

text: is

text: also

text: an

text: isomor-

text: phism.

text: The

text: composition

text: i∗δ∗

text: is

text: injective

text: because

text: it

text: is

text: induced

text: by

text: a

text: section

text: of

text: the

text: bundle

text: M

text: M

text: Since

text: δ∗

text: is

text: an

text: isomorphism,

text: the

text: map

text: i∗

text: is

text: Γ

text: also

text: injective.

text: Suppose

text: there

text: is

text: a

text: homotopy

text: section

text: c

text: M

text: of

text: M.

text: Then

text: i∗

text: is

text: surjective

text: (because

text: it

text: has

text: a

text: section

text: of

text: the

text: form

text: (x,

text: y)

text: (x,

text: cy))

text: and

text: then

text: all

text: Z

text: homology

text: groups

text: in

text: the

text: to

text: So,

text: the

text: map

text: Γ

text: Γ

text: (x,

text: y)

text: (x,

text: cy),

text: is

text: an

text: H

text: -isomorphism,

text: and

text: hence

text: the

text: ﬂipped

text: map

text: (x,

text: y)

text: is,

text: n

text: as

text: So,

text: the

text: composite

text: x

text: (x,

text: x)

text: (x,

text: cx)

text: (cx,

text: cx)

text: is

text: an

text: H

text: n

text: isomorphism.

text: But,

text: this

text: composite

text: map

text: factors

text: as

text: M

text: so

text: it

text: is

text: Γ

text: zero

text: on

text: H

text: because

text: bounds

text: M.

text: This

text: gives

text: a

text: contradiction,

text: so

text: there

text: is

text: n

text: no

text: homotopy

text: section

text: c.

text: Proposition

text: is

text: a

text: restatement

text: of

text: Theorem

text: so

text: we

text: are

text: now

text: done

text: with

text: the

text: proof

text: of

text: that

text: theorem.

text: Note

text: that

text: the

text: same

text: argument

text: proves

text: the

text: following

text: variant

text: of

text: Theorem

text: If

text: is

text: connected,

text: (M,

text: is

text: a

text: ﬁlling

text: and

text: the

text: π

text: of

text: has

text: a

text: small

text: singular

text: model,

text: then

text: (M,

text: is

text: incompressible.

text: The

text: point

text: here

text: is

text: that

text: M

text: doesn’t

text: need

text: to

text: be

text: π

text: -injective

text: as

text: long

text: as

text: the

text: cover

text: that

text: has

text: a

text: small

text: singular

text: model

text: comes

text: from

text: the

text: ﬁlling

text: M.

text: The

text: curve

text: complex

text: is

text: small

text: (Proof

text: of

text: Proposition

text: We

text: need

text: to

text: verify

text: inequality

text: for

text: the

text: curve

text: complex

text: C.

text: Let

text: (σ)

text: be

text: Γ

text: the

text: pointwise

text: stabilizer

text: of

text: a

text: simplex

text: σ

text: in

text: C.

text: It

text: is

text: a

text: ﬁnite

text: index

text: subgroup

text: of

text: the

text: stabilizer

text: St

text: (σ).

text: Therefore,

text: (σ)

text: (τ)

text: is

text: a

text: ﬁnite

text: index

text: subgroup

text: Γ

text: Γ

text: Γ

text: of

text: St

text: (σ)

text: St

text: (τ).

text: Since

text: the

text: group

text: Γ

text: is

text: torsion-free,

text: these

text: two

text: groups

text: have

text: Γ

text: Γ

text: the

text: same

text: homological

text: dimension.

text: So,

text: we

text: can

text: use

text: poinwise

text: stabilizers

text: when

text: verifying

text: In

text: the

text: case

text: of

text: the

text: curve

text: complex

text: the

text: inequality

text: for

text: pointwise

text: stabilizers

text: we

text: need

text: to

text: prove

text: has

text: the

text: following

text: form.

text: Lemma

text: Let

text: A

text: be

text: a

text: collection

text: of

text: disjoint

text: simple

text: closed

text: curves

text: and

text: B

text: another

text: collection

text: of

text: disjoint

text: simple

text: closed

text: curves,

text: such

text: that

text: no

text: two

text: curves

text: in

text: A

text: B

text: are

text: homotopic.

text: Then

text: hdim(Stab

text: (A

text: B))

text: A

text: B

text: Γ

text: (x,

text: y)

text: (y,

text: x)

text: is

text: an

text: isomorphism.

text: ##LTLine##


text: 16 GRIGORI AVRAMIDI (cid:88) length(E∗) ∩   dim(N/(St(F∗) N)) + (1 + length(F ∗)) = (cid:88) i  i=0  length(E∗) ∩ dim(N/(St(F∗) N)) +  length(F ∗) + length(E∗) + 1 i (cid:88)  i=0  length(E∗) ≥ length(F∗0) + length(F ∗) + length(E∗) + 1 i i=0 = length(F∗) + length(E∗) + 1 | | | | = ( σ + 1) + ( τ + 1) + 1. We can rewrite this conclusion as | | | | ≤ R − (36) dim(G/K) + σ + τ dim(GL( m)/ SO(m)) 3 R − (37) = dim(SL( m)/ SO(m)) 2 (38) < dim ∂, where ∂ is the Borel-Serre boundary. Next we will prove the following claim. Z For any torsionfree subgroup Γ < SL( m) we have ∩ ≤ (39) hdim(St (E∗) St (F∗)) dim(G/K). Γ Γ (cid:101) Together with (36) this claim implies that the complex of ﬂags is a small → singular model for the Γ-cover ∂ ∂ of the Borel-Serre boundary. We will need the following simple lemma. ∈ Q Lemma 15. Suppose γ GL( m) is an element whose characteristic poly- − ± ⊂ nomial p (t) := det(t γ) has integer coeﬃcients and det(γ) = 1. If V Q γ | ± | ± m is a γ-invariant subspace then det(γ ) = 1 and det(γ Q ) = 1. V ( m/V ) Proof. First, since p (t) is a monic polynomial with integer coeﬃcients, any γ ± ± root λ of p (t) is an algebraic integer. Second, p (0) = det(γ) = 1. γ · ± γ Therefore p (t) = t q(t) 1 for some monic polynomial with integer coeﬃ- γ − − ± cients q(t). The inverse λ 1 can be expressed as λ 1 = q(λ), so it is also an algebraic integer.

text: GRIGORI

text: AVRAMIDI

text: length(E∗)

text: dim(N/(St(F∗)

text: N))

text: length(F

text: i

text: length(E∗)

text: dim(N/(St(F∗)

text: N))

text: length(F

text: length(E∗)

text: i

text: length(E∗)

text: length(F

text: length(E∗)

text: i

text: length(F∗)

text: length(E∗)

text: σ

text: τ

text: We

text: can

text: rewrite

text: this

text: conclusion

text: as

text: R

text: dim(G/K)

text: σ

text: τ

text: dim(GL(

text: m)/

text: SO(m))

text: R

text: dim(SL(

text: m)/

text: SO(m))

text: dim

text: where

text: is

text: the

text: Borel-Serre

text: boundary.

text: Next

text: we

text: will

text: prove

text: the

text: following

text: claim.

text: Z

text: For

text: any

text: torsionfree

text: subgroup

text: Γ

text: SL(

text: m)

text: we

text: have

text: hdim(St

text: (E∗)

text: St

text: (F∗))

text: dim(G/K).

text: Γ

text: Γ

text: Together

text: with

text: this

text: claim

text: implies

text: that

text: the

text: complex

text: of

text: ﬂags

text: is

text: a

text: small

text: singular

text: model

text: for

text: the

text: Γ-cover

text: of

text: the

text: Borel-Serre

text: boundary.

text: We

text: will

text: need

text: the

text: following

text: simple

text: lemma.

text: Q

text: Lemma

text: Suppose

text: γ

text: GL(

text: m)

text: is

text: an

text: element

text: whose

text: characteristic

text: poly-

text: nomial

text: p

text: (t)

text: det(t

text: γ)

text: has

text: integer

text: coeﬃcients

text: and

text: det(γ)

text: If

text: V

text: Q

text: γ

text: m

text: is

text: a

text: γ-invariant

text: subspace

text: then

text: det(γ

text: and

text: det(γ

text: Q

text: V

text: m/V

text: Proof.

text: First,

text: since

text: p

text: (t)

text: is

text: a

text: monic

text: polynomial

text: with

text: integer

text: coeﬃcients,

text: any

text: γ

text: root

text: λ

text: of

text: p

text: (t)

text: is

text: an

text: algebraic

text: integer.

text: Second,

text: p

text: det(γ)

text: γ

text: γ

text: Therefore

text: p

text: (t)

text: t

text: q(t)

text: for

text: some

text: monic

text: polynomial

text: with

text: integer

text: coeﬃ-

text: γ

text: cients

text: q(t).

text: The

text: inverse

text: λ

text: can

text: be

text: expressed

text: as

text: λ

text: q(λ),

text: so

text: it

text: is

text: also

text: an

text: algebraic

text: integer.


text: INCOMPRESSIBLE FILLINGS OF MANIFOLDS 3 Simply connected manifolds do not have small singular models. At the other extreme, a single point is a small singular model for the universal cover of any aspherical manifold. More interestingly, (some) Borel-Serre boundaries of locally symmetric manifolds and Harvey-Ivanov boundaries of moduli spaces are a source of small singular models. (cid:101) This space has fundamental group Γ = F d. Let ∂ be the boundary of → 2 M. The Γ-cover of the boundary ∂ ∂ has a small singular model that (cid:104) (cid:105) × · · · × is a d-fold join of inﬁnite discrete sets. Explicitly, let F d = a , b (cid:104) (cid:105) (cid:104) (cid:105) (cid:104) (cid:105) 2 1 1 (cid:104) (cid:105) a , b . Let X := a , b / [a , b ] be the discrete, transitive left a , b - d d i i i i i i i space8 whose stabilizers are conjugates of the cyclic group generated by the − − commutator [a , b ] = a b a 1b 1. It is well known that the d-fold join of i i i i i i these spaces (cid:101) ∗ · · · ∗ X X 1 d is a singular model for ∂ and not hard to check that it is small. Z \ R The space of ﬂat m-tori of unit volume is the orbifold SL( m) SL( m)/ SO(m). Z Let Γ < SL( m) be a ﬁnite index, torsion free subgroup and denote by M(cid:102)˙ th(cid:101)e \ R quotient Γ SL( m)/ SO(m). It can be compactiﬁed to a compact manifold- with-boundary (M,(cid:101)∂) (the Borel-Serre compactiﬁcation). Its Γ-cover (M, ∂) ∗ ∨ − Q is homotopy equivalent to ( , Sm 2), and the complex of ﬂags in m is a singular model for ∂. (cid:101) Q Proposition 4. The ﬂag complex of m is a small singular model for ∂. ≥ Let Σ be a closed orientable surface of genus g 2, and denote by Γ g a ﬁnite index torsion free subgroup of the extended mapping class group π Homeo(Σ ). Let M˙ be the ﬁnite (orbifold) cover of moduli space9 cor- 0 g respondin(cid:102)g t(cid:101)o the subgroup Γ. It is can be compactiﬁed to a compact manifold-with-boundary (M, ∂) (t(cid:101)he Harvey-Ivanov compactiﬁcation). Its ∗ ∨ − Γ-cover (M, ∂) is homotopy equivalent to ( , S2g 2), and the curve com- plex of Σg is a singular model for ∂. (cid:101) Proposition 5. The curve complex is a small singular model for ∂. ≥ Z Suppose m 4 and Γ < SL( m) is a ﬁnite index torsion free subgroup. Then Theorem 3, Proposition 4 and the Margulis normal subgroup theo- rem imply that the only potentially compressible ﬁllings of the Borel-Serre \ R boundary ∂(Γ SL( m)/ SO(m)) are simply connected (see subsection ??). (cid:104) (cid:105) (cid:104)(cid:104) (cid:105)(cid:105) 8Note that this is not the abelianization a , b / [a , b ] . i i M±i i 9In this paper, moduli space will mean the space of hyperbolic structures on the ≥ g closed, orientable, genus g 2 surface. It is double covered by the space of oriented hyperbolic structures. ##LTLine##

text: INCOMPRESSIBLE

text: FILLINGS

text: OF

text: MANIFOLDS

text: Simply

text: connected

text: manifolds

text: do

text: not

text: have

text: small

text: singular

text: models.

text: At

text: the

text: other

text: extreme,

text: a

text: single

text: point

text: is

text: a

text: small

text: singular

text: model

text: for

text: the

text: universal

text: cover

text: of

text: any

text: aspherical

text: manifold.

text: More

text: interestingly,

text: (some)

text: Borel-Serre

text: boundaries

text: of

text: locally

text: symmetric

text: manifolds

text: and

text: Harvey-Ivanov

text: boundaries

text: of

text: moduli

text: spaces

text: are

text: a

text: source

text: of

text: small

text: singular

text: models.

text: This

text: space

text: has

text: fundamental

text: group

text: Γ

text: F

text: d.

text: Let

text: be

text: the

text: boundary

text: of

text: M.

text: The

text: Γ-cover

text: of

text: the

text: boundary

text: has

text: a

text: small

text: singular

text: model

text: that

text: is

text: a

text: d-fold

text: join

text: of

text: inﬁnite

text: discrete

text: sets.

text: Explicitly,

text: let

text: F

text: d

text: a

text: b

text: a

text: b

text: Let

text: X

text: a

text: b

text: [a

text: b

text: be

text: the

text: discrete,

text: transitive

text: left

text: a

text: b

text: d

text: d

text: i

text: i

text: i

text: i

text: i

text: i

text: i

text: whose

text: stabilizers

text: are

text: conjugates

text: of

text: the

text: cyclic

text: group

text: generated

text: by

text: the

text: commutator

text: [a

text: b

text: a

text: b

text: a

text: It

text: is

text: well

text: known

text: that

text: the

text: d-fold

text: join

text: of

text: i

text: i

text: i

text: i

text: i

text: i

text: these

text: spaces

text: X

text: X

text: d

text: is

text: a

text: singular

text: model

text: for

text: and

text: not

text: hard

text: to

text: check

text: that

text: it

text: is

text: small.

text: Z

text: R

text: The

text: space

text: of

text: ﬂat

text: m-tori

text: of

text: unit

text: volume

text: is

text: the

text: orbifold

text: SL(

text: m)

text: SL(

text: m)/

text: SO(m).

text: Z

text: Let

text: Γ

text: SL(

text: m)

text: be

text: a

text: ﬁnite

text: index,

text: torsion

text: free

text: subgroup

text: and

text: denote

text: by

text: R

text: quotient

text: Γ

text: SL(

text: m)/

text: SO(m).

text: It

text: can

text: be

text: compactiﬁed

text: to

text: a

text: compact

text: manifold-

text: with-boundary

text: (the

text: Borel-Serre

text: compactiﬁcation).

text: Its

text: Γ-cover

text: (M,

text: Q

text: is

text: homotopy

text: equivalent

text: to

text: Sm

text: and

text: the

text: complex

text: of

text: ﬂags

text: in

text: m

text: is

text: a

text: singular

text: model

text: for

text: Q

text: Proposition

text: The

text: ﬂag

text: complex

text: of

text: m

text: is

text: a

text: small

text: singular

text: model

text: for

text: Let

text: Σ

text: be

text: a

text: closed

text: orientable

text: surface

text: of

text: genus

text: g

text: and

text: denote

text: by

text: Γ

text: g

text: a

text: ﬁnite

text: index

text: torsion

text: free

text: subgroup

text: of

text: the

text: extended

text: mapping

text: class

text: group

text: π

text: Homeo(Σ

text: Let

text: M˙

text: be

text: the

text: ﬁnite

text: (orbifold)

text: cover

text: of

text: moduli

text: cor-

text: g

text: the

text: subgroup

text: Γ.

text: It

text: is

text: can

text: be

text: compactiﬁed

text: to

text: a

text: compact

text: manifold-with-boundary

text: (M,

text: Harvey-Ivanov

text: compactiﬁcation).

text: Its

text: Γ-cover

text: (M,

text: is

text: homotopy

text: equivalent

text: to

text: and

text: the

text: curve

text: com-

text: plex

text: of

text: Σg

text: is

text: a

text: singular

text: model

text: for

text: Proposition

text: The

text: curve

text: complex

text: is

text: a

text: small

text: singular

text: model

text: for

text: Z

text: Suppose

text: m

text: and

text: Γ

text: SL(

text: m)

text: is

text: a

text: ﬁnite

text: index

text: torsion

text: free

text: subgroup.

text: Then

text: Theorem

text: Proposition

text: and

text: the

text: Margulis

text: normal

text: subgroup

text: theo-

text: rem

text: imply

text: that

text: the

text: only

text: potentially

text: compressible

text: ﬁllings

text: of

text: the

text: Borel-Serre

text: R

text: boundary

text: ∂(Γ

text: SL(

text: m)/

text: SO(m))

text: are

text: simply

text: connected

text: (see

text: subsection

text: that

text: this

text: is

text: not

text: the

text: abelianization

text: a

text: b

text: [a

text: b

text: i

text: i

text: M±i

text: i

text: this

text: paper,

text: moduli

text: space

text: will

text: mean

text: the

text: space

text: of

text: hyperbolic

text: structures

text: on

text: the

text: g

text: closed,

text: orientable,

text: genus

text: g

text: surface.

text: It

text: is

text: double

text: covered

text: by

text: the

text: space

text: of

text: oriented

text: hyperbolic

text: structures.

text: ##LTLine##


text: (cid:101) INCOMPRESSIBLE FILLINGS OF MANIFOLDS 21 then the isometry group I of the lifted metric h on the universal cover is the extended mapping class group. (cid:101) (cid:101) Proof. Since h is lifted from moduli space, the extended mapping class group acts by isometries of h. Farb and Weinberger show in Theor(cid:102)em 1.2 of [4] that the isometry group I contains the extended mapping class group as M± → ˙ a ﬁnite index subgroup so we get an orbifold cover M/I. By the g minimal orbifold theorem this cover is trivial, i.e. the isometry group is the (cid:3) extended mapping class group. Remark. In [2] these two corollaries were proved using L2-betti numbers instead of small singular models. 9. The obstruction α (Description of α and proof of M (cid:101)M (cid:102) (cid:101) Proposition 9) − → There(cid:102)are no obstru(cid:101)ctions to building a Γ-map f : M(q 1) ∂ that is the(cid:101)identity on the boundary. Such a map deﬁnes a Γ-equivariant cochain → o : C (M) H − (∂) = D by sending each cell c to the homology class f q q 1 of f(∂c). This cochain is ∗ze∈ro on th(cid:101)e boundary, an(cid:101)d it(cid:102)is actua(cid:101)lly a cocycle, so it gives an element [o ] Hq(M, ∂; D) in the relativ(cid:101)e cohomology group. f → It is the obstruction(cid:101) to (cid:102)extending f(cid:101)to a Γ-map g : M(q) (cid:101) ∂ th−at is still the identity on the boundary, after possibly changing f on the (q 1)-cells (cid:48) − → to ano(cid:101)ther Γ-map f : M(q 1) ∂. Such an extension g does not exist → (it would give a homotopy section g of ∂(q) (cid:44) M(q)(cid:101), and would imply that ∗ H − (∂) = 0) so the cohomology class [o ] is non-zero. Moreover, this q 1 f class does not depend on the ch(cid:101)oice (cid:102)of initial map f, and so we simply call ∗ it α . Its image in the absolute cohomology Hq(M; D) is denoted α . It M M is the obstruction to extending f to M(q) after possibly changing it on some − (interior or boundary) (q 1)-chains. Proof of Proposition 9. Poincare duality and (41) imply that ∼ − (50) Hq 1(∂; D) = H − (∂; D) = 0. n q ∗ (cid:54) Since α = 0 the long exact homology sequence M − → → Hq 1(∂; D) Hq(M, ∂; D) Hq(M; D), || (51) ∗ (cid:55)→ 0 α α M M ##LTLine## ##LTLine##

text: INCOMPRESSIBLE

text: FILLINGS

text: OF

text: MANIFOLDS

text: then

text: the

text: isometry

text: group

text: I

text: of

text: the

text: lifted

text: metric

text: h

text: on

text: the

text: universal

text: cover

text: is

text: the

text: extended

text: mapping

text: class

text: group.

text: Proof.

text: Since

text: h

text: is

text: lifted

text: from

text: moduli

text: space,

text: the

text: extended

text: mapping

text: class

text: group

text: acts

text: by

text: isometries

text: of

text: h.

text: Farb

text: and

text: Weinberger

text: show

text: in

text: of

text: that

text: the

text: isometry

text: group

text: I

text: contains

text: the

text: extended

text: mapping

text: class

text: group

text: as

text: M±

text: a

text: ﬁnite

text: index

text: subgroup

text: so

text: we

text: get

text: an

text: orbifold

text: cover

text: M/I.

text: By

text: the

text: g

text: minimal

text: orbifold

text: theorem

text: this

text: cover

text: is

text: trivial,

text: i.e.

text: the

text: isometry

text: group

text: is

text: the

text: extended

text: mapping

text: class

text: group.

text: Remark.

text: In

text: these

text: two

text: corollaries

text: were

text: proved

text: using

text: numbers

text: instead

text: of

text: small

text: singular

text: models.

text: The

text: obstruction

text: α

text: (Description

text: of

text: α

text: and

text: proof

text: of

text: M

text: Proposition

text: no

text: to

text: building

text: a

text: Γ-map

text: f

text: M(q

text: that

text: is

text: on

text: the

text: boundary.

text: Such

text: a

text: map

text: deﬁnes

text: a

text: Γ-equivariant

text: cochain

text: o

text: C

text: (M)

text: H

text: D

text: by

text: sending

text: each

text: cell

text: c

text: to

text: the

text: homology

text: class

text: f

text: q

text: q

text: of

text: f(∂c).

text: This

text: cochain

text: is

text: ∗ze∈ro

text: on

text: boundary,

text: a

text: cocycle,

text: so

text: it

text: gives

text: an

text: element

text: [o

text: Hq(M,

text: D)

text: in

text: the

text: cohomology

text: group.

text: f

text: It

text: is

text: the

text: to

text: a

text: Γ-map

text: g

text: M(q)

text: th−at

text: is

text: still

text: the

text: identity

text: on

text: the

text: boundary,

text: after

text: possibly

text: changing

text: f

text: on

text: the

text: (q

text: to

text: Γ-map

text: f

text: M(q

text: Such

text: an

text: extension

text: g

text: does

text: not

text: exist

text: (it

text: would

text: give

text: a

text: homotopy

text: section

text: g

text: of

text: ∂(q)

text: and

text: would

text: imply

text: that

text: H

text: so

text: the

text: cohomology

text: class

text: [o

text: is

text: non-zero.

text: Moreover,

text: this

text: q

text: f

text: class

text: does

text: not

text: depend

text: on

text: the

text: initial

text: map

text: f,

text: and

text: so

text: we

text: simply

text: call

text: it

text: α

text: Its

text: image

text: in

text: the

text: absolute

text: cohomology

text: Hq(M;

text: D)

text: is

text: denoted

text: α

text: It

text: M

text: M

text: is

text: the

text: obstruction

text: to

text: extending

text: f

text: to

text: M(q)

text: after

text: possibly

text: changing

text: it

text: on

text: some

text: (interior

text: or

text: boundary)

text: (q

text: Proof

text: of

text: Proposition

text: Poincare

text: duality

text: and

text: imply

text: that

text: Hq

text: D)

text: H

text: D)

text: n

text: q

text: Since

text: α

text: the

text: long

text: exact

text: homology

text: sequence

text: M

text: Hq

text: D)

text: Hq(M,

text: D)

text: Hq(M;

text: D),

text: α

text: α

text: M

text: M

text: ##LTLine##

text: ##LTLine##


text: 10 GRIGORI AVRAMIDI We conclude from the second bullet that the left hand side of the spectral ≥ − sequence (21) vanishes for i + j n 1 and consequently × (22) H≥Γ − (C C, ∆) = 0. n 1 This ﬁnishes the proof of the Proposition. 2. No homotopy section (Proof of Theorem 3) → Next, we use Proposition(cid:101)10 to obstruct homotopy sections of ∂ (cid:44) M. (cid:101) → → Proposition 11. Suppose ∂ ∂ is a connected Γ-cover and let ∂ (cid:44) M be → a π -injective ﬁlling. If ∂ has a small singular model C, then ∂ (cid:44) M has 1 no homotopy section. (cid:101) (cid:101) Proof. Let C be a small singular model for ∂. Then there is a homotopy → equivalence h : ∂ C that is also a Γ-map. Such a map induces an H∗Γ- isomorphism (cid:101) ∼ h∼∗ (23) H∗(∂(cid:101)) =(cid:101)H∗Γ(∂) = H∗Γ(C). × × → × The product map h h : ∂ ∂ C C is also a homotopy equivalence and a Γ-map, so it induces an H∗Γ-isomorphism (cid:101) (cid:101) (cid:101) (cid:101) × ∼ × (h×∼h)∗ × (24) H∗(∂ ∂) = H∗Γ(∂ ∂) = H∗Γ(C C). Γ Now, let (cid:101) (cid:101) → × δ : ∂ ∂ ∂, Γ (cid:55)→ x (x, x) be the diagonal map. Look at the commutative diagram (cid:101) (cid:101) (cid:101) (cid:102) ∼ HΓ− (C × C) = H − (∂ × ∂) →i∗ H − (∂ × M) n 1 ↑ n 1 ↑ Γ (cid:37) n 1 Γ s∗ ∼ δ∗ (25) HΓ− (C) = H − (∂) n 1 n ||1 Z

text: GRIGORI

text: AVRAMIDI

text: We

text: conclude

text: from

text: the

text: second

text: bullet

text: that

text: the

text: left

text: hand

text: side

text: of

text: the

text: spectral

text: sequence

text: vanishes

text: for

text: i

text: j

text: n

text: and

text: consequently

text: H≥Γ

text: (C

text: C,

text: n

text: This

text: ﬁnishes

text: the

text: proof

text: of

text: the

text: Proposition.

label: section

text: No

text: homotopy

text: section

text: (Proof

text: of

text: Theorem

text: Next,

text: we

text: use

text: to

text: obstruct

text: homotopy

text: sections

text: of

text: M.

text: Proposition

text: Suppose

text: is

text: a

text: connected

text: Γ-cover

text: and

text: let

text: M

text: be

text: a

text: π

text: -injective

text: ﬁlling.

text: If

text: has

text: a

text: small

text: singular

text: model

text: C,

text: then

text: M

text: has

text: no

text: homotopy

text: section.

text: Proof.

text: Let

text: C

text: be

text: a

text: small

text: singular

text: model

text: for

text: Then

text: there

text: is

text: a

text: homotopy

text: equivalence

text: h

text: C

text: that

text: is

text: also

text: a

text: Γ-map.

text: Such

text: a

text: map

text: induces

text: an

text: H∗Γ-

text: isomorphism

text: h∼∗

text: H∗Γ(C).

text: The

text: product

text: map

text: h

text: h

text: C

text: C

text: is

text: also

text: a

text: homotopy

text: equivalence

text: and

text: a

text: Γ-map,

text: so

text: it

text: induces

text: an

text: H∗Γ-isomorphism

text: (h×∼h)∗

text: H∗(∂

text: H∗Γ(∂

text: H∗Γ(C

text: C).

text: Γ

text: Now,

text: let

text: δ

text: Γ

text: x

text: (x,

text: x)

text: be

text: the

text: diagonal

text: map.

text: Look

text: at

text: the

text: commutative

text: diagram

text: HΓ−

text: (C

text: C)

text: H

text: →i∗

text: H

text: M)

text: n

text: n

text: Γ

text: n

text: Γ

text: s∗

text: δ∗

text: HΓ−

text: (C)

text: H

text: n

text: n

text: Z


text: PT ∗ -symmetric trimer systems † L. Jin1, PT We studied parity-time ( ) symmetric trimer systems that feature open and closed boundaries. The exceptional point is three-state coalescence at zero energy because of chiral symmetry in the open trimer; however, two-state coalescence appears in the closed trimer aﬀected by the magnetic PT ﬂux enclosed, which suppresses the transition. Dynamics at the phase transition point were also qualitatively studied. For three-state coalescence, the probability of an initial state increases in a power law. The highest order is four; however, it can be reduced to two for the state that is only relevant to one associated state. For two-state coalescence in the trimer ring, we found four typical dynamical behaviors of state probability: (i) unchanged, (ii) oscillation, (iii) quadratic increase, and (iv) quadratic increase with oscillation. The scattering properties of the trimer systems were also studied, particularly, asymmetric wave emissions at spectral singularities. PACS numbers: 11.30.Er, 42.25.-p, 03.65.Vf, 42.60.Da PT I. INTRODUCTION dynamics at the -symmetric phase transition (i.e., exceptional) point of trimer systems, particularly in the PT presence of eﬀective magnetic ﬂux. From the eigen spec- Parity-time ( ) symmetric systems possess intrigu- PT trum, we obtained the -symmetric phase transition ing features that derive from non-Hermiticity and the PT PT point and revealed that the uniformly coupled trimer symmetry. The -symmetric systems have been chain with balanced gain and loss at its ends has one extensively investigated, both theoretically [1–20] and ex- PT exceptional point, where three eigenstates coalesce at perimentally [21–26]. The -symmetry breaking [21– zero energy. In the trimer ring, a nonreciprocal cou- 23], nonreciprocal reﬂectionless [24], and coherent per- PT pling connects the head and tail of the trimer chain fect absorber [25–27] have been observed. In 2014, - and the nonreciprocal coupling factors correspond to a symmetric systems have been realized by on-chip de- gauge invariant ﬁeld, which can be seen as an eﬀective vices, known as two coupled whispering gallery mode PT magnetic ﬂux. For the trimer ring, the -symmetric ring microresonators [28–30]. These resonators have high phase transition point can be a two or three eigenstates quality factors, and a gain in one resonator, induced coalescence. To determined the dynamical features at by pumping the doped ions, can balance losses in both PT the exceptional point in the trimer systems, we exam- resonators. The realized -symmetric coupled waveg- ined the time evolution of diﬀerent initial states. For uides and resonators are mainly described by the two- PT the trimer chain, the probability increases at the excep- site models, which act as a -symmetric dimer [31]. tional point in a power law. Because of the three-state The two-site dimers are the most concise and simple PT PT coalescence (EP3) [38], the highest order in the power -symmetric systems, however, all the symmetry law is four; however, this probability increases quadrat- breaking [21], power oscillation [22], and gain induced ically at the EP3 when the initial state is only related large nonlinearity [23, 28, 30] were found in these sys- to the eigenstate and the ﬁrst associated state. In the tems. Previous study demonstrated the diﬀerences of PT PT trimer ring, the -symmetric phase transition point -symmetry and pseudo-Hermiticity in non-Hermitian is a two-state coalescence (EP2) except for the eﬀective trimer chain systems [32]. In an optomechanical system ∈ Z magnetic ﬂux, which is Φ = 2nπ + π/2 (n ). For the coupled with an active resonator, high-order EPs are use- EP2, the coalesced two eigenenergy and the third one are ful for low-power mechanical cooling [34]. Triple-cavity all nonzero. The evolved probability of an initial state supermodes was analysised based on thye coupled mode PT has four typical behaviors, (i) unchanged, (ii) oscillation, theory [33]. Besides, -symmetric oligomers were in- (iii) quadratic increase, and (iv) quadratic increase with vestigated in nonlinearity systems, the solution and non- oscillation. Notably, the oscillation periods are the same, linear dynamics were examined [35, 36]. The asymmetric because they are coalesced eigenenergy dependent, i.e., scattering properties for left and right propagation were PT 2π/E . We also investigated the scattering properties of demonstrated for nonlinear -symmetric oligomers em- 2 the trimer systems, particularly, the spectral singulari- bedded in linear lead; the plane waves are dynamically ties and asymmetric wave emissions under the inﬂuence unstable except in the vicinity of the linear limit; how- of nonreciprocal coupling. ever, the asymmetric transmissions persist for Gaussian wave packets [37]. The remainder of this paper is organized as follows. In PT PT This paper focuses on the -symmetric trimer. We Sec. , we present the trimer systems, the -symmetric PT investigated two -symmetric trimer systems under phase diagram, and the spectrum. In Sec. , we investi- diﬀerent boundary conditions (one with an open chain gate the dynamics at the exceptional point. In Sec. , we form and the other with a closed ring form) and ana- explore the scattering properties of the trimer systems, lytically solved their Hamiltonians. We concentrate on especially the spectral singularities and asymmetric wave

text: PT

label: title

text: -symmetric

text: trimer

text: systems

text: L.

label: author

text: PT

label: abstract

text: We

text: studied

text: parity-time

text: symmetric

text: trimer

text: systems

text: that

text: feature

text: open

text: and

text: closed

text: boundaries.

text: The

text: exceptional

text: point

text: is

text: three-state

text: coalescence

text: at

text: zero

text: energy

text: because

text: of

text: chiral

text: symmetry

text: in

text: the

text: open

text: trimer;

text: however,

text: two-state

text: coalescence

text: appears

text: in

text: the

text: closed

text: trimer

text: aﬀected

text: by

text: the

text: magnetic

text: PT

text: ﬂux

text: enclosed,

text: which

text: suppresses

text: the

text: transition.

text: Dynamics

text: at

text: the

text: phase

text: transition

text: point

text: were

text: also

text: qualitatively

text: studied.

text: For

text: three-state

text: coalescence,

text: the

text: probability

text: of

text: an

text: initial

text: state

text: increases

text: in

text: a

text: power

text: law.

text: The

text: highest

text: order

text: is

text: four;

text: however,

text: it

text: can

text: be

text: reduced

text: to

text: two

text: for

text: the

text: state

text: that

text: is

text: only

text: relevant

text: to

text: one

text: associated

text: state.

text: For

text: two-state

text: coalescence

text: in

text: the

text: trimer

text: ring,

text: we

text: found

text: four

text: typical

text: dynamical

text: behaviors

text: of

text: state

text: probability:

text: (i)

text: unchanged,

text: (ii)

text: oscillation,

text: (iii)

text: quadratic

text: increase,

text: and

text: (iv)

text: quadratic

text: increase

text: with

text: oscillation.

text: The

text: scattering

text: properties

text: of

text: the

text: trimer

text: systems

text: were

text: also

text: studied,

text: particularly,

text: asymmetric

text: wave

text: emissions

text: at

text: spectral

text: singularities.

text: PACS

text: numbers:

text: PT

text: I.

text: INTRODUCTION

text: dynamics

text: at

text: the

text: -symmetric

text: phase

text: transition

text: (i.e.,

text: exceptional)

text: point

text: of

text: trimer

text: systems,

text: particularly

text: in

text: the

text: PT

text: presence

text: of

text: eﬀective

text: magnetic

text: ﬂux.

text: From

text: the

text: eigen

text: spec-

text: Parity-time

text: symmetric

text: systems

text: possess

text: intrigu-

text: PT

text: trum,

text: we

text: obtained

text: the

text: -symmetric

text: phase

text: transition

text: ing

text: features

text: that

text: derive

text: from

text: non-Hermiticity

text: and

text: the

text: PT

text: PT

text: point

text: and

text: revealed

text: that

text: the

text: uniformly

text: coupled

text: trimer

text: symmetry.

text: The

text: -symmetric

text: systems

text: have

text: been

text: chain

text: with

text: balanced

text: gain

text: and

text: loss

text: at

text: its

text: ends

text: has

text: one

text: extensively

text: investigated,

text: both

text: theoretically

text: and

text: ex-

text: PT

text: exceptional

text: point,

text: where

text: three

text: eigenstates

text: coalesce

text: at

text: perimentally

text: The

text: -symmetry

text: breaking

text: zero

text: energy.

text: In

text: the

text: trimer

text: ring,

text: a

text: nonreciprocal

text: cou-

text: nonreciprocal

text: reﬂectionless

text: and

text: coherent

text: per-

text: PT

text: pling

text: connects

text: the

text: head

text: and

text: tail

text: of

text: the

text: trimer

text: chain

text: fect

text: absorber

text: have

text: been

text: observed.

text: In

text: and

text: the

text: nonreciprocal

text: coupling

text: factors

text: correspond

text: to

text: a

text: symmetric

text: systems

text: have

text: been

text: realized

text: by

text: on-chip

text: de-

text: gauge

text: invariant

text: ﬁeld,

text: which

text: can

text: be

text: seen

text: as

text: an

text: eﬀective

text: vices,

text: known

text: as

text: two

text: coupled

text: whispering

text: gallery

text: mode

text: PT

text: magnetic

text: ﬂux.

text: For

text: the

text: trimer

text: ring,

text: the

text: -symmetric

text: ring

text: microresonators

text: These

text: resonators

text: have

text: high

text: phase

text: transition

text: point

text: can

text: be

text: a

text: two

text: or

text: three

text: eigenstates

text: quality

text: factors,

text: and

text: a

text: gain

text: in

text: one

text: resonator,

text: induced

text: coalescence.

text: To

text: determined

text: the

text: dynamical

text: features

text: at

text: by

text: pumping

text: the

text: doped

text: ions,

text: can

text: balance

text: losses

text: in

text: both

text: PT

text: the

text: exceptional

text: point

text: in

text: the

text: trimer

text: systems,

text: we

text: exam-

text: resonators.

text: The

text: realized

text: -symmetric

text: coupled

text: waveg-

text: ined

text: the

text: time

text: evolution

text: of

text: diﬀerent

text: initial

text: states.

text: For

text: uides

text: and

text: resonators

text: are

text: mainly

text: described

text: by

text: the

text: two-

text: PT

text: the

text: trimer

text: chain,

text: the

text: probability

text: increases

text: at

text: the

text: excep-

text: site

text: models,

text: which

text: act

text: as

text: a

text: -symmetric

text: dimer

text: tional

text: point

text: in

text: a

text: power

text: law.

text: Because

text: of

text: the

text: three-state

text: The

text: two-site

text: dimers

text: are

text: the

text: most

text: concise

text: and

text: simple

text: PT

text: PT

text: coalescence

text: the

text: highest

text: order

text: in

text: the

text: power

text: -symmetric

text: systems,

text: however,

text: all

text: the

text: symmetry

text: law

text: is

text: four;

text: however,

text: this

text: probability

text: increases

text: quadrat-

text: breaking

text: power

text: oscillation

text: and

text: gain

text: induced

text: ically

text: at

text: the

text: when

text: the

text: initial

text: state

text: is

text: only

text: related

text: large

text: nonlinearity

text: were

text: found

text: in

text: these

text: sys-

text: to

text: the

text: eigenstate

text: and

text: the

text: ﬁrst

text: associated

text: state.

text: In

text: the

text: tems.

text: Previous

text: study

text: demonstrated

text: the

text: diﬀerences

text: of

text: PT

text: PT

text: trimer

text: ring,

text: the

text: -symmetric

text: phase

text: transition

text: point

text: -symmetry

text: and

text: pseudo-Hermiticity

text: in

text: non-Hermitian

text: is

text: a

text: two-state

text: coalescence

text: except

text: for

text: the

text: eﬀective

text: trimer

text: chain

text: systems

text: In

text: an

text: optomechanical

text: system

text: Z

text: magnetic

text: ﬂux,

text: which

text: is

text: Φ

text: (n

text: For

text: the

text: coupled

text: with

text: an

text: active

text: resonator,

text: high-order

text: EPs

text: are

text: use-

text: the

text: coalesced

text: two

text: eigenenergy

text: and

text: the

text: third

text: one

text: are

text: ful

text: for

text: low-power

text: mechanical

text: cooling

text: Triple-cavity

text: all

text: nonzero.

text: The

text: evolved

text: probability

text: of

text: an

text: initial

text: state

text: supermodes

text: was

text: analysised

text: based

text: on

text: thye

text: coupled

text: mode

text: PT

text: has

text: four

text: typical

text: behaviors,

text: (i)

text: unchanged,

text: (ii)

text: oscillation,

text: theory

text: Besides,

text: -symmetric

text: oligomers

text: were

text: in-

text: (iii)

text: quadratic

text: increase,

text: and

text: (iv)

text: quadratic

text: increase

text: with

text: vestigated

text: in

text: nonlinearity

text: systems,

text: the

text: solution

text: and

text: non-

text: oscillation.

text: Notably,

text: the

text: oscillation

text: periods

text: are

text: the

text: same,

text: linear

text: dynamics

text: were

text: examined

text: The

text: asymmetric

text: because

text: they

text: are

text: coalesced

text: eigenenergy

text: dependent,

text: i.e.,

text: scattering

text: properties

text: for

text: left

text: and

text: right

text: propagation

text: were

text: PT

text: We

text: also

text: investigated

text: the

text: scattering

text: properties

text: of

text: demonstrated

text: for

text: nonlinear

text: -symmetric

text: oligomers

text: em-

text: the

text: trimer

text: systems,

text: particularly,

text: the

text: spectral

text: singulari-

text: bedded

text: in

text: linear

text: lead;

text: the

text: plane

text: waves

text: are

text: dynamically

text: ties

text: and

text: asymmetric

text: wave

text: emissions

text: under

text: the

text: inﬂuence

text: unstable

text: except

text: in

text: the

text: vicinity

text: of

text: the

text: linear

text: limit;

text: how-

text: of

text: nonreciprocal

text: coupling.

text: ever,

text: the

text: asymmetric

text: transmissions

text: persist

text: for

text: Gaussian

text: wave

text: packets

text: The

text: remainder

text: of

text: this

text: paper

text: is

text: organized

text: as

text: follows.

text: In

text: PT

text: PT

text: This

text: paper

text: focuses

text: on

text: the

text: -symmetric

text: trimer.

text: We

text: Sec.

text: we

text: present

text: the

text: trimer

text: systems,

text: the

text: -symmetric

text: PT

text: investigated

text: two

text: -symmetric

text: trimer

text: systems

text: under

text: phase

text: diagram,

text: and

text: the

text: spectrum.

text: In

text: Sec.

text: we

text: investi-

text: diﬀerent

text: boundary

text: conditions

text: (one

text: with

text: an

text: open

text: chain

text: gate

text: the

text: dynamics

text: at

text: the

text: exceptional

text: point.

text: In

text: Sec.

text: we

text: form

text: and

text: the

text: other

text: with

text: a

text: closed

text: ring

text: form)

text: and

text: ana-

text: explore

text: the

text: scattering

text: properties

text: of

text: the

text: trimer

text: systems,

text: lytically

text: solved

text: their

text: Hamiltonians.

text: We

text: concentrate

text: on

text: especially

text: the

text: spectral

text: singularities

text: and

text: asymmetric

text: wave


text: 3 FIG. 3. (Color online) (a) Critical gain and loss of Eq. 5 at EP2, and (b) the spectrum at EP2 as functions of magnetic ﬂux Φ. The green dotted line in (b) represents the two coa- lesced energies, and the parameters are κ = 1 and J = 1/2. PT -symmetric region shrinks when γ increases. For weak gain and loss γ < J, the trajectory of exceptional point PT forms an island with broken symmetry [Fig. 2(b)]. PT The isolated broken -symmetric region in parameter space was previously found in a dimerized photonic crys- PT PT tals [41], where two kinds of transitions were found: FIG. 2. (Color online) Magnetic ﬂux suppresses the tran- PT sition. (a) Phase diagram of the tPriTmer ring in the κ and (Pi)T reentering exact -symmetric phase from broken γ plane. The yellow region is the -symmetric phase for -symmetric phase at higher non-Hermiticity; and (ii) PT Φ = 0. The -symmetric region becomes large as magnetic coalescence of EPs from the Brillouin zone center and PT ﬂux increases, the blue region is the additional -symmetric boundary generates higher order of EPs in the interior region for Φ = πP/4T. All the colored regions are for Φ = π/2, of the Brillouin zone [41]. By contrast, we have similar which shows the -symmetric region at its maximum. (b-d) conclusions related to parameters κ and Φ. There are PT Phase diagram of the trimer ring in the Φ and κ plane. The PT also two kinds of transitions in the trimer ring. (i) √ green regions are the -symmetric phase for (b) γ/J = 1/2, As the coupling strength κ increases, the system crosses (c) γ/J = 1, and (d) γ/J = 3. PT PT broken -symmetric phase from exact -symmetric phase and reenters it [Fig. 2(b)]; (ii) EPs from magnetic ﬂux Φ = 0 and Φ = π coalesce and form higher order region has additional areas highlighted in green compared ± ∈ Z EPs at Φ = π/2 [Figs. 2(c) and 2(d)]. At gain and with Φ = 2nπ + π/4 (n ). The trimer ring has the PT loss γ = J, the trimer ring is an EP2 exceptional point largest region with exact -symmetric phase. All of at κ = 0 for any magnetic ﬂux, and the trimer ring is the areas in Fig. 2 ﬁlled with colors represent the ex- PT PT in the exact -symmetric phase at Φ = nπ + π/2 for act -symmetric phase, whereas the white areas are (cid:54) | | PT PT κ = 0 [Fig. 2(c)]. However, for γ/J > 1, the - the broken -symmetric p∈haZse for half magnetic ﬂux symmetric is most robust at Φ = nπ + π/2 (n ∈ Z√). The quantum Φ = 2nπ + π/2 (n ). PT | | trimer is -symmetric at κ/J > 1 for γ/J = 3. As The critical balanced gain and loss rate is the small- PT √ Fig. 2(d) reveals, the trimer ring is in the broken - est at coupling κ = 0 and it increases as the coupling κ | | √ PT symmetric region for coupling κ/J < 1 and in the exact in the form of γ = J2 + 2κ2. At this -symmetric PT | | c -symmetric region for κ/J > 2.43. Additionally, the phase transition point (γ = J2 + 2κ2), three eigen- PT c -symmetric region expands as magnetic ﬂux increases states of the trimer ring coalesce (except for the trimer ∈ Z from Φ = 2nπ to Φ = 2nπ + π/2 (n ) for the cou- reduced to a dimer at coupling κ = 0 (γ = J), which | | PT c pling in between 1 < κ/J < 2.43, and subsequently is an EP2). In Figs. 2(b-d), the -symmetric phase shrinks as magnetic ﬂux increases from Φ = 2nπ + π/2 diag|ram |in the parameter plane of Φ and κ is plotted. to Φ = (2n + 1)π (n ∈ Z). For γ/J < 0, as shown in Figs. 2(b,c), we acquire four PT PT typical rules: (i) the -symmetric∈pZhase is the most In the -symmetric syst(cid:80)em, the coalesced states en- fragile at magnetic ﬂux Φ = nπ (n ) and κ/J = 1; countered are mostly the extensively investigated EP2, PT (ii) the exact -symmetric region widens as the mag- which leads to a probability increase under a power law netic ﬂux increases; (iiPi)Tat coupling κ = 0, magnetic with a highest order of two, 2n=0 αntn [42]. In the open √ ﬂux does not aﬀect the -symmetric of the trimer ring, chain form trimer shown in Fig. 1(a), the only exceptional PT and the trimer ring is in the exact -symmetric phase; point is an EP3 at γ = 2κ. EP3 is a high-order coales- PT c and (iv) the symmetry is robust at magnetic ﬂux cence, where the coalesced states consist of one eigenstate ∈ Z (cid:80) Φ = nπ + π/2 (n ) and the trimer ring is in the and two associated states. The probability in the system PT exact -symmetric phase. All four rules remain un- increases under a power law, and the evolved probability changed as γ increases from 0 to J; however, the exact is a polynomial function of time, i.e., 4 α tn. No- n=0 n ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: FIG.

text: (Color

label: caption

text: online)

text: (a)

text: Critical

text: gain

text: and

text: loss

text: of

text: Eq.

text: at

text: and

text: (b)

text: the

text: spectrum

text: at

text: as

text: functions

text: of

text: magnetic

text: ﬂux

text: Φ.

text: The

text: green

text: dotted

text: line

text: in

text: (b)

text: represents

text: the

text: two

text: coa-

text: lesced

text: energies,

text: and

text: the

text: parameters

text: are

text: κ

text: and

text: J

text: PT

text: -symmetric

text: region

text: shrinks

text: when

text: γ

text: increases.

text: For

text: weak

text: gain

text: and

text: loss

text: γ

text: J,

text: the

text: trajectory

text: of

text: exceptional

text: point

text: PT

text: forms

text: an

text: island

text: with

text: broken

text: symmetry

text: [Fig.

text: PT

text: The

text: isolated

text: broken

text: -symmetric

text: region

text: in

text: parameter

text: space

text: was

text: previously

text: found

text: in

text: a

text: dimerized

text: photonic

text: crys-

text: PT

text: PT

text: tals

text: where

text: two

text: kinds

text: of

text: transitions

text: were

text: found:

text: FIG.

text: (Color

text: online)

text: Magnetic

text: ﬂux

text: suppresses

text: the

text: tran-

text: PT

text: sition.

text: (a)

text: Phase

text: diagram

text: of

text: the

text: tPriTmer

text: ring

text: in

text: the

text: κ

text: and

text: (Pi)T

text: reentering

text: exact

text: -symmetric

text: phase

text: from

text: broken

text: γ

text: plane.

text: The

text: yellow

text: region

text: is

text: the

text: -symmetric

text: phase

text: for

text: -symmetric

text: phase

text: at

text: higher

text: non-Hermiticity;

text: and

text: (ii)

text: PT

text: Φ

text: The

text: -symmetric

text: region

text: becomes

text: large

text: as

text: magnetic

text: coalescence

text: of

text: EPs

text: from

text: the

text: Brillouin

text: zone

text: center

text: and

text: PT

text: ﬂux

text: increases,

text: the

text: blue

text: region

text: is

text: the

text: additional

text: -symmetric

text: boundary

text: generates

text: higher

text: order

text: of

text: EPs

text: in

text: the

text: interior

text: region

text: for

text: Φ

text: All

text: the

text: colored

text: regions

text: are

text: for

text: Φ

text: of

text: the

text: Brillouin

text: zone

text: By

text: contrast,

text: we

text: have

text: similar

text: which

text: shows

text: the

text: -symmetric

text: region

text: at

text: its

text: maximum.

text: (b-d)

text: conclusions

text: related

text: to

text: parameters

text: κ

text: and

text: Φ.

text: There

text: are

text: PT

text: Phase

text: diagram

text: of

text: the

text: trimer

text: ring

text: in

text: the

text: Φ

text: and

text: κ

text: plane.

text: The

text: PT

text: also

text: two

text: kinds

text: of

text: transitions

text: in

text: the

text: trimer

text: ring.

text: (i)

text: green

text: regions

text: are

text: the

text: -symmetric

text: phase

text: for

text: (b)

text: γ/J

text: As

text: the

text: coupling

text: strength

text: κ

text: increases,

text: the

text: system

text: crosses

text: (c)

text: γ/J

text: and

text: (d)

text: γ/J

text: PT

text: PT

text: broken

text: -symmetric

text: phase

text: from

text: exact

text: -symmetric

text: phase

text: and

text: reenters

text: it

text: [Fig.

text: (ii)

text: EPs

text: from

text: magnetic

text: ﬂux

text: Φ

text: and

text: Φ

text: π

text: coalesce

text: and

text: form

text: higher

text: order

text: region

text: has

text: additional

text: areas

text: highlighted

text: in

text: green

text: compared

text: Z

text: EPs

text: at

text: Φ

text: [Figs.

text: and

text: At

text: gain

text: and

text: with

text: Φ

text: (n

text: The

text: trimer

text: ring

text: has

text: the

text: PT

text: loss

text: γ

text: J,

text: the

text: trimer

text: ring

text: is

text: an

text: exceptional

text: point

text: largest

text: region

text: with

text: exact

text: -symmetric

text: phase.

text: All

text: of

text: at

text: κ

text: for

text: any

text: magnetic

text: ﬂux,

text: and

text: the

text: trimer

text: ring

text: is

text: the

text: areas

text: in

text: Fig.

text: ﬁlled

text: with

text: colors

text: represent

text: the

text: ex-

text: PT

text: PT

text: in

text: the

text: exact

text: -symmetric

text: phase

text: at

text: Φ

text: nπ

text: for

text: act

text: -symmetric

text: phase,

text: whereas

text: the

text: white

text: areas

text: are

text: PT

text: PT

text: κ

text: [Fig.

text: However,

text: for

text: γ/J

text: the

text: the

text: broken

text: -symmetric

text: p∈haZse

text: for

text: half

text: magnetic

text: ﬂux

text: symmetric

text: is

text: most

text: robust

text: at

text: Φ

text: nπ

text: (n

text: Z√).

text: The

text: quantum

text: Φ

text: (n

text: PT

text: trimer

text: is

text: -symmetric

text: at

text: κ/J

text: for

text: γ/J

text: As

text: The

text: critical

text: balanced

text: gain

text: and

text: loss

text: rate

text: is

text: the

text: small-

text: PT

text: Fig.

text: reveals,

text: the

text: trimer

text: ring

text: is

text: in

text: the

text: broken

text: est

text: at

text: coupling

text: κ

text: and

text: it

text: increases

text: as

text: the

text: coupling

text: κ

text: PT

text: symmetric

text: region

text: for

text: coupling

text: κ/J

text: and

text: in

text: the

text: exact

text: in

text: the

text: form

text: of

text: γ

text: At

text: this

text: -symmetric

text: PT

text: c

text: -symmetric

text: region

text: for

text: κ/J

text: Additionally,

text: the

text: phase

text: transition

text: point

text: (γ

text: three

text: eigen-

text: PT

text: c

text: -symmetric

text: region

text: expands

text: as

text: magnetic

text: ﬂux

text: increases

text: states

text: of

text: the

text: trimer

text: ring

text: coalesce

text: (except

text: for

text: the

text: trimer

text: Z

text: from

text: Φ

text: to

text: Φ

text: (n

text: for

text: the

text: cou-

text: reduced

text: to

text: a

text: dimer

text: at

text: coupling

text: κ

text: (γ

text: J),

text: which

text: PT

text: c

text: pling

text: in

text: between

text: κ/J

text: and

text: subsequently

text: is

text: an

text: In

text: Figs.

text: the

text: -symmetric

text: phase

text: shrinks

text: as

text: magnetic

text: ﬂux

text: increases

text: from

text: Φ

text: diag|ram

text: |in

text: the

text: parameter

text: plane

text: of

text: Φ

text: and

text: κ

text: is

text: plotted.

text: to

text: Φ

text: (n

text: Z).

text: For

text: γ/J

text: as

text: shown

text: in

text: Figs.

text: we

text: acquire

text: four

text: PT

text: PT

text: typical

text: rules:

text: (i)

text: the

text: -symmetric∈pZhase

text: is

text: the

text: most

text: In

text: the

text: -symmetric

text: the

text: coalesced

text: states

text: en-

text: fragile

text: at

text: magnetic

text: ﬂux

text: Φ

text: nπ

text: (n

text: and

text: κ/J

text: countered

text: are

text: mostly

text: the

text: extensively

text: investigated

text: PT

text: (ii)

text: the

text: exact

text: -symmetric

text: region

text: widens

text: as

text: the

text: mag-

text: which

text: leads

text: to

text: a

text: probability

text: increase

text: under

text: a

text: power

text: law

text: netic

text: ﬂux

text: increases;

text: (iiPi)Tat

text: coupling

text: κ

text: magnetic

text: with

text: a

text: highest

text: order

text: of

text: two,

text: αntn

text: In

text: the

text: open

text: ﬂux

text: does

text: not

text: aﬀect

text: the

text: -symmetric

text: of

text: the

text: trimer

text: ring,

text: chain

text: form

text: trimer

text: shown

text: in

text: Fig.

text: the

text: only

text: exceptional

text: PT

text: and

text: the

text: trimer

text: ring

text: is

text: in

text: the

text: exact

text: -symmetric

text: phase;

text: point

text: is

text: an

text: at

text: γ

text: is

text: a

text: high-order

text: coales-

text: PT

text: c

text: and

text: (iv)

text: the

text: symmetry

text: is

text: robust

text: at

text: magnetic

text: ﬂux

text: cence,

text: where

text: the

text: coalesced

text: states

text: consist

text: of

text: one

text: eigenstate

text: Z

text: Φ

text: nπ

text: (n

text: and

text: the

text: trimer

text: ring

text: is

text: in

text: the

text: and

text: two

text: associated

text: states.

text: The

text: probability

text: in

text: the

text: system

text: PT

text: exact

text: -symmetric

text: phase.

text: All

text: four

text: rules

text: remain

text: un-

text: increases

text: under

text: a

text: power

text: law,

text: and

text: the

text: evolved

text: probability

text: changed

text: as

text: γ

text: increases

text: from

text: to

text: J;

text: however,

text: the

text: exact

text: is

text: a

text: polynomial

text: function

text: of

text: time,

text: i.e.,

text: α

text: tn.

text: No-

text: n

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##