text: Asymptotic properties of extremal Kähler metrics of Poincaré type In this direction, the so-called Yau-Tian-Donaldson conjecture predicts that in the algebraic case, the existence of extremal Kähler metrics is equivalent to a stability condition, close to the Geometric Invariant Theory, on the polarised manifold: Conjecture 1 Let (X, L) be a compact polarised manifold. Then there exists an extremal Kähler metric in if and only if is K-stable relatively to a c (L) (X, L) 1 maximal torus of . Aut0(X, L) This conjecture, ﬁrst designed for Kähler-Einstein metrics on Fano manifolds [Yau93, Tia97], was reformulated [Don01] for constant scalar curvature Kähler metrics (an important special case of extremal metrics), and ﬁnally adapted to extremal Kähler metrics [Mab04, Szé07]. This problem is still widely open in the “if” direction, except for the notable case of its specialisation to Kähler-Einstein metrics on Fano manifolds, see [CDS12a, CDS12b, CDS13] and [Tia12]. Within the scope of ﬁnding necessary conditions for the existence of extremal metrics, this article provides constraints to the existence of extremal Kähler met- rics with cusp singularities along a divisor in a compact Kähler manifold. Cusp singularities are compatible with the extremal condition, in the sense that such singular canonical metrics have already been produced [Szé06]; they appear more speciﬁcally along a continuity path between stable and unstable polarisations, when following smooth extremal metrics. We believe in this respect that extremal Käh- ler metrics with cusp singularities might be of crucial interest in the study of Conjecture 1, as particular degenerations of smooth extremal metrics. Following [Auv11, Auv13] for the deﬁnition of the class of metrics we are in- vestigating, ﬁx a simple normal crossing divisor in a compact Kähler mani- D fold , , of of polydiscs of holomorphic coordinates (X, J, ω ) dimC X = m X U X ∩ { · · · } ∈ of radius , such that for some (z1, . . . , zm) 1 U D = z1 zk = 0 k = k(U) { }. 2 0, . . . , m D(cid:80)eﬁnition 2 Let ω(cid:80)be a smooth (1, 1)-form on X\D. We say that ω is a Poincaré type Kähler metric if for all and as above, is quasi-isometric to the product U k ω k | i|dzj∧dz|j | + m idzj ∧ dzj, and has bounded derivatives at any order witjh=1rezsjp2elcotg2t(ozjt)h(cid:0)is mo(cid:80)jd=ekl+o1n \ . (cid:1) U D We say moreover that has class if , where is smooth ω [ω ] ω = ω + ddcϕ ϕ \ O − | | X X on X D, ϕ = 1 + k log[ log( zj )] , and ϕ has bounded derivatives at any positive order for the mojd=e1l metric, in the above charts. Notice that this deﬁnition allows a rather loose behaviour near the divisor, in the sense that one can easily produce Poincaré type metrics such that their restrictions to directions parallel to the divisor does not converge near the divisor. Our ﬁrst main result states nonetheless that such a convergence does occur for ⊂ extremal Poincaré type Kähler metrics, when is smooth: D X 2 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Asymptotic

text: properties

text: of

text: extremal

text: Kähler

text: metrics

text: of

text: Poincaré

text: type

text: In

text: this

text: direction,

text: the

text: so-called

text: Yau-Tian-Donaldson

text: conjecture

text: predicts

text: that

text: in

text: the

text: algebraic

text: case,

text: the

text: existence

text: of

text: extremal

text: Kähler

text: metrics

text: is

text: equivalent

text: to

text: a

text: stability

text: condition,

text: close

text: to

text: the

text: Geometric

text: Invariant

text: Theory,

text: on

text: the

text: polarised

text: manifold:

text: Conjecture

text: Let

text: (X,

text: L)

text: be

text: a

text: compact

text: polarised

text: manifold.

text: Then

text: there

text: exists

text: an

text: extremal

text: Kähler

text: metric

text: in

text: if

text: and

text: only

text: if

text: is

text: K-stable

text: relatively

text: to

text: a

text: c

text: (L)

text: (X,

text: L)

text: maximal

text: torus

text: of

text: L)

text: This

text: conjecture,

text: ﬁrst

text: designed

text: for

text: Kähler-Einstein

text: metrics

text: on

text: Fano

text: manifolds

text: was

text: reformulated

text: for

text: constant

text: scalar

text: curvature

text: Kähler

text: metrics

text: (an

text: important

text: special

text: case

text: of

text: extremal

text: metrics),

text: and

text: ﬁnally

text: adapted

text: to

text: extremal

text: Kähler

text: metrics

text: This

text: problem

text: is

text: still

text: widely

text: open

text: in

text: the

text: “if”

text: direction,

text: except

text: for

text: the

text: notable

text: case

text: of

text: its

text: specialisation

text: to

text: Kähler-Einstein

text: metrics

text: on

text: Fano

text: manifolds,

text: see

text: and

text: Within

text: the

text: scope

text: of

text: ﬁnding

text: necessary

text: conditions

text: for

text: the

text: existence

text: of

text: extremal

text: metrics,

text: this

text: article

text: provides

text: constraints

text: to

text: the

text: existence

text: of

text: extremal

text: Kähler

text: met-

text: rics

text: with

text: cusp

text: singularities

text: along

text: a

text: divisor

text: in

text: a

text: compact

text: Kähler

text: manifold.

text: Cusp

text: singularities

text: are

text: compatible

text: with

text: the

text: extremal

text: condition,

text: in

text: the

text: sense

text: that

text: such

text: singular

text: canonical

text: metrics

text: have

text: already

text: been

text: produced

text: they

text: appear

text: more

text: speciﬁcally

text: along

text: a

text: continuity

text: path

text: between

text: stable

text: and

text: unstable

text: polarisations,

text: when

text: following

text: smooth

text: extremal

text: metrics.

text: We

text: believe

text: in

text: this

text: respect

text: that

text: extremal

text: Käh-

text: ler

text: metrics

text: with

text: cusp

text: singularities

text: might

text: be

text: of

text: crucial

text: interest

text: in

text: the

text: study

text: of

text: Conjecture

text: as

text: particular

text: degenerations

text: of

text: smooth

text: extremal

text: metrics.

text: Following

text: for

text: the

text: deﬁnition

text: of

text: the

text: class

text: of

text: metrics

text: we

text: are

text: in-

text: vestigating,

text: ﬁx

text: a

text: simple

text: normal

text: crossing

text: divisor

text: in

text: a

text: compact

text: Kähler

text: mani-

text: D

text: fold

text: of

text: of

text: polydiscs

text: of

text: holomorphic

text: coordinates

text: (X,

text: J,

text: ω

text: dimC

text: X

text: m

text: X

text: U

text: X

text: of

text: radius

text: such

text: that

text: for

text: some

text: zm)

text: U

text: D

text: zk

text: k

text: k(U)

text: m

text: Let

text: a

text: smooth

text: on

text: X\D.

text: We

text: say

text: that

text: ω

text: is

text: a

text: Poincaré

text: type

text: Kähler

text: metric

text: if

text: for

text: all

text: and

text: as

text: above,

text: is

text: quasi-isometric

text: to

text: the

text: product

text: U

text: k

text: ω

text: k

text: i|dzj∧dz|j

text: m

text: idzj

text: dzj,

text: and

text: has

text: bounded

text: derivatives

text: at

text: any

text: order

text: U

text: D

text: We

text: say

text: moreover

text: that

text: has

text: class

text: if

text: where

text: is

text: smooth

text: ω

text: [ω

text: ω

text: ω

text: ddcϕ

text: ϕ

text: O

text: X

text: X

text: on

text: X

text: D,

text: ϕ

text: k

text: log[

text: log(

text: zj

text: and

text: ϕ

text: has

text: bounded

text: derivatives

text: at

text: any

text: positive

text: order

text: for

text: the

text: metric,

text: in

text: the

text: above

text: charts.

text: Notice

text: that

text: this

text: deﬁnition

text: allows

text: a

text: rather

text: loose

text: behaviour

text: near

text: the

text: divisor,

text: in

text: the

text: sense

text: that

text: one

text: can

text: easily

text: produce

text: Poincaré

text: type

text: metrics

text: such

text: that

text: their

text: restrictions

text: to

text: directions

text: parallel

text: to

text: the

text: divisor

text: does

text: not

text: converge

text: near

text: the

text: divisor.

text: Our

text: ﬁrst

text: main

text: result

text: states

text: nonetheless

text: that

text: such

text: a

text: convergence

text: does

text: occur

text: for

text: extremal

text: Poincaré

text: type

text: Kähler

text: metrics,

text: when

text: is

text: smooth:

text: D

text: X

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Asymptotic properties of extremal Kähler metrics of Poincaré type (cid:82) |D | LDemma≡1.6 For ϕ as in Theorem 1.1, ∆∗×(Y \E) ϕ(vϕ) 2ϕ et volϕ = 0, and thus . (v ) 0 ϕ ϕ (cid:90) F (cid:90) Proof of Lemma 1.6. — Denote by the function ϕ (cid:55)−→ ∧ s v Dϕ 2dϑ = 4π v Dϕ { } ϕ s { }× \ ϕ s t=s s (Y E) (this holds by S -invariance under the ﬁrst integral); one moment’s thought – use 1 the ﬂow along – gives: F F · for all ∈ R. Now (cid:90) Z = ∂t + ϑ∂ϑ ϕ(·×+a,·) \= ϕ( + a) a formula (5) – or rather its analogue on – can then be rewritten as: ∆ (Y E) s (cid:90) 0 = etv L (v ) volϕ (cid:10) (cid:11) ϕ ϕ ϕ × \ ∆s (Y E) D F − F − = d(etv ), δϕ (v ) volϕ + (s) ( s). × \ ϕ ϕ ϕ ϕ ϕ ϕ ∆s (Y E) (cid:90) Similarly, (cid:10) (cid:11) D(cid:90) d(etv ),δϕ (v ) v(cid:10)olϕ (cid:11) × \ ϕ ϕ ϕ ϕ ∆s (Y E) D D G − G − = (etv ), (v ) volϕ + (s) ( s), × \ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ∆s (Y E) if G denotes on R the func(cid:90)tion (cid:90) ϕ (cid:55)−→ ∧ s dϕ 2dϑ = 4π dϕ, { } s { }× \ s t=s s (Y E) (cid:104) where (cid:0) (cid:1) − dϕ :=e−t D (v ) ∇ϕ(e(cid:0)tv ), ∂ (ωsϕ(cid:1)−)m 1 (cid:105) t ϕ ϕ ϕ t (m 1)! − − D ∇ · | ∧ ∧ (ωϕ)m 2 (cid:0) (v ) ϕ(etv ),(cid:1) dc v s− ϕ ϕ ϕ Y Y ϕ (m 2)! − D ∇ ∇ (ωϕ)m 1 = (v ) (v ϕt +(cid:0) ϕv ), ∂ s−(cid:1) ϕ ϕ ϕ ϕ t (m 1)! − − D ∇ ∇ · | ∧ ∧ (ωϕ)m 2 on \ (v ) v ϕt + ϕv , dc v s− Y E. ϕ ϕ ϕ ϕ Y Y ϕ (m 2)! 14 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Asymptotic

label: paragraph

text: properties

text: of

text: extremal

text: Kähler

text: metrics

text: of

text: Poincaré

text: type

text: |D

text: For

text: ϕ

text: as

text: in

text: Theorem

text: ∆∗×(Y

text: \E)

text: ϕ(vϕ)

text: 2ϕ

text: et

text: volϕ

text: and

text: thus

text: (v

text: ϕ

text: ϕ

text: F

text: Proof

text: of

text: Lemma

text: Denote

text: by

text: the

text: function

text: ϕ

text: s

text: v

text: Dϕ

text: 4π

text: v

text: Dϕ

text: ϕ

text: s

text: ϕ

text: s

text: t=s

text: s

text: (Y

text: E)

text: (this

text: holds

text: by

text: S

text: -invariance

text: under

text: the

text: ﬁrst

text: integral);

text: one

text: moment’s

text: thought

text: use

text: the

text: ﬂow

text: along

text: gives:

text: F

text: F

text: for

text: all

text: R.

text: Now

text: Z

text: ∂t

text: ϑ∂ϑ

text: ϕ(·×+a,·)

text: ϕ(

text: a)

text: a

text: formula

text: or

text: rather

text: its

text: analogue

text: on

text: can

text: then

text: be

text: rewritten

text: as:

text: (Y

text: E)

text: s

text: etv

text: L

text: (v

text: volϕ

text: ϕ

text: ϕ

text: ϕ

text: ∆s

text: (Y

text: E)

text: D

text: F

text: F

text: d(etv

text: δϕ

text: (v

text: volϕ

text: (s)

text: s).

text: ϕ

text: ϕ

text: ϕ

text: ϕ

text: ϕ

text: ϕ

text: ∆s

text: (Y

text: E)

text: Similarly,

text: d(etv

text: ),δϕ

text: (v

text: ϕ

text: ϕ

text: ϕ

text: ϕ

text: ∆s

text: (Y

text: E)

text: D

text: D

text: G

text: G

text: (etv

text: (v

text: volϕ

text: (s)

text: s),

text: ϕ

text: ϕ

text: ϕ

text: ϕ

text: ϕ

text: ϕ

text: ϕ

text: ∆s

text: (Y

text: E)

text: if

text: G

text: denotes

text: on

text: R

text: the

text: ϕ

text: s

text: dϕ

text: 4π

text: dϕ,

text: s

text: s

text: t=s

text: s

text: (Y

text: E)

text: where

text: dϕ

text: :=e−t

text: D

text: (v

text: t

text: ϕ

text: ϕ

text: ϕ

text: t

text: (m

text: D

text: (ωϕ)m

text: (v

text: ϕ(etv

text: dc

text: v

text: s−

text: ϕ

text: ϕ

text: ϕ

text: Y

text: Y

text: ϕ

text: (m

text: D

text: (ωϕ)m

text: (v

text: (v

text: ϕt

text: ϕv

text: ϕ

text: ϕ

text: ϕ

text: ϕ

text: t

text: (m

text: D

text: (ωϕ)m

text: on

text: (v

text: v

text: ϕt

text: ϕv

text: dc

text: v

text: s−

text: Y

text: E.

text: ϕ

text: ϕ

text: ϕ

text: ϕ

text: Y

text: Y

text: ϕ

text: (m

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Asymptotic properties of extremal Kähler metrics of Poincaré type We conclude as (cid:80)follows: let be a nonnegative smooth function with compact χ support in and with integral . We deﬁne the time-dependent holomorphic (0, 1) 1 vector ﬁeld ∞ − , and the associated ﬂow . We set Z := χ(t j)Z σ (cid:36) := ∗ for atll ; thejs=ej0are metricjs of constant scalar curvature, in some ﬁtxed σ − (cid:36) t net igj0hboj0urhood of L . One easily checks that(cid:0) indeed (cid:36)(cid:1) = (cid:36) for all j ≥ j , t=j j ≥ 0 as well as the claimed asymptotic properties of . For instance, for , ((cid:36) ) t j L − L − t t · 0 ; since is bounded at ∂ (cid:36) = (cid:36) = χ(t [t]) (cid:36) = χ(t [t])d (cid:36) (Z , ) (cid:36) ant ytorderZ,t(cid:107) t (cid:107) is conZt[rt]ollted by | |, whicht is[t]controlled byt ∂ (cid:36) Z η(2ε + δ ) · t t Cκ,α ∈ { [t] } k([t]) ∞[t] (where is deﬁned by ), hence goes to as goes to . k( ) (cid:96) j , . . . , j − 0 t ∈ k((cid:96)) k((cid:96))∈1 Finally, as Z = 0 for t [0, j ], (cid:36) = (cid:36) L ; now (cid:36) was arbitrarily chosen in L so thatt (cid:107)ω − (cid:36) (cid:107) 0 ≤ 0ε . Buj0t given (cid:36) ∈ Lj0, there exists t so that (cid:107)ω − (cid:36) (cid:107) ≤t=jε0 , andj0 foCrκ,αall t ≥0 t , d (ω , L ) < ε ; the assertio0n on the gent0iricityj0ofCκ,α foll0ows by applying th0e pCrκe,vαiouts constru0ction to for (cid:36) (ω − ) which one can k0eep the same . t+t0 j0 (cid:3) j 0 3 Constant scalar curvature Poincaré type Kähler metrics on the complement of a divisor (cid:80) We ﬁx in this part a compact Kähler manifold , and a simple normal (X, ω ) X crossing divisor D = N D (the D are the smooth irreducible components). j=1 j j 3.1 Basic tools and statements of the results Reminder: ﬁbration near the divisor, and previous results. — The details concerning following material, necessary for what follows, can be found in [Auv11, Auv13]. To ﬁx ideas, assume ﬁrst that is smooth, and even reduced to one D component. One ca(cid:0)n endow a(cid:1)tubular neighbourhood N of with an S -action D 1 and an S -invariant projection N → , and constructAan S -invariant function 1 p : D 1 such that − | | OA − at any order for Poincaré type metrics on t t = log log( σ 2) + (e t) \ ∈ { } | · | X D, where σ O([D]) is such that D = σ = 0 , and any smooth hermitian metric on . Up to adjusting N and ∈ R, we get this way a circle ﬁbration [D] A N \ → ∞ × A (cid:0) (cid:1) , for which we construct a connexion 1-form , q = (t, p) : D [A, + ) D η such that A − O − at any order – notice the analogy with the formula dct = 2e tη + (e t) − on the model ∗. Setting moreover − − | | JCdt = 2e tdϑ ∆ ω := ω ddc log log( σ 2) | · | X with a good choice of – which does not aﬀect what precedes – we have that ω is of Poincaré type, and near , D (11) ∧ − ∗ O − ω = dt 2e tη + p ω + (e t) D | at any order, where . This can be generalised when the divisor has ω = ω D X D several components, and the ﬁbrations respect the crossings when they exist; we 22 ##LTLine##

text: Asymptotic

text: properties

text: of

text: extremal

text: Kähler

text: metrics

text: of

text: Poincaré

text: type

text: We

text: conclude

text: as

text: let

text: be

text: a

text: nonnegative

text: smooth

text: function

text: with

text: compact

text: χ

text: support

text: in

text: and

text: with

text: integral

text: We

text: deﬁne

text: the

text: time-dependent

text: holomorphic

text: vector

text: ﬁeld

text: and

text: the

text: associated

text: ﬂow

text: We

text: set

text: Z

text: χ(t

text: j)Z

text: σ

text: for

text: atll

text: metricjs

text: of

text: constant

text: scalar

text: curvature,

text: in

text: some

text: ﬁtxed

text: σ

text: t

text: net

text: of

text: L

text: One

text: easily

text: checks

text: indeed

text: for

text: all

text: j

text: j

text: t=j

text: j

text: as

text: well

text: as

text: the

text: claimed

text: asymptotic

text: properties

text: of

text: For

text: instance,

text: for

text: t

text: j

text: L

text: L

text: t

text: t

text: since

text: is

text: bounded

text: at

text: χ(t

text: [t])

text: χ(t

text: [t])d

text: (Z

text: ant

text: t

text: is

text: conZt[rt]ollted

text: by

text: whicht

text: is[t]controlled

text: byt

text: Z

text: η(2ε

text: δ

text: t

text: t

text: Cκ,α

text: [t]

text: k([t])

text: ∞[t]

text: (where

text: is

text: deﬁned

text: by

text: hence

text: goes

text: to

text: as

text: goes

text: to

text: k(

text: j

text: j

text: t

text: Finally,

text: as

text: Z

text: for

text: t

text: j

text: L

text: now

text: was

text: arbitrarily

text: chosen

text: in

text: L

text: so

text: thatt

text: 0ε

text: given

text: there

text: exists

text: t

text: so

text: that

text: foCrκ,αall

text: t

text: ≥0

text: t

text: d

text: (ω

text: L

text: ε

text: the

text: on

text: the

text: by

text: applying

text: pCrκe,vαiouts

text: to

text: for

text: (ω

text: which

text: one

text: can

text: the

text: same

text: j

text: Constant

text: scalar

text: curvature

text: Poincaré

text: type

text: Kähler

text: metrics

text: on

text: the

text: complement

text: of

text: a

text: divisor

text: We

text: ﬁx

text: in

text: this

text: part

text: a

text: compact

text: Kähler

text: manifold

text: and

text: a

text: simple

text: normal

text: (X,

text: ω

text: X

text: crossing

text: divisor

text: D

text: N

text: D

text: (the

text: D

text: are

text: the

text: smooth

text: irreducible

text: components).

text: j

text: j

text: Basic

text: tools

text: and

text: statements

text: of

text: the

text: results

text: Reminder:

text: ﬁbration

text: near

text: the

text: divisor,

text: and

text: previous

text: results.

text: The

text: details

text: concerning

text: following

text: material,

text: necessary

text: for

text: what

text: follows,

text: can

text: be

text: found

text: in

text: To

text: ﬁx

text: ideas,

text: assume

text: ﬁrst

text: that

text: is

text: smooth,

text: and

text: even

text: reduced

text: to

text: one

text: D

text: component.

text: One

text: endow

text: neighbourhood

text: N

text: of

text: with

text: an

text: S

text: -action

text: D

text: and

text: an

text: S

text: -invariant

text: projection

text: N

text: and

text: constructAan

text: S

text: -invariant

text: function

text: p

text: D

text: such

text: that

text: OA

text: at

text: any

text: order

text: for

text: Poincaré

text: type

text: metrics

text: on

text: t

text: t

text: log

text: log(

text: σ

text: (e

text: t)

text: X

text: D,

text: where

text: σ

text: O([D])

text: is

text: such

text: that

text: D

text: σ

text: and

text: any

text: smooth

text: hermitian

text: metric

text: on

text: Up

text: to

text: adjusting

text: N

text: and

text: R,

text: we

text: get

text: this

text: way

text: a

text: circle

text: ﬁbration

text: [D]

text: A

text: N

text: A

text: for

text: which

text: we

text: construct

text: a

text: connexion

text: q

text: (t,

text: p)

text: D

text: [A,

text: D

text: η

text: such

text: that

text: A

text: O

text: at

text: any

text: order

text: notice

text: the

text: analogy

text: with

text: the

text: formula

text: dct

text: tη

text: (e

text: t)

text: on

text: the

text: model

text: Setting

text: moreover

text: JCdt

text: tdϑ

text: ω

text: ω

text: ddc

text: log

text: log(

text: σ

text: X

text: with

text: a

text: good

text: choice

text: of

text: which

text: does

text: not

text: aﬀect

text: what

text: precedes

text: we

text: have

text: that

text: ω

text: is

text: of

text: Poincaré

text: type,

text: and

text: near

text: D

text: O

text: ω

text: dt

text: tη

text: p

text: ω

text: (e

text: t)

text: D

text: at

text: any

text: order,

text: where

text: This

text: can

text: be

text: generalised

text: when

text: the

text: divisor

text: has

text: ω

text: ω

text: D

text: X

text: D

text: several

text: components,

text: and

text: the

text: ﬁbrations

text: respect

text: the

text: crossings

text: when

text: they

text: exist;

text: we

text: ##LTLine##


text: Asymptotic properties of extremal Kähler metrics of Poincaré type 3. is a critical weight for L , and if denotes a basis of δ = 0 0 ∗× (f , . . . , f ) L , then there exists ∆sucDh that 1 r ker δ > 0 ωD (cid:0) 0 (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (20) L0 ∗× : Cκ+4,α [0, ∞) × D ⊕ span χ(t)f −→ Cκ,α [0, ∞) × D ∆ D 0,δ j δ is an isomorphism for all ∈ and any ∈ N × . δ (0, δ ) (κ, α) (0, 1) 0 Remark 3.9 1. Observe that as L0 ∗× is inv(cid:0)ariant b(cid:1)y translation in the t direc- tion, we can translate these stateme∆ntsDon ∞ × for any . [A, ) D A 2. The map (20) is well-deﬁned, as L L on { ≥ } 0 ∗× χ(t)f = (f ) = 0 t 1 for . ∆ D j ωD j j = 1, . . . , r (cid:82) Proof of Lemma 3.8. — We start by points 1. and 2. We will use several times ∞ | | ∞ the following inequality: if and , then (cid:90) f(0) = 0 (cid:90)f 2e2δtdt < 0 ∞ ∞ | | ≥ | | (21) ∂ f 2e2δtdt δ2 f 2e2δtdt t 0 0 (cid:90) (cid:90) (cid:90) (see [Biq97, Lemme 6.1]); this follows from Cauchy-Schwarz inequality applied to ∞ ∞ ∞ 1 0 = ∂ (f2e2δt)dt = (f∂ f)e2δtdt + δ f2e2δtdt t t 2 0 0 0 where(cid:0) is sm(cid:1)ooth(cid:0)with co(cid:1)mpact support an(cid:0)d vanis(cid:1)hes at , plus a density f t = 0 argument. Now, to simplify expressions, we denote L by L; we recall that 0 ∗× L(cid:90)= 1 ∂ − ∂2 2 + ∂t − ∂2 + L + ∆ ◦ ∂t − ∂2 , ∆so tDhat 2 t t t ωD ωD t (cid:90) (cid:90) L u ue2δt dt vol = D ∞ × [0, ) D 1 (cid:90) − (cid:90) − u(∂ ∂2)2u e2δtdt vol + u(∂ ∂2)u e2δtdt vol 2 ∞ × t t D ∞ × t t D [0, ) D [0, ) D L − + u u e2δtdt vol + u(∂ ∂2)∆ u e2δtdt vol . (cid:82) ∞ × ωD D ∞ × t t(cid:0) D (cid:1) D [0, ) D [0, ) D We deal successively with the diﬀerent summands to estimate the positivity of ∞ × uLu e2δtdt vol . We assume from now on that u ∈ L2,4 [0, ∞] × D . [0, ) D D 0,δ 33 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Asymptotic

text: properties

text: of

text: extremal

text: Kähler

text: metrics

text: of

text: Poincaré

text: type

label: list

text: is

text: a

text: critical

text: weight

text: for

text: L

text: and

text: if

text: denotes

text: a

text: basis

text: of

text: δ

text: (f

text: f

text: L

text: then

text: there

text: exists

text: ∆sucDh

text: that

text: r

text: ker

text: δ

text: ωD

label: equation

text: D

text: span

text: χ(t)f

text: Cκ,α

text: D

text: D

text: j

text: δ

text: is

text: an

text: isomorphism

text: for

text: all

text: and

text: any

text: N

text: δ

text: δ

text: (κ,

text: α)

text: Remark

text: Observe

text: that

text: as

text: is

text: translation

text: in

text: the

text: t

text: direc-

text: tion,

text: we

text: can

text: translate

text: these

text: stateme∆ntsDon

text: for

text: any

text: [A,

text: D

text: A

text: The

text: map

text: is

text: well-deﬁned,

text: as

text: L

text: L

text: on

text: χ(t)f

text: (f

text: t

text: for

text: D

text: j

text: ωD

text: j

text: j

text: r

text: Proof

text: of

text: Lemma

text: We

text: start

text: by

text: points

text: and

text: We

text: will

text: use

text: several

text: times

text: the

text: following

text: inequality:

text: if

text: and

text: then

text: f

text: δ2

text: f

text: t

text: (see

text: Lemme

text: this

text: follows

text: from

text: Cauchy-Schwarz

text: inequality

text: applied

text: to

text: (f∂

text: δ

text: t

text: t

text: is

text: support

text: at

text: plus

text: a

text: density

text: f

text: t

text: argument.

text: Now,

text: to

text: simplify

text: expressions,

text: we

text: denote

text: L

text: by

text: L;

text: we

text: recall

text: that

text: ∂2

text: ∂t

text: ∂2

text: L

text: ∂t

text: ∂2

text: ∆so

text: tDhat

text: t

text: t

text: t

text: ωD

text: ωD

text: t

text: L

text: u

text: dt

text: vol

text: D

text: D

text: u(∂

text: vol

text: u(∂

text: vol

text: t

text: t

text: D

text: t

text: t

text: D

text: D

text: D

text: L

text: u

text: u

text: vol

text: u(∂

text: ∂2)∆

text: u

text: vol

text: ωD

text: D

text: t

text: D

text: D

text: D

text: D

text: We

text: deal

text: successively

text: with

text: the

text: diﬀerent

text: summands

text: to

text: estimate

text: the

text: positivity

text: of

text: uLu

text: vol

text: We

text: assume

text: from

text: now

text: on

text: that

text: u

text: D

text: D

text: D

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Asymptotic properties of extremal Kähler metrics of Poincaré type Proof. — Let us settle the linear independence assertion. We chose as a ω | | D limit point of , and built so that , which means that ω + ddc (Π ϕ) ω ω = ω there exists gDoing tDo ∞0so that | tendDs to Din . Moreover (t ) (d Π ϕ) { } 0 C0(D) · (cid:80) (cid:96) (cid:2)· (cid:0) · (cid:1)D−(cid:3)(cid:12) 0 t=(cid:80)·t(cid:96) · O − at any Π (Z ϕ) = Z (Π ϕ) + Z (Π⊥ϕ) (cid:12) Π⊥(Z ϕ) = Z (Π ϕ) + (e t) 0 j j 0 j j j 0 o(cid:80)rder, fo(cid:2)r (cid:0) . Cons(cid:1)e(cid:3)quently, for(cid:80)any , as the are tangent to j = 1, . . . , r (λ ) ≤ ≤ Z D, (σ˜)∗ r λ (σ˜)∗ f + Π (Z · ϕ) (cid:0) → jr1 jλ rf(cid:1) as (cid:96) goejs to ∞. Now if r λ (σ˜j=)∗1 fj + Π (Zj · ϕ) 0 →j 0, thent=t(cid:96) r λ fj=1= j0,jhence λ = ·(cid:0)· · = λ =(cid:1) 0. j=N1otjice that j(22) 0is ejquivalent to L j=1 j j · O −1 . We getrthese f + Π (Z ϕ) = (e t) ϕ j 0 j latter equations in a way similar to the starting point of this part, L ∂ (Π ϕ) = O − . Namely, being ﬁxed, we choose in a neighbourhood oϕf cot ord0inates (e t) j X U around some point of where is given by , we extend (z1, . . . , zm) D D z1 = 0 | ∇ in independently of and denote this holomorphic extension by ZZ˜j;Dw=e havωDe:fjZ˜ =UZ + O(e−t) at a(cid:0)nyz1order.(cid:1)Since(cid:0) s(ω ) is cons(cid:1)tant, j j j ϕ · L − (cid:0)L (cid:1) ˜ 0 = Z s(ω ) = ∆ Λ ω 2 ω , (cid:37)(ω ) . j ϕ ϕ ϕ Z˜j ϕ Z˜j ϕ ϕ ϕ We will thus be done if we prove that L ω = ddc p∗f +(cid:0)Π (Z˜ · ϕ) (cid:1)on U up to some O(e−t) at any order, as L = 1∆2Z˜j+ ϕ((cid:37) , ddc·) , jand a0s rejplacing Z˜ by Z (which is globally deﬁned arounϕd )2 inϕthe eϕxpressioϕn L ˜ · onljy givesj D Π (Z ϕ) rise to an error term which is O − at any order. ϕ 0 j (e t) (cid:0) Now (cid:1) − O − near , where extends in inde- ω = ω˜ + ddc(ϕ t) + (e t) D ω˜ ω U ϕ D D D L pendently of , and with the at any order. Thus by Cartan’s formula, z1 O ω = d ω˜D(oZ˜nj, ·) +byLZd˜jedﬁdnci(tϕio−n t)o+f O(cid:0)(,e−atn)d. Othbasetrv(cid:1)e th∗at ω(cid:0)D(Zj, ·) = g(cid:1)DO(JD−∇ωDnZfe˜jja,r·ϕ) =. dc f D Z dc(p f ) = dc f + (e t) D MDo(cid:0)rjeover L ddc(ϕ −(cid:1) t) = ddcj Z˜ · (ϕ − t) = djdc Π (Z˜X ·j ϕ) , as dt(Z˜ ) and Π⊥(Z˜ · ϕ) aZr˜je O(e−t) at any orderj. Summing these estim0 atjes thus gives L j ω = ddc p∗jf + Π (Z˜ · ϕ) + O(e−t) near D on U, as wanted. Z˜j(cid:0) ϕ (cid:4) j (cid:1) 0 j − To complete the proof of Proposition 3.7, just set ψ := χ(A t)(σ˜)∗ f + · , for . j j (cid:3) Π (Z ϕ) j = 1, . . . , r 0 j 4 The extremal case ∗ × \ 4.1 Extremal Kähler metrics on the model ∆ (Y E) ∗ × \ 4.1.1 Potentials of extremal Kähler metrics of Poincaré type on ∆ (Y E) We come back in this section to the point of view of Part , and recall that − ∧ − , that is a ﬁxed Kähler metric of Poincaré type ω ∗ = ddct = dt 2e tdϑ ω \ of∆class on \ , and that Y E on ∗ × \ . [ω ] Y E ω = ω ∗ + ω \ ∆ (Y E) Y 0 ∆ Y E 39 ##LTLine## ##LTLine##

text: Asymptotic

text: properties

text: of

text: extremal

text: Kähler

text: metrics

text: of

text: Poincaré

text: type

text: Proof.

text: Let

text: us

text: settle

text: the

text: linear

text: independence

text: assertion.

text: We

text: chose

text: as

text: a

text: ω

text: D

text: limit

text: point

text: of

text: and

text: built

text: so

text: that

text: which

text: means

text: that

text: ω

text: ddc

text: (Π

text: ϕ)

text: ω

text: ω

text: ω

text: there

text: exists

text: gDoing

text: tDo

text: that

text: tendDs

text: to

text: Din

text: Moreover

text: (t

text: (d

text: Π

text: ϕ)

text: O

text: at

text: any

text: Π

text: (Z

text: ϕ)

text: Z

text: (Π

text: ϕ)

text: Z

text: (Π⊥ϕ)

text: Π⊥(Z

text: ϕ)

text: Z

text: (Π

text: ϕ)

text: (e

text: t)

text: j

text: j

text: j

text: j

text: j

text: as

text: the

text: are

text: tangent

text: to

text: j

text: r

text: (λ

text: Z

text: D,

text: (σ˜)∗

text: r

text: λ

text: (σ˜)∗

text: f

text: Π

text: (Z

text: ϕ)

text: jλ

text: as

text: goejs

text: to

text: Now

text: if

text: r

text: λ

text: fj

text: Π

text: (Zj

text: ϕ)

text: →j

text: r

text: λ

text: λ

text: λ

text: that

text: ejquivalent

text: to

text: L

text: j

text: j

text: O

text: −1

text: We

text: getrthese

text: f

text: Π

text: (Z

text: ϕ)

text: (e

text: t)

text: ϕ

text: j

text: j

text: latter

text: equations

text: in

text: a

text: way

text: similar

text: to

text: the

text: starting

text: point

text: of

text: this

text: part,

text: L

text: (Π

text: ϕ)

text: O

text: Namely,

text: being

text: ﬁxed,

text: we

text: choose

text: in

text: a

text: neighbourhood

text: oϕf

text: cot

text: (e

text: t)

text: j

text: X

text: U

text: around

text: some

text: point

text: of

text: where

text: is

text: given

text: by

text: we

text: extend

text: zm)

text: D

text: D

text: in

text: independently

text: of

text: and

text: denote

text: this

text: holomorphic

text: extension

text: by

text: ZZ˜j;Dw=e

text: havωDe:fjZ˜

text: =UZ

text: O(e−t)

text: at

text: s(ω

text: is

text: j

text: j

text: j

text: ϕ

text: L

text: Z

text: s(ω

text: Λ

text: ω

text: ω

text: j

text: ϕ

text: ϕ

text: ϕ

text: Z˜j

text: ϕ

text: Z˜j

text: ϕ

text: ϕ

text: ϕ

text: We

text: will

text: thus

text: be

text: done

text: if

text: we

text: prove

text: that

text: L

text: ω

text: ddc

text: p∗f

text: (Z˜

text: ϕ)

text: U

text: up

text: to

text: some

text: O(e−t)

text: at

text: any

text: order,

text: as

text: L

text: ddc·)

text: jand

text: rejplacing

text: Z˜

text: by

text: Z

text: (which

text: is

text: globally

text: deﬁned

text: arounϕd

text: inϕthe

text: eϕxpressioϕn

text: L

text: onljy

text: givesj

text: D

text: Π

text: (Z

text: ϕ)

text: rise

text: to

text: an

text: error

text: term

text: which

text: is

text: O

text: at

text: any

text: order.

text: ϕ

text: j

text: (e

text: t)

text: Now

text: O

text: near

text: where

text: extends

text: in

text: inde-

text: ω

text: ω˜

text: ddc(ϕ

text: t)

text: (e

text: t)

text: D

text: ω˜

text: ω

text: U

text: ϕ

text: D

text: D

text: D

text: L

text: pendently

text: of

text: and

text: with

text: the

text: at

text: any

text: order.

text: Thus

text: by

text: Cartan’s

text: formula,

text: O

text: ω

text: d

text: ω˜D(oZ˜nj,

text: +byLZd˜jedﬁdnci(tϕio−n

text: t)o+f

text: th∗at

text: dc

text: f

text: D

text: Z

text: dc(p

text: f

text: dc

text: f

text: (e

text: t)

text: D

text: L

text: ddc(ϕ

text: t)

text: ddcj

text: Z˜

text: (ϕ

text: t)

text: djdc

text: Π

text: (Z˜X

text: ·j

text: ϕ)

text: as

text: dt(Z˜

text: and

text: Π⊥(Z˜

text: ϕ)

text: aZr˜je

text: O(e−t)

text: at

text: any

text: orderj.

text: Summing

text: these

text: atjes

text: thus

text: gives

text: L

text: j

text: ω

text: ddc

text: p∗jf

text: Π

text: (Z˜

text: ϕ)

text: O(e−t)

text: near

text: D

text: on

text: U,

text: as

text: wanted.

text: ϕ

text: j

text: j

text: To

text: complete

text: the

text: proof

text: of

text: Proposition

text: just

text: set

text: ψ

text: χ(A

text: t)(σ˜)∗

text: f

text: for

text: j

text: j

text: Π

text: (Z

text: ϕ)

text: j

text: r

text: j

text: The

text: extremal

text: case

text: Extremal

text: Kähler

text: metrics

text: on

text: the

text: model

text: (Y

text: E)

text: Potentials

text: of

text: extremal

text: Kähler

text: metrics

text: of

text: Poincaré

text: type

text: on

text: (Y

text: E)

text: We

text: come

text: back

text: in

text: this

text: section

text: to

text: the

text: point

text: of

text: view

text: of

text: Part

text: and

text: recall

text: that

text: that

text: is

text: a

text: ﬁxed

text: Kähler

text: metric

text: of

text: Poincaré

text: type

text: ω

text: ddct

text: dt

text: tdϑ

text: ω

text: of∆class

text: on

text: and

text: that

text: Y

text: E

text: on

text: [ω

text: Y

text: E

text: ω

text: ω

text: ω

text: (Y

text: E)

text: Y

text: Y

text: E

text: ##LTLine##

text: ##LTLine##


text: Asymptotic properties of extremal Kähler metrics of Poincaré type ∇ where we set , and recall that . s = s(cid:82)(ω ) (cid:82)K = ϕs ϕ ϕ ϕ ϕ Remark 4.6 When s = s, one recovers(cid:82)the obstruction s > s in this way: if ≡ ϕ Dj− s = s, K 0, and \ s ∆ etj ωϕm = s \ ∆ etj ωϕm = 4πs Vol(D ). One ϕ ϕ X D ϕ ϕ m! X D ϕ m! j can moreover rewrite constraint (23) as: \ L (etj)ωϕm < 0. X D ϕ m! Proof of Proposition 4.5. — We ﬁrst make an easy but crucial observation about the statement. If indeed is bounded for Poincaré type metrics (see [Auv13, ddcetj Part 5]), making integrable for the volume form , the analogue about ∆ etj ωm · deserves a slϕight explanation. Now, as an holomϕ orphic vector ﬁeld on K etj L2 ϕ\ , extends smoothly through , and its normal component vanishes along X D K D ϕ (cid:0) (cid:1) the divisor ( [Auv11], proof of Lemma 5.2). In other words, in an open set of U local coordinates centred at some point of such that is given by (z1, . . . , zm) D D · · · j j , can be written as , with z1 = 0 K Re z1f ∂ + f ∂ + + f ∂ f , . . . , f holomorphϕic on . Moreover in , 1 ∂z1 −2 ∂z2 | | m ∂zwm ith a1 smootmh U U etj = log( z1 2) + f f · − j,U · j,U function by construction, and therefore, . From this K etj = Re(f ) + K f (cid:0) · (cid:1) ϕ \ 1 ϕ j,U we get that is bounded near , hence on , and is thus for . K etj D X D L1 ωm This beingϕsaid, let us come to thejproof of the proposition itself. By Remϕ ark 4.4, on − { } × \ , one has: p 1 t (D E ) j k j j (24) ω = a dt ∧ 2e−tjη + p∗ωϕ + ε(t , z), ϕ k j j j tk k ≥ ≥ · where: a c (c such that ω 2cω), the ωϕ := ω + ddc [Π ϕ(t , )] are uniformly bokunded on \ aϕt any order in tthke PoinDcjaré senD(cid:0)sje a0n,jd poksitivel(cid:1)y (D E ) bounded below on compactj subj sets, and all the , ≥ , tend to 0 in ∞ \ ∂(cid:96) ε (cid:96) 0 C (D E ) as goes to ∞. Consequently, a direct computatitoj n yields, on − { }×loc \j j : k p 1 t (D E ) j k j j (25) s = −2a dt ∧ 2e−tjη + p∗(cid:37)(ωϕ ) + ε (t , z), ϕ k j j j tk 1 k with tending to 0 in the same sense as above. We can now write, for (cid:90) ε ε s 1 { ≤ } \ large and seeing as a subset of : t s X D j − (cid:90) ∧ ωm 1 (cid:90) s ddcetj (cid:16) ϕ− (cid:17) ϕ { ≤ } (m 1)! tj s − − (cid:90) ∧ ωm 1 (cid:90) − ∧ ∧ ωm 1 = d s dcetj ϕ− ds dcetj ϕ− ϕ ϕ { ≤ } (m 1)! { ≤ } (m 1)! tj s − tj s ∧ ωm 1 − · ωm = s dcetj ϕ− K etj ϕ , ϕ ϕ { } (m 1)! { ≤ } m! tj=s tj s 43 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Asymptotic

text: properties

text: of

text: extremal

text: Kähler

text: metrics

text: of

text: Poincaré

text: type

text: where

text: we

text: set

text: and

text: recall

text: that

text: s

text: ϕs

text: ϕ

text: ϕ

text: ϕ

text: ϕ

text: Remark

text: When

text: s

text: s,

text: one

text: obstruction

text: s

text: s

text: in

text: this

text: way:

text: if

text: ϕ

text: Dj−

text: s

text: s,

text: K

text: and

text: s

text: etj

text: ωϕm

text: s

text: etj

text: ωϕm

text: Vol(D

text: One

text: ϕ

text: ϕ

text: X

text: D

text: ϕ

text: ϕ

text: m!

text: X

text: D

text: ϕ

text: m!

text: j

text: can

text: moreover

text: rewrite

text: constraint

text: as:

text: L

text: (etj)ωϕm

text: X

text: D

text: ϕ

text: m!

text: Proof

text: of

text: Proposition

text: We

text: ﬁrst

text: make

text: an

text: easy

text: but

text: crucial

text: observation

text: about

text: the

text: statement.

text: If

text: indeed

text: is

text: bounded

text: for

text: Poincaré

text: type

text: metrics

text: (see

text: ddcetj

text: Part

text: making

text: integrable

text: for

text: the

text: volume

text: form

text: the

text: analogue

text: about

text: etj

text: ωm

text: deserves

text: a

text: slϕight

text: explanation.

text: Now,

text: as

text: an

text: holomϕ

text: orphic

text: vector

text: ﬁeld

text: on

text: K

text: etj

text: ϕ\

text: extends

text: smoothly

text: through

text: and

text: its

text: normal

text: component

text: vanishes

text: along

text: X

text: D

text: K

text: D

text: ϕ

text: the

text: divisor

text: proof

text: of

text: Lemma

text: In

text: other

text: words,

text: in

text: an

text: open

text: set

text: of

text: U

text: local

text: coordinates

text: centred

text: at

text: some

text: point

text: of

text: such

text: that

text: is

text: given

text: by

text: zm)

text: D

text: D

text: j

text: j

text: can

text: be

text: written

text: as

text: with

text: K

text: Re

text: f

text: f

text: f

text: f

text: holomorphϕic

text: on

text: Moreover

text: in

text: −2

text: m

text: ∂zwm

text: ith

text: smootmh

text: U

text: U

text: etj

text: log(

text: f

text: f

text: j,U

text: j,U

text: function

text: by

text: construction,

text: and

text: therefore,

text: From

text: this

text: K

text: etj

text: Re(f

text: K

text: f

text: ϕ

text: ϕ

text: j,U

text: we

text: get

text: that

text: is

text: bounded

text: near

text: hence

text: on

text: and

text: is

text: thus

text: for

text: K

text: etj

text: D

text: X

text: D

text: ωm

text: This

text: beingϕsaid,

text: let

text: us

text: come

text: to

text: thejproof

text: of

text: the

text: proposition

text: itself.

text: By

text: Remϕ

text: ark

text: on

text: one

text: has:

text: p

text: t

text: (D

text: E

text: j

text: k

text: j

text: j

text: ω

text: a

text: dt

text: p∗ωϕ

text: ε(t

text: z),

text: ϕ

text: k

text: j

text: j

text: j

text: tk

text: k

text: where:

text: a

text: c

text: (c

text: such

text: that

text: ω

text: the

text: ωϕ

text: ω

text: ddc

text: [Π

text: ϕ(t

text: are

text: uniformly

text: bokunded

text: on

text: aϕt

text: any

text: order

text: in

text: tthke

text: PoinDcjaré

text: (D

text: E

text: bounded

text: below

text: on

text: compactj

text: subj

text: sets,

text: and

text: all

text: the

text: tend

text: to

text: in

text: ε

text: C

text: (D

text: E

text: as

text: goes

text: to

text: Consequently,

text: a

text: direct

text: computatitoj

text: n

text: yields,

text: on

text: }×loc

text: \j

text: j

text: k

text: p

text: t

text: (D

text: E

text: j

text: k

text: j

text: j

text: s

text: dt

text: ε

text: (t

text: z),

text: ϕ

text: k

text: j

text: j

text: j

text: tk

text: k

text: with

text: tending

text: to

text: in

text: the

text: same

text: sense

text: as

text: above.

text: We

text: can

text: now

text: write,

text: for

text: ε

text: ε

text: s

text: large

text: and

text: seeing

text: as

text: a

text: subset

text: of

text: t

text: s

text: X

text: D

text: j

text: ωm

text: s

text: ddcetj

text: ϕ−

text: ϕ

text: (m

text: tj

text: s

text: ωm

text: ωm

text: d

text: s

text: dcetj

text: ϕ−

text: ds

text: dcetj

text: ϕ−

text: ϕ

text: ϕ

text: (m

text: (m

text: tj

text: s

text: tj

text: s

text: ωm

text: ωm

text: s

text: dcetj

text: ϕ−

text: K

text: etj

text: ϕ

text: ϕ

text: ϕ

text: (m

text: m!

text: tj=s

text: tj

text: s

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: Asymptotic properties of extremal Kähler metrics of Poincaré type (cid:82) (cid:82) where we used Stokes’ theorem to pass from the integral of shape in the { ≤ } dα second line to the integral of shape in the third line (there is ntjo isnterference { } α with the other here, as the in ptjl=asy, and its diﬀerential, are ), and applied D (cid:82) α (cid:82) L1 (cid:96) ∇ ∧ ∧ − · a h(cid:82)ermitian identity to Kϕ = ϕsϕ to replace (cid:82)dsϕ dcetj (ωmϕm−11)! by Kϕ (cid:82)etj ωmϕm! in ∧ − − the last summand. Now { ≤ } s ddcetj ωϕm− 1 = { ≤ } s ∆ etj volϕ tends to − tj s ϕ∞ (m 1)! t·j s ϕ ϕ · \ s ∆ etj volϕ as s goes to ; likewise, { ≤ } K (cid:82)etj ωϕm tends to \ K X D aϕs ϕ goes to ∞. Finally, taking tj asnd uϕsing mth! e asymptotXicsD (2ϕ4) etj volϕ s (cid:82) s = t k ∧ − and (25) with the evoked uniform bounds, we get that { } s dcetj ωϕm− 1 = 4≥πs−Dj Vol(Dj),−he4nπceDtjh\Eejrae−ksu1lvto. lωtϕk +o(1) as k goes to ∞,tja=ntdk thϕe last in(tmegr1a)!l (cid:3)is c 1 Vol(D ) j Remark 4.7 We also get from this proof that (a ) tends to the inverse of the k k left-hand-side of (23), say, that depends only on , this for all sequence . a ω (t ) j (cid:55)→ ϕ · − k We hence sharpen Remark 4.4 by saying that: tends to t ∂ Π ϕ(t, ) (1 a ) in ∞ \ . tj 0,j j C (D E ) loc j j 4.2.3 Asymptotics of extremal Kähler metrics of Poincaré type (smooth divisor) When is smooth, we have a perfect analogue of Theorem 3.1: D Theorem 4.8 Assuming D(cid:0) smooth, let ω (cid:1) be an extremal metric of Poincaré type \ ϕ on , of class . One has near the asymptotics of Riemannian metrics: X D [ω ] D X j − ∗ O − g = a dt2 + (2e tjη)2 + p g + (e δtj) ϕ j j Dj · · | with an extremal Kähler metric on , such that , g (D , J ) [g (J , )] = [ω ] and whDejre is the inverse of the left-jhanDdj side of (23); tDhjis hDoj lds for allDj. a > 0 j j Proof. — The ingredients are the same as in the constant scalar curvature case. Namely, 1. starting from an equation analogous to (13), 2. we twist it after parametrising judiciously the “family of almost extremal metrics” (ωϕ) , and then t t 3. lead the appropriate asymptotic analysis of the arising Lichnerowicz-modelled operator. Before examining those three points, and more precisely why they adapt to the extremal situation, we assume that the divisor is reduced to one component, (cid:0) (cid:1) and we proceed to the following normalisation: call the inverse of a = a (ω ) the left hand-side of (23); up to working with − , whDich neDar ϕ can be written a 1ω D as − − − − − − – recaDll tϕhat − –, we a 1ω ddct + ddc a 1ϕ (a 1 1)t ω = ω + ddc(ϕ t) can Dassume that D, as D for all ϕ , by (23). We then a = 1 a (λω ) = λa (ω ) λ > 0 D D ϕ D ϕ recover, by Remarks 4.4 and 4.7, the extremal analogue of Proposition 3.3. The ﬁrst point goes as follows. Again we work on an open subset of coordi- U nates centred at some point of , such that is the local equation of (z1, . . . , zm) D z1 44 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Asymptotic

text: properties

text: of

text: extremal

text: Kähler

text: metrics

text: of

text: Poincaré

text: type

text: where

text: we

text: used

text: Stokes’

text: theorem

text: to

text: pass

text: from

text: the

text: integral

text: of

text: shape

text: in

text: the

text: dα

text: second

text: line

text: to

text: the

text: integral

text: of

text: shape

text: in

text: the

text: third

text: line

text: (there

text: is

text: ntjo

text: isnterference

text: α

text: with

text: the

text: other

text: here,

text: as

text: the

text: in

text: ptjl=asy,

text: and

text: its

text: diﬀerential,

text: are

text: and

text: applied

text: D

text: α

text: a

text: identity

text: to

text: Kϕ

text: ϕsϕ

text: to

text: replace

text: dcetj

text: by

text: Kϕ

text: ωmϕm!

text: in

text: the

text: last

text: summand.

text: Now

text: s

text: ddcetj

text: ωϕm−

text: s

text: etj

text: volϕ

text: tends

text: to

text: tj

text: s

text: ϕ∞

text: (m

text: t·j

text: s

text: ϕ

text: ϕ

text: s

text: etj

text: volϕ

text: as

text: s

text: goes

text: to

text: likewise,

text: K

text: ωϕm

text: tends

text: to

text: K

text: X

text: D

text: aϕs

text: ϕ

text: goes

text: to

text: Finally,

text: taking

text: tj

text: asnd

text: uϕsing

text: mth!

text: e

text: asymptotXicsD

text: etj

text: volϕ

text: s

text: s

text: t

text: k

text: and

text: with

text: the

text: evoked

text: uniform

text: bounds,

text: we

text: get

text: that

text: s

text: dcetj

text: ωϕm−

text: lωtϕk

text: as

text: k

text: goes

text: to

text: ∞,tja=ntdk

text: thϕe

text: last

text: c

text: Vol(D

text: j

text: Remark

text: We

text: also

text: get

text: from

text: this

text: proof

text: that

text: (a

text: tends

text: to

text: the

text: inverse

text: of

text: the

text: k

text: k

text: left-hand-side

text: of

text: say,

text: that

text: depends

text: only

text: on

text: this

text: for

text: all

text: sequence

text: a

text: ω

text: (t

text: j

text: ϕ

text: k

text: We

text: hence

text: sharpen

text: Remark

text: by

text: saying

text: that:

text: tends

text: to

text: t

text: Π

text: ϕ(t,

text: a

text: in

text: tj

text: j

text: C

text: (D

text: E

text: loc

text: j

text: j

text: Asymptotics

text: of

text: extremal

text: Kähler

text: metrics

text: of

text: Poincaré

text: type

text: (smooth

text: divisor)

text: When

text: is

text: smooth,

text: we

text: have

text: a

text: perfect

text: analogue

text: of

text: Theorem

text: D

text: Theorem

text: Assuming

text: smooth,

text: let

text: ω

text: be

text: an

text: extremal

text: metric

text: of

text: Poincaré

text: type

text: ϕ

text: on

text: of

text: class

text: One

text: has

text: near

text: the

text: asymptotics

text: of

text: Riemannian

text: metrics:

text: X

text: D

text: [ω

text: D

text: X

text: j

text: O

text: g

text: a

text: p

text: g

text: (e

text: δtj)

text: ϕ

text: j

text: j

text: Dj

text: with

text: an

text: extremal

text: Kähler

text: metric

text: on

text: such

text: that

text: g

text: (D

text: J

text: [g

text: (J

text: [ω

text: and

text: whDejre

text: is

text: the

text: inverse

text: of

text: the

text: left-jhanDdj

text: side

text: of

text: tDhjis

text: hDoj

text: lds

text: for

text: allDj.

text: a

text: j

text: j

text: Proof.

text: The

text: ingredients

text: are

text: the

text: same

text: as

text: in

text: the

text: constant

text: scalar

text: curvature

text: case.

text: Namely,

text: starting

text: from

text: an

text: equation

text: analogous

text: to

text: we

text: twist

text: it

text: after

text: parametrising

text: judiciously

text: the

text: “family

text: of

text: almost

text: extremal

text: metrics”

text: (ωϕ)

text: and

text: then

text: t

text: t

text: lead

text: the

text: appropriate

text: asymptotic

text: analysis

text: of

text: the

text: arising

text: Lichnerowicz-modelled

text: operator.

text: Before

text: examining

text: those

text: three

text: points,

text: and

text: more

text: precisely

text: why

text: they

text: adapt

text: to

text: the

text: extremal

text: situation,

text: we

text: assume

text: that

text: the

text: divisor

text: is

text: reduced

text: to

text: one

text: component,

text: and

text: we

text: proceed

text: to

text: the

text: following

text: normalisation:

text: call

text: the

text: inverse

text: of

text: a

text: a

text: (ω

text: the

text: left

text: hand-side

text: of

text: up

text: to

text: working

text: with

text: whDich

text: neDar

text: ϕ

text: can

text: be

text: written

text: a

text: 1ω

text: D

text: as

text: recaDll

text: tϕhat

text: we

text: a

text: 1ω

text: ddct

text: ddc

text: a

text: 1ϕ

text: (a

text: ω

text: ω

text: ddc(ϕ

text: t)

text: can

text: Dassume

text: that

text: D,

text: as

text: D

text: for

text: all

text: ϕ

text: by

text: We

text: then

text: a

text: a

text: (λω

text: λa

text: (ω

text: λ

text: D

text: D

text: ϕ

text: D

text: ϕ

text: recover,

text: by

text: Remarks

text: and

text: the

text: extremal

text: analogue

text: of

text: Proposition

text: The

text: ﬁrst

text: point

text: goes

text: as

text: follows.

text: Again

text: we

text: work

text: on

text: an

text: open

text: subset

text: of

text: coordi-

text: U

text: nates

text: centred

text: at

text: some

text: point

text: of

text: such

text: that

text: is

text: the

text: local

text: equation

text: of

text: zm)

text: D

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Asymptotic properties of extremal Kähler metrics of Poincaré type D D Knowing that , the assertion on now directly comes from (u ) = 0 (v ) the deﬁnitions of D ϕ ϕ∇ − and , andϕLeϕibniz rule. (cid:3) = ( ϕ) d v = etu ϕ ϕ ϕ D ∈ Finiteness of a weighted L2 norm of (v(cid:82)). — For g(cid:12)eneral (cid:12)φ K , we only know that D (φ˙) is bounded on ∆∗ × (Y \Eϕ); sϕince et ωφm is(cid:12) mutua(cid:12)lly bounded with the cbyeliﬁnndirtφieca–l hveorleu|m· e| fdoremnotdets∧thdeϑn∧or(ωmmYm−\−c1E1)o!m, p∆u∗t×e(dY \wmEi!)teht Dφ. (Cφ˙o)m2φbωmφmi!nihnagsenqouarleiatiseosn(1to) g φ φ and (2), we claim that this is indeed the case for such that is constant: ϕ s(ω ) (cid:90) ϕ (cid:12) (cid:12) Lemma 1.4 For ϕ as in Theorem 1(cid:12).1, (cid:12) D ∞ et (v ) 2 volϕ < , ∗× \ ϕ ϕ ϕ ∆ (Y E) − w(cid:82) here volϕ =(cid:12) ωϕm , an(cid:12) d where w(cid:82)e recall the notation: v = ϕ˙ 1. (cid:12) m! (cid:12) ϕ Proof of Lemma 1.4. — For ≥ , set {| | ≤ } ⊂ ∗. First we relate s 0 ∆ := t s ∆ × \ et D (v ) 2 volϕ to × \ etv Ls (v ) volϕ, where volϕ = ωϕm ; notice th∆osu(gYh Et)hat thϕe lϕattϕer integral∆iss a(YlwEa)ys 0ϕ, bϕy Lϕemma 1.2. Assume momm! entarily that E(cid:90) is empty, so that we wo(cid:90)rk on ∆ × Y . From (1) and as L = δϕδϕD , one s ϕ ϕ has: (cid:90) L (cid:10) D (cid:11) 0 = etv (v ) volϕ = etv δϕδϕ (v ) volϕ ϕ ϕ ϕ ϕ ϕ ϕ × × ∆s Y ∆s Y (cid:90) (cid:90) D = d(etv ), δϕ (v ) volϕ × ϕ ϕ ϕ ϕ ∆s Y ∧ − − − ∧ + esDϕ 2e sdϑ e sDϕ− 2esdϑ, { } s { − } s t=s t= s where − − D (cid:0) (ω(cid:1)ϕ)m 1 − D | ∧ ∧ (ωϕ)m 2 Dϕ := δϕ (v )(∂ ) t − δϕ (v ) dc v t − , t ϕ ϕ t (m 1)! ϕ ϕ Y Y ϕ (m 2)! · with . Here we use Stokes’ theorem, and the Kähler ωϕ = ω \ + ddc ϕ(s, ) identitises (cid:104)α, βY (cid:105)Eωm =Yα ∧ Jβ ∧ ωm−−1 and (δα)ωm = −dcα ∧ ωm−−1 for 1-forms, as well as ω = ωω±ϕm! + d ϕ˙ ∧ 2e∓sd(mϑ o1)n! slices {t =m±! s}, to comp(mute1)t!he boundary integrals. Oϕbserve sthat Ythe and − cancel each other in these terms, and there- es e s fore the integrands are bounded – for the metric , say – independently g \ + dϑ2 O Y E of . Conse(cid:90)quently the boundary int(cid:90)egrals are , that is: s (cid:10) (1) (cid:11) (5) L D O 0 = etv (v ) volϕ = d(etv ), δϕ (v ) volϕ + (1), × ϕ ϕ ϕ × ϕ ϕ ϕ ϕ ∆s Y ∆s Y 10 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Asymptotic

text: properties

text: of

text: extremal

text: Kähler

text: metrics

text: of

text: Poincaré

text: type

text: D

text: D

text: Knowing

text: that

text: the

text: assertion

text: on

text: now

text: directly

text: comes

text: from

text: (u

text: (v

text: the

text: deﬁnitions

text: of

text: D

text: ϕ

text: ϕ∇

text: and

text: andϕLeϕibniz

text: rule.

text: ϕ)

text: d

text: v

text: etu

text: ϕ

text: ϕ

text: ϕ

text: D

text: Finiteness

text: of

text: a

text: weighted

text: norm

text: of

text: For

text: K

text: we

text: only

text: know

text: that

text: D

text: (φ˙)

text: is

text: bounded

text: on

text: (Y

text: \Eϕ);

text: sϕince

text: et

text: ωφm

text: bounded

text: with

text: the

text: cbyeliﬁnndirtφieca–l

text: hveorleu|m·

text: e|

text: p∆u∗t×e(dY

text: \wmEi!)teht

text: Dφ.

text: g

text: φ

text: φ

text: and

text: we

text: claim

text: that

text: this

text: is

text: indeed

text: the

text: case

text: for

text: such

text: that

text: is

text: constant:

text: ϕ

text: s(ω

text: ϕ

text: Lemma

text: For

text: ϕ

text: as

text: in

text: Theorem

text: D

text: et

text: (v

text: volϕ

text: ϕ

text: ϕ

text: ϕ

text: (Y

text: E)

text: here

text: volϕ

text: ωϕm

text: d

text: where

text: recall

text: the

text: notation:

text: v

text: ϕ˙

text: m!

text: ϕ

text: Proof

text: of

text: Lemma

text: For

text: set

text: First

text: we

text: relate

text: s

text: t

text: s

text: et

text: D

text: (v

text: volϕ

text: to

text: etv

text: Ls

text: (v

text: volϕ,

text: where

text: volϕ

text: ωϕm

text: notice

text: th∆osu(gYh

text: Et)hat

text: thϕe

text: lϕattϕer

text: integral∆iss

text: a(YlwEa)ys

text: bϕy

text: Lϕemma

text: Assume

text: momm!

text: entarily

text: that

text: is

text: empty,

text: so

text: that

text: we

text: on

text: Y

text: From

text: and

text: as

text: L

text: δϕδϕD

text: one

text: s

text: ϕ

text: ϕ

text: has:

text: L

text: D

text: etv

text: (v

text: volϕ

text: etv

text: δϕδϕ

text: (v

text: volϕ

text: ϕ

text: ϕ

text: ϕ

text: ϕ

text: ϕ

text: ϕ

text: ∆s

text: Y

text: ∆s

text: Y

text: D

text: d(etv

text: δϕ

text: (v

text: volϕ

text: ϕ

text: ϕ

text: ϕ

text: ϕ

text: ∆s

text: Y

text: esDϕ

text: sdϑ

text: e

text: sDϕ−

text: s

text: s

text: t=s

text: t=

text: s

text: where

text: D

text: D

text: (ωϕ)m

text: Dϕ

text: δϕ

text: (v

text: t

text: δϕ

text: (v

text: dc

text: v

text: t

text: t

text: ϕ

text: ϕ

text: t

text: (m

text: ϕ

text: ϕ

text: Y

text: Y

text: ϕ

text: (m

text: with

text: Here

text: we

text: use

text: Stokes’

text: theorem,

text: and

text: the

text: Kähler

text: ωϕ

text: ω

text: ddc

text: ϕ(s,

text: identitises

text: βY

text: =Yα

text: Jβ

text: ωm−−1

text: and

text: (δα)ωm

text: −dcα

text: ωm−−1

text: for

text: as

text: well

text: as

text: ω

text: ωω±ϕm!

text: d

text: ϕ˙

text: slices

text: {t

text: =m±!

text: s},

text: to

text: boundary

text: integrals.

text: Oϕbserve

text: sthat

text: Ythe

text: and

text: cancel

text: each

text: other

text: in

text: these

text: terms,

text: and

text: there-

text: es

text: e

text: s

text: fore

text: the

text: integrands

text: are

text: bounded

text: for

text: the

text: metric

text: say

text: independently

text: g

text: O

text: Y

text: E

text: of

text: the

text: boundary

text: are

text: that

text: is:

text: s

text: L

text: D

text: O

text: etv

text: (v

text: volϕ

text: d(etv

text: δϕ

text: (v

text: volϕ

text: ϕ

text: ϕ

text: ϕ

text: ϕ

text: ϕ

text: ϕ

text: ϕ

text: ∆s

text: Y

text: ∆s

text: 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: 3.2 Random initial data 3.2.1 Minimization of the number of interphases Here, in Figures (8) and (9) the discrete components of u are randomly calculated following a uniform − ∀ ∈ 0 law on [ 1, 1] and we start from α0(t) = 0, t [0, T ]. As we see, here again, in all cases, the global procedure, illustrating the eﬀective numerical controllability by the boundaries and the robustness of the approach, since the data are very oscillating. Figure 8: (cid:15) = 0.01, ∆t = 0.01, T = 0.8, u = rand, N = 127 0 13 ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure##

label: section

text: Random

text: initial

text: data

text: Minimization

text: of

text: the

text: number

text: of

text: interphases

text: Here,

text: in

text: Figures

text: and

text: the

text: discrete

text: components

text: of

text: u

text: are

text: randomly

text: calculated

text: following

text: a

text: uniform

text: law

text: on

text: and

text: we

text: start

text: from

text: t

text: T

text: As

text: we

text: see,

text: here

text: again,

text: in

text: all

text: cases,

text: the

text: global

text: procedure,

text: illustrating

text: the

text: eﬀective

text: numerical

text: controllability

text: by

text: the

text: boundaries

text: and

text: the

text: robustness

text: of

text: the

text: approach,

text: since

text: the

text: data

text: are

text: very

text: oscillating.

text: Figure

label: caption

text: ∆t

text: T

text: u

text: rand,

text: N

text: ##LTFigure##

label: figure

font: default

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##


text: · · · When the diﬀerent phases U , i = 1, , m are known a priori, the merit function can be deﬁned I more precisely as (cid:88) m (cid:107) − (cid:107) F (u) = ω u U 2. (7) i i i=1 − ± For instance, with m = 2 and f(u) = u(u2 1), the two stable phases are U = 1 and the merit 1,2 function associated to U = +1 is 1 (cid:107) − (cid:107) (cid:90) F (u) = u 1 2, which can be considered directly in the continuous case with, e.g., L2 norm, giving rise to the merit 1 − function F (u) = (u(x) 1)2dx. 0 2.3 Global scheme We denote by u (t) the approximation of u(x , t) generated by the semi discrete scheme i i (cid:0) (cid:1) − − · · · du (t) 2u (t) u − (t) u (t) 1 − for i = 1, , N i + i i 1 i+1 + u (t) u2(t) 1 = 0 t > 0, (8) dt h2 (cid:15)2 i i − u (t) = u (t) hα(t) t > 0, (9) 0 1 u (t) = u (t) t > 0, (10) N+1 N u (0) = v . (11) i i in which we have implemented Neumann boundary condition to calculate the values u (t) and N+1 u (t). 0 As a time marching scheme, we will use a semi implicit one, in order to have a good stability: it is important since, as we will see hereafter, the calculation of optimal α(t) requires a great number of numerical solutions of this system and a not too small time step ∆t must be used. We ﬁx a value for the ﬁnal time T , T being large enough to obtain a steady state: in practice the solution converges toward equilibrium relatively fastly for small values of (cid:15), which is the case here. The · · · − time interval [0, T ] is subdivided into M subintervals of length ∆t, [t , t ], k = 0, M 1, with k k+1 t = k∆t and t = T . We note uM the numerical approximation to u¯. k M A ﬁrst idea is to compute α(t) as a piecewise constant function in time, say (cid:88) − M 1 α(t) = α χ . (12) k [t ,t ] k k+1 k=0 We remark here that calculating α(t) as a piecewise constant function, is a simple and stable ap- proach: other techniques allowing orthogonal polynomial (such as Fourier or Laguerre) could be used but the strong decreasing of the Fourier coeﬃcients make the problem ill-conditioned. Also, an heuristic method can be used for accelerating the numerical convergence: once computed αM(t) for a given M and ∆t, one can repeat the computation for 2M time steps (with ∆t/2), in order to get a more precise result α2M(t), starting from a mid-point interpolated value. 6 ##LTLine## ##LTLine## ##LTLine##

text: When

text: the

text: diﬀerent

text: phases

text: U

text: i

text: m

text: are

text: known

text: a

text: priori,

text: the

text: merit

text: function

text: can

text: be

text: deﬁned

text: I

text: more

text: precisely

text: as

text: m

text: F

text: (u)

text: ω

text: u

text: U

text: i

text: i

text: For

text: instance,

text: with

text: m

text: and

text: f(u)

text: the

text: two

text: stable

text: phases

text: are

text: U

text: and

text: the

text: merit

text: function

text: associated

text: to

text: U

text: is

text: F

text: (u)

text: u

text: which

text: can

text: be

text: considered

text: directly

text: in

text: the

text: continuous

text: case

text: with,

text: e.g.,

text: norm,

text: giving

text: rise

text: to

text: the

text: merit

text: function

text: F

text: (u)

text: (u(x)

text: Global

text: scheme

text: We

text: denote

text: by

text: u

text: (t)

text: the

text: approximation

text: of

text: u(x

text: t)

text: generated

text: by

text: the

text: semi

text: discrete

text: scheme

text: i

text: i

text: du

text: (t)

text: (t)

text: u

text: (t)

text: u

text: (t)

text: for

text: i

text: N

text: i

text: i

text: i

text: u

text: (t)

text: t

text: dt

text: i

text: i

text: u

text: (t)

text: u

text: (t)

text: hα(t)

text: t

text: u

text: (t)

text: u

text: (t)

text: t

text: N

text: u

text: v

text: i

text: i

text: in

text: which

text: we

text: have

text: implemented

text: Neumann

text: boundary

text: condition

text: to

text: calculate

text: the

text: values

text: u

text: (t)

text: and

text: u

text: (t).

text: As

text: a

text: time

text: marching

text: scheme,

text: we

text: will

text: use

text: a

text: semi

text: implicit

text: one,

text: in

text: order

text: to

text: have

text: a

text: good

text: stability:

text: it

text: is

text: important

text: since,

text: as

text: we

text: will

text: see

text: hereafter,

text: the

text: calculation

text: of

text: optimal

text: α(t)

text: requires

text: a

text: great

text: number

text: of

text: numerical

text: solutions

text: of

text: this

text: system

text: and

text: a

text: not

text: too

text: small

text: time

text: step

text: ∆t

text: must

text: be

text: used.

text: We

text: ﬁx

text: a

text: value

text: for

text: the

text: ﬁnal

text: time

text: T

text: T

text: being

text: large

text: enough

text: to

text: obtain

text: a

text: steady

text: state:

text: in

text: practice

text: the

text: solution

text: converges

text: toward

text: equilibrium

text: relatively

text: fastly

text: for

text: small

text: values

text: of

text: which

text: is

text: the

text: case

text: here.

text: The

text: time

text: interval

text: T

text: is

text: subdivided

text: into

text: M

text: subintervals

text: of

text: length

text: ∆t,

text: [t

text: t

text: k

text: M

text: with

text: k

text: t

text: k∆t

text: and

text: t

text: T

text: We

text: note

text: uM

text: the

text: numerical

text: approximation

text: to

text: u¯.

text: k

text: M

text: A

text: ﬁrst

text: idea

text: is

text: to

text: compute

text: α(t)

text: as

text: a

text: piecewise

text: constant

text: function

text: in

text: time,

text: say

text: M

text: α(t)

text: α

text: χ

text: k

text: [t

text: ,t

text: k

text: We

text: remark

text: here

text: that

text: calculating

text: α(t)

text: as

text: a

text: piecewise

text: constant

text: function,

text: is

text: a

text: simple

text: and

text: stable

text: ap-

text: proach:

text: other

text: techniques

text: allowing

text: orthogonal

text: polynomial

text: (such

text: as

text: Fourier

text: or

text: Laguerre)

text: could

text: be

text: used

text: but

text: the

text: strong

text: decreasing

text: of

text: the

text: Fourier

text: coeﬃcients

text: make

text: the

text: problem

text: ill-conditioned.

text: Also,

text: an

text: heuristic

text: method

text: can

text: be

text: used

text: for

text: accelerating

text: the

text: numerical

text: convergence:

text: once

text: computed

text: αM(t)

text: for

text: a

text: given

text: M

text: and

text: ∆t,

text: one

text: can

text: repeat

text: the

text: computation

text: for

text: time

text: steps

text: (with

text: in

text: order

text: to

text: get

text: a

text: more

text: precise

text: result

text: starting

text: from

text: a

text: mid-point

text: interpolated

text: value.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: (cid:16) (cid:17) ≤ ∆t has the discrete maximum property (M is a M-matrix) and since Uk 0 and < 1, we have k (cid:15)2 − − ≤ 1 ∆t Uk ∆t(Uk)2 0 so that (cid:15)2 (cid:15)2 ≤ Uk+1 0. We proceed in a similar way for Uk. As, we will see hereafter in the numerical simulations, this is observed for moderate values of α. It is important to underline that it is crucial that the numerical scheme heritates of the intrinsic properties of the equation (maximum principle, asymptotic behavior) for calculating a numerical control: if not, non-physical control can be determined[6]. 2.4.2 Choice of the optimization method As pointed out before, the principal key of this approach is the choice of the optimization method to compute α(t). This is not an easy task since the merit function is not diﬀerentiable in the L1 case; generally the gradient is not available as in the L2 case. Hence, gradient methods cannot be used. We have then to address to derivative free optimization algorithms [2]. In order to illutrate the numerical cacutation of optimal α, we use Matlab build-in house solver, such as x = fminsearch(fun,x0) or x = fminunc(fun,x0) where fun is the objective function which in our case is built numerically by solving the Allen–Cahn equation up to t = T with α as an input. 3 Numerical Results The minimization procedure used is fminunc available in from the optimization toolbox of Matlab ∀ ∈ computational software, [7] . We ﬁx (cid:15) and we start from α0(t) = 0, t [0, T ]. We hereafter display results for diﬀerent values of (cid:15) and T . The merit function is N(uM). 3.1 Regular ﬁxed initial data - numerical convergence Starting from a regular initial data allows to observe numerical convergence of the optimal control function α(t) as the number of discretisation points N increases, and as, for a ﬁxed T , the time step ∆t decreases. We can see, in Figures (4)-(5)-(6)-(7) the good coherence of the results for ﬁxed values of (cid:15) = 0.01 and T when varying ∆t and the number N of grid points. In all the cases, the global procedure allows minimizing the number of interphases or stripes. 8 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: ∆t

text: has

text: the

text: discrete

text: maximum

text: property

text: (M

text: is

text: a

text: M-matrix)

text: and

text: since

text: Uk

text: and

text: we

text: have

text: k

text: ∆t

text: Uk

text: so

text: that

text: We

text: proceed

text: in

text: a

text: similar

text: way

text: for

text: Uk.

text: As,

text: we

text: will

text: see

text: hereafter

text: in

text: the

text: numerical

text: simulations,

text: this

text: is

text: observed

text: for

text: moderate

text: values

text: of

text: α.

text: It

text: is

text: important

text: to

text: underline

text: that

text: it

text: is

text: crucial

text: that

text: the

text: numerical

text: scheme

text: heritates

text: of

text: the

text: intrinsic

text: properties

text: of

text: the

text: equation

text: (maximum

text: principle,

text: asymptotic

text: behavior)

text: for

text: calculating

text: a

text: numerical

text: control:

text: if

text: not,

text: non-physical

text: control

text: can

text: be

text: Choice

text: of

text: the

text: optimization

text: method

text: As

text: pointed

text: out

text: before,

text: the

text: principal

text: key

text: of

text: this

text: approach

text: is

text: the

text: choice

text: of

text: the

text: optimization

text: method

text: to

text: compute

text: α(t).

text: This

text: is

text: not

text: an

text: easy

text: task

text: since

text: the

text: merit

text: function

text: is

text: not

text: diﬀerentiable

text: in

text: the

text: case;

text: generally

text: the

text: gradient

text: is

text: not

text: available

text: as

text: in

text: the

text: case.

text: Hence,

text: gradient

text: methods

text: cannot

text: be

text: used.

text: We

text: have

text: then

text: to

text: address

text: to

text: derivative

text: free

text: optimization

text: algorithms

text: In

text: order

text: to

text: illutrate

text: the

text: numerical

text: cacutation

text: of

text: optimal

text: α,

text: we

text: use

text: Matlab

text: build-in

text: house

text: solver,

text: such

text: as

text: x

text: or

text: x

text: where

text: fun

text: is

text: the

text: objective

text: function

text: which

text: in

text: our

text: case

text: is

text: built

text: numerically

text: by

text: solving

text: the

text: Allen–Cahn

text: equation

text: up

text: to

text: t

text: T

text: with

text: α

text: as

text: an

text: input.

text: Numerical

text: Results

text: The

text: minimization

text: procedure

text: used

text: is

text: fminunc

text: available

text: in

text: from

text: the

text: optimization

text: toolbox

text: of

text: Matlab

text: computational

text: software,

text: We

text: ﬁx

text: and

text: we

text: start

text: from

text: t

text: T

text: We

text: hereafter

text: display

text: results

text: for

text: diﬀerent

text: values

text: of

text: and

text: T

text: The

text: merit

text: function

text: is

text: N(uM).

text: Regular

text: ﬁxed

text: initial

text: data

text: numerical

text: convergence

text: Starting

text: from

text: a

text: regular

text: initial

text: data

text: allows

text: to

text: observe

text: numerical

text: convergence

text: of

text: the

text: optimal

text: control

text: function

text: α(t)

text: as

text: the

text: number

text: of

text: discretisation

text: points

text: N

text: increases,

text: and

text: as,

text: for

text: a

text: ﬁxed

text: T

text: the

text: time

text: step

text: ∆t

text: decreases.

text: We

text: can

text: see,

text: in

text: Figures

text: the

text: good

text: coherence

text: of

text: the

text: results

text: for

text: ﬁxed

text: values

text: of

text: and

text: T

text: when

text: varying

text: ∆t

text: and

text: the

text: number

text: N

text: of

text: grid

text: points.

text: In

text: all

text: the

text: cases,

text: the

text: global

text: procedure

text: allows

text: minimizing

text: the

text: number

text: of

text: interphases

text: or

text: stripes.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Note on Poincaré type Kähler metrics and Futaki characters vector ﬁeld on , parallel to . Conversely, any holomorphic vector ﬁeld on X D X \ parallel to gives on a vector ﬁeld bounded at any order for any Poincaré D X D \ type metric on . X D We now provide a Hodge decomposition of holomorphic vector ﬁelds parallel \ to with respect to Poincaré type Kähler metrics on , analogous to the D X D decomposition of the compact setting: ∈ ∈ Proposition 1.2 Let Z hD, and let ω = ω + ddcϕ M D . Let ξZ be the dual 1-form of Z with respect to ω//(·, J·). Then X [ωX] ϕ (cid:0) (cid:1) (cid:0) (cid:1) · − · (3) ξZ = ξZ + d fZ + Z ϕ + dc hZ (JZ) ϕ ϕ harm ωX ωX \ on X D, (cid:90)with the same harmonic (cid:90)part ξZ as in the compact case, and this (cid:0) (cid:1) (cid:0)harm (cid:1) decomposition is unique. Moreover, · − · fZ + Z ϕ ωm = hZ (JZ) ϕ ωm = 0. \ ωX \ ωX X D X D The uniqueness we state here is understood as follows: if ξZ = α+dβ+dcγ with \ ϕ α harmonic on X D, and α, β, γ bounded for ω of Poincaré type, then α = ξZ , · − · harm and β = fZ + Z ϕ and γ = hZ (JZ) ϕ up to a constant. This justiﬁes: ωX ωX Notation 1.3 With the notations of Proposition 1.2, we set · − · fZ = fZ + Z ϕ and hZ = hZ (JZ) ϕ. ω ωX ω ωX Proof of Proposition 1.2. — With the notations of the statement, we ﬁrst prove \ that equality (3) holds on . This identity is purely local; it is thus suﬃcient X D to establish it for any Kähler metric equal to in the neighbourhood of any given ω \ point of X D. More concretely, as ω = ω + ddcϕ is of Poincaré type, local → −∞ X R → R analysis provides that near . Consider a convex function , ϕ D χ : ≤ − ≥ ≤ (cid:48) ≤ with if , if – and thus for all . Given χ(t) = 0 t 1 χ(t) = t t 1 0 χ (t) 1 t ∈ R (cid:0) (cid:1) , one now easily checks that K ◦ ω := ω + ddc χ (ϕ + K) K X { ≥ − } \ is a smooth metric on , equal to on (compact in ), and to X ω ϕ 1 K X D { ≤ − } { − } on . Now (3) follows on by the smooth case of ω ϕ (K + 1) ϕ > 1 K X \ → ∞ Hodge decomposition applied to , thus on all by letting . ω X D K K 6 ##LTLine##

text: Note

text: on

text: Poincaré

text: type

text: Kähler

text: metrics

text: and

text: Futaki

text: characters

text: vector

text: ﬁeld

text: on

text: parallel

text: to

text: Conversely,

text: any

text: holomorphic

text: vector

text: ﬁeld

text: on

text: X

text: D

text: X

text: parallel

text: to

text: gives

text: on

text: a

text: vector

text: ﬁeld

text: bounded

text: at

text: any

text: order

text: for

text: any

text: Poincaré

text: D

text: X

text: D

text: type

text: metric

text: on

text: X

text: D

text: We

text: now

text: provide

text: a

text: Hodge

text: decomposition

text: of

text: holomorphic

text: vector

text: ﬁelds

text: parallel

text: to

text: with

text: respect

text: to

text: Poincaré

text: type

text: Kähler

text: metrics

text: on

text: analogous

text: to

text: the

text: D

text: X

text: D

text: decomposition

text: of

text: the

text: compact

text: setting:

text: Proposition

text: Let

text: Z

text: hD,

text: and

text: let

text: ω

text: ω

text: ddcϕ

text: M

text: D

text: Let

text: ξZ

text: be

text: the

text: dual

text: of

text: Z

text: with

text: respect

text: to

text: ω//(·,

text: J·).

text: Then

text: X

text: [ωX]

text: ϕ

text: ξZ

text: ξZ

text: d

text: fZ

text: Z

text: ϕ

text: dc

text: hZ

text: (JZ)

text: ϕ

text: ϕ

text: harm

text: ωX

text: ωX

text: on

text: X

text: D,

text: the

text: same

text: harmonic

text: ξZ

text: as

text: in

text: the

text: compact

text: case,

text: and

text: this

text: decomposition

text: is

text: unique.

text: Moreover,

text: fZ

text: Z

text: ϕ

text: ωm

text: hZ

text: (JZ)

text: ϕ

text: ωm

text: ωX

text: ωX

text: X

text: D

text: X

text: D

text: The

text: uniqueness

text: we

text: state

text: here

text: is

text: understood

text: as

text: follows:

text: if

text: ξZ

text: α+dβ+dcγ

text: with

text: ϕ

text: α

text: harmonic

text: on

text: X

text: D,

text: and

text: α,

text: β,

text: γ

text: bounded

text: for

text: ω

text: of

text: Poincaré

text: type,

text: then

text: α

text: ξZ

text: harm

text: and

text: β

text: fZ

text: Z

text: ϕ

text: and

text: γ

text: hZ

text: (JZ)

text: ϕ

text: up

text: to

text: a

text: constant.

text: This

text: justiﬁes:

text: ωX

text: ωX

text: Notation

text: With

text: the

text: notations

text: of

text: Proposition

text: we

text: set

text: fZ

text: fZ

text: Z

text: ϕ

text: and

text: hZ

text: hZ

text: (JZ)

text: ϕ.

text: ω

text: ωX

text: ω

text: ωX

text: Proof

text: of

text: Proposition

text: With

text: the

text: notations

text: of

text: the

text: statement,

text: we

text: ﬁrst

text: prove

text: that

text: equality

text: holds

text: on

text: This

text: identity

text: is

text: purely

text: local;

text: it

text: is

text: thus

text: suﬃcient

text: X

text: D

text: to

text: establish

text: it

text: for

text: any

text: Kähler

text: metric

text: equal

text: to

text: in

text: the

text: neighbourhood

text: of

text: any

text: given

text: ω

text: point

text: of

text: X

text: D.

text: More

text: concretely,

text: as

text: ω

text: ω

text: ddcϕ

text: is

text: of

text: Poincaré

text: type,

text: local

text: X

text: R

text: R

text: analysis

text: provides

text: that

text: near

text: Consider

text: a

text: convex

text: function

text: ϕ

text: D

text: χ

text: with

text: if

text: if

text: and

text: thus

text: for

text: all

text: Given

text: χ(t)

text: t

text: χ(t)

text: t

text: t

text: χ

text: (t)

text: t

text: R

text: one

text: now

text: easily

text: checks

text: that

text: K

text: ω

text: ω

text: ddc

text: χ

text: (ϕ

text: K)

text: K

text: X

text: is

text: a

text: smooth

text: metric

text: on

text: equal

text: to

text: on

text: (compact

text: in

text: and

text: to

text: X

text: ω

text: ϕ

text: K

text: X

text: D

text: on

text: Now

text: follows

text: on

text: by

text: the

text: smooth

text: case

text: of

text: ω

text: ϕ

text: (K

text: ϕ

text: K

text: X

text: Hodge

text: decomposition

text: applied

text: to

text: thus

text: on

text: all

text: by

text: letting

text: ω

text: X

text: D

text: K

text: K

text: ##LTLine##


text: 4 clock in the parameter regime considered above. The reason for this is that we can choose a small h-value and still accumulate a phase shift large enough to observe. To second order in h, however, we start to see a discrep- ancy between the cavity clock and a pointlike one. This diﬀerence is due to both the ﬁnite extension of the cav- ity and the fact that non-uniform acceleration leads to mode-mixing and particle creation, eventually resulting in a diﬀerent phase shift for the ﬁrst mode. First, neglect- ing the latter eﬀects, we note that a cavity clock diﬀers from a pointlike one during acceleration only. During an accelerated segment, the proper time elapsed according to the cavity clock is θ L arcsinh (at /c) FIG. 3: Time dilation. Relative phase shift between Rob’s τa = a = a , (8) cav ω c 2 arctanh (h/2) and Alice’s cavities in an experimentally feasible regime. The 1 parameter values used are t = 1 ns, t = 0 and L = 1.1 a i while the corresponding expression for the pointlike cm, leading to an eﬀective cavity displacement of 1.7 mm and a maximal velocity of 0.014c. With t = 4 ns and the clock, obtained by integrating Rob’s proper time over t trajectory being repeated 500 times, the total travel time is the trajectory, is 2µs. Left inset: Diﬀerence between the time dilation shown c by the cavity clock and a pointlike clock as a function of L, τa = arcsinh (at /c). (9) point a a normalized to the total time dilation between Alice and Rob. The blue (red) curve is excluding (including) the eﬀects of Thus, the ratio of the proper times is given by mode-mixing and particle creation. Right inset: Diﬀerence in time dilation between the cases with and without particle τcaav = (h/2) = 1 − h2 + O(h4), (10) creation, again normalized to the total time dilation. The τpaoint arctanh (h/2) 12 parameter va×lues used in the inset plots are ta = 1 ns, ti = 0 and a = 1.7 1015 m/s2. which is smaller than one and decreases with h. This means that an extended clock is slowed down during ac- celeration. The larger the clock, the slower its rate. The the phase of a coherent state inside the cavity as a clock eﬀects of mode-mixing and particle creation depend only pointer, we ﬁnd that the time dilation produces a phase on changes in acceleration and are encoded in the second- shift of up to 130 degrees, which is clearly in the mea- order terms of the Bogoliubov coeﬃcients (A6) and (A7). surable regime. We also note that at high accelerations In order to observe the higher order eﬀects, we need to the extension of the clock becomes relevant: time dila- use larger h-values. In earlier cQED experiments [12], tion increases with the clock’s spatial dimension. This eﬀective accelerations up to 4 × 1017 m/s2 have been opens up an avenue for the experimental exploration of achieved. With such accelerations, though, the time t the diﬀerences between a pointlike [3, 4, 6] and a physi- a would have to be very short, making the eﬀective motion cally extended clock. In the near future, we foresee that of the mirrors diﬃcult to control. What we can do in- other quantum eﬀects on clock accuracy and time dila- stead is to increase h by using larger cavities. The inset tion can be explored using squeezed cavity states. By plots of ﬁgure 3 show the shift in time dilation due to the analysing for the ﬁrst time the twin paradox in a frame- diﬀerent eﬀects, as a function of L. In order to observe work of quantum ﬁeld theory with boundary conditions the eﬀects of clock size, we can choose L = 6 cm, which corresponding to relativistic motion, we are able to study is easily realizible in the cQED setup. In this regime, the theoretically the interplay of quantum eﬀects, such as the clock size is clearly the dominant eﬀect and would con- dynamical Casimir eﬀect, in a paradigmatic relativistic tribute with an additional phase shift of 3 degrees, which eﬀect such as time dilation. In this way we take a step is possible to resolve in the measurement stage. For even further in our knowledge on the overlap between quan- larger L, the other eﬀects start to become important, tum theory and relativity. with mode-mixing being the dominant one. This can be Appendix A: Bogoliubov coeﬃcients clearly seen in the right inset, where we plot the diﬀer- ence in phase shift between the cases with and without To determine the Bogoliubov coeﬃcients A and mn particle creation. B in equation (6) we use techniques developed in mn V. CONCLUSIONS [16]. The Bogoliubov coeﬃcients relating the modes of an inertial observer to those of a uniformly accelerating In conclusion, we have shown that using state-of-the- observer are, expressed as Klein-Gordon inner products ∗ art superconducting circuit technology, the twin paradox [17], α = (v , u ) and β = −(v , u ), where u mn m n mn m n n can be demonstrated in a ground-based experiment at and v are given by (2) and (5). α and β account m mn mn velocities approaching 1.4% of the speed of light. Using for mode-mixing and particle creation, respectively. The ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: clock

text: in

text: the

text: parameter

text: regime

text: considered

text: above.

text: The

text: reason

text: for

text: this

text: is

text: that

text: we

text: can

text: choose

text: a

text: small

text: h-value

text: and

text: still

text: accumulate

text: a

text: phase

text: shift

text: large

text: enough

text: to

text: observe.

text: To

text: second

text: order

text: in

text: h,

text: however,

text: we

text: start

text: to

text: see

text: a

text: discrep-

text: ancy

text: between

text: the

text: cavity

text: clock

text: and

text: a

text: pointlike

text: one.

text: This

text: diﬀerence

text: is

text: due

text: to

text: both

text: the

text: ﬁnite

text: extension

text: of

text: the

text: cav-

text: ity

text: and

text: the

text: fact

text: that

text: non-uniform

text: acceleration

text: leads

text: to

text: mode-mixing

text: and

text: particle

text: creation,

text: eventually

text: resulting

text: in

text: a

text: diﬀerent

text: phase

text: shift

text: for

text: the

text: ﬁrst

text: mode.

text: First,

text: neglect-

text: ing

text: the

text: latter

text: eﬀects,

text: we

text: note

text: that

text: a

text: cavity

text: clock

text: diﬀers

text: from

text: a

text: pointlike

text: one

text: during

text: acceleration

text: only.

text: During

text: an

text: accelerated

text: segment,

text: the

text: proper

text: time

text: elapsed

text: according

text: to

text: the

text: cavity

text: clock

text: is

text: θ

text: L

text: arcsinh

text: (at

text: /c)

text: FIG.

text: Time

text: dilation.

text: Relative

text: phase

text: shift

text: between

text: Rob’s

text: τa

text: a

text: a

text: cav

text: ω

text: c

text: arctanh

text: and

text: Alice’s

text: cavities

text: in

text: an

text: experimentally

text: feasible

text: regime.

text: The

text: parameter

text: values

text: used

text: are

text: t

text: ns,

text: t

text: and

text: L

text: a

text: i

text: while

text: the

text: corresponding

text: expression

text: for

text: the

text: pointlike

text: cm,

text: leading

text: to

text: an

text: eﬀective

text: cavity

text: displacement

text: of

text: mm

text: and

text: a

text: maximal

text: velocity

text: of

text: With

text: t

text: ns

text: and

text: the

text: clock,

text: obtained

text: by

text: integrating

text: Rob’s

text: proper

text: time

text: over

text: t

text: trajectory

text: being

text: repeated

text: times,

text: the

text: total

text: travel

text: time

text: is

text: the

text: trajectory,

text: is

text: Left

text: inset:

text: Diﬀerence

text: between

text: the

text: time

text: dilation

text: shown

text: c

text: by

text: the

text: cavity

text: clock

text: and

text: a

text: pointlike

text: clock

text: as

text: a

text: function

text: of

text: L,

text: τa

text: arcsinh

text: (at

text: /c).

text: point

text: a

text: a

text: normalized

text: to

text: the

text: total

text: time

text: dilation

text: between

text: Alice

text: and

text: Rob.

text: The

text: blue

text: (red)

text: curve

text: is

text: excluding

text: (including)

text: the

text: eﬀects

text: of

text: Thus,

text: the

text: ratio

text: of

text: the

text: proper

text: times

text: is

text: given

text: by

text: mode-mixing

text: and

text: particle

text: creation.

text: Right

text: inset:

text: Diﬀerence

text: in

text: time

text: dilation

text: between

text: the

text: cases

text: with

text: and

text: without

text: particle

text: τcaav

text: creation,

text: again

text: normalized

text: to

text: the

text: total

text: time

text: dilation.

text: The

text: τpaoint

text: arctanh

text: parameter

text: va×lues

text: used

text: in

text: the

text: inset

text: plots

text: are

text: ta

text: ns,

text: ti

text: and

text: a

text: which

text: is

text: smaller

text: than

text: one

text: and

text: decreases

text: with

text: h.

text: This

text: means

text: that

text: an

text: extended

text: clock

text: is

text: slowed

text: down

text: during

text: ac-

text: celeration.

text: The

text: larger

text: the

text: clock,

text: the

text: slower

text: its

text: rate.

text: The

text: the

text: phase

text: of

text: a

text: coherent

text: state

text: inside

text: the

text: cavity

text: as

text: a

text: clock

text: eﬀects

text: of

text: mode-mixing

text: and

text: particle

text: creation

text: depend

text: only

text: pointer,

text: we

text: ﬁnd

text: that

text: the

text: time

text: dilation

text: produces

text: a

text: phase

text: on

text: changes

text: in

text: acceleration

text: and

text: are

text: encoded

text: in

text: the

text: second-

text: shift

text: of

text: up

text: to

text: degrees,

text: which

text: is

text: clearly

text: in

text: the

text: mea-

text: order

text: terms

text: of

text: the

text: Bogoliubov

text: coeﬃcients

text: and

text: surable

text: regime.

text: We

text: also

text: note

text: that

text: at

text: high

text: accelerations

text: In

text: order

text: to

text: observe

text: the

text: higher

text: order

text: eﬀects,

text: we

text: need

text: to

text: the

text: extension

text: of

text: the

text: clock

text: becomes

text: relevant:

text: time

text: dila-

text: use

text: larger

text: h-values.

text: In

text: earlier

text: cQED

text: experiments

text: tion

text: increases

text: with

text: the

text: clock’s

text: spatial

text: dimension.

text: This

text: eﬀective

text: accelerations

text: up

text: to

text: have

text: been

text: opens

text: up

text: an

text: avenue

text: for

text: the

text: experimental

text: exploration

text: of

text: achieved.

text: With

text: such

text: accelerations,

text: though,

text: the

text: time

text: t

text: the

text: diﬀerences

text: between

text: a

text: pointlike

text: and

text: a

text: physi-

text: a

text: would

text: have

text: to

text: be

text: very

text: short,

text: making

text: the

text: eﬀective

text: motion

text: cally

text: extended

text: clock.

text: In

text: the

text: near

text: future,

text: we

text: foresee

text: that

text: of

text: the

text: mirrors

text: diﬃcult

text: to

text: control.

text: What

text: we

text: can

text: do

text: in-

text: other

text: quantum

text: eﬀects

text: on

text: clock

text: accuracy

text: and

text: time

text: dila-

text: stead

text: is

text: to

text: increase

text: h

text: by

text: using

text: larger

text: cavities.

text: The

text: inset

text: tion

text: can

text: be

text: explored

text: using

text: squeezed

text: cavity

text: states.

text: By

text: plots

text: of

text: ﬁgure

text: show

text: the

text: shift

text: in

text: time

text: dilation

text: due

text: to

text: the

text: analysing

text: for

text: the

text: ﬁrst

text: time

text: the

text: twin

text: paradox

text: in

text: a

text: frame-

text: diﬀerent

text: eﬀects,

text: as

text: a

text: function

text: of

text: L.

text: In

text: order

text: to

text: observe

text: work

text: of

text: quantum

text: ﬁeld

text: theory

text: with

text: boundary

text: conditions

text: the

text: eﬀects

text: of

text: clock

text: size,

text: we

text: can

text: choose

text: L

text: cm,

text: which

text: corresponding

text: to

text: relativistic

text: motion,

text: we

text: are

text: able

text: to

text: study

text: is

text: easily

text: realizible

text: in

text: the

text: cQED

text: setup.

text: In

text: this

text: regime,

text: the

text: theoretically

text: the

text: interplay

text: of

text: quantum

text: eﬀects,

text: such

text: as

text: the

text: clock

text: size

text: is

text: clearly

text: the

text: dominant

text: eﬀect

text: and

text: would

text: con-

text: dynamical

text: Casimir

text: eﬀect,

text: in

text: a

text: paradigmatic

text: relativistic

text: tribute

text: with

text: an

text: additional

text: phase

text: shift

text: of

text: degrees,

text: which

text: eﬀect

text: such

text: as

text: time

text: dilation.

text: In

text: this

text: way

text: we

text: take

text: a

text: step

text: is

text: possible

text: to

text: resolve

text: in

text: the

text: measurement

text: stage.

text: For

text: even

text: further

text: in

text: our

text: knowledge

text: on

text: the

text: overlap

text: between

text: quan-

text: larger

text: L,

text: the

text: other

text: eﬀects

text: start

text: to

text: become

text: important,

text: tum

text: theory

text: and

text: relativity.

text: with

text: mode-mixing

text: being

text: the

text: dominant

text: one.

text: This

text: can

text: be

text: Appendix

text: A:

text: Bogoliubov

text: coeﬃcients

text: clearly

text: seen

text: in

text: the

text: right

text: inset,

text: where

text: we

text: plot

text: the

text: diﬀer-

text: ence

text: in

text: phase

text: shift

text: between

text: the

text: cases

text: with

text: and

text: without

text: To

text: determine

text: the

text: Bogoliubov

text: coeﬃcients

text: A

text: and

text: mn

text: particle

text: creation.

text: B

text: in

text: equation

text: we

text: use

text: techniques

text: developed

text: in

text: mn

text: V.

text: CONCLUSIONS

text: The

text: Bogoliubov

text: coeﬃcients

text: relating

text: the

text: modes

text: of

text: an

text: inertial

text: observer

text: to

text: those

text: of

text: a

text: uniformly

text: accelerating

text: In

text: conclusion,

text: we

text: have

text: shown

text: that

text: using

text: state-of-the-

text: observer

text: are,

text: expressed

text: as

text: Klein-Gordon

text: inner

text: products

text: art

text: superconducting

text: circuit

text: technology,

text: the

text: twin

text: paradox

text: α

text: (v

text: u

text: and

text: β

text: −(v

text: u

text: where

text: u

text: mn

text: m

text: n

text: mn

text: m

text: n

text: n

text: can

text: be

text: demonstrated

text: in

text: a

text: ground-based

text: experiment

text: at

text: and

text: v

text: are

text: given

text: by

text: and

text: α

text: and

text: β

text: account

text: m

text: mn

text: mn

text: velocities

text: approaching

text: of

text: the

text: speed

text: of

text: light.

text: Using

text: for

text: mode-mixing

text: and

text: particle

text: creation,

text: respectively.

text: The

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##