幾何問題奇點集的形成與結構

行政院國家科學委員會補助專題研究計畫成果報告

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幾何問題奇點集的形成與結構

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計畫類別：□個別型計畫　　□整合型計畫

計畫編號：NSC　89-2115-M-009-018

執行期間：　88 年 8 月 1 日至 89 年 7 月 31 日

計畫主持人：王夏聲

共同主持人：

本成果報告包括以下應繳交之附件：

□赴國外出差或研習心得報告一份

□赴大陸地區出差或研習心得報告一份

□出席國際學術會議心得報告及發表之論文各一份

□國際合作研究計畫國外研究報告書一份

執行單位：國立交通大學應用數學系

中　華　民　國　89 年　10　月 30　日

行政院國家科學委員會專題研究計畫成果報告

幾何問題奇點集的形成與結構

The Str uctur e of Singular Sets in Some Geometr ic Pr oblems

計畫編號：NSC 89-2115-M-009-018

執行期限：88 年 8 月 1 日至 89 年 7 月 31 日

主持人：王夏聲

執行機構：國立交通大學應用數學系

Email：swang@math.nctu.edu.tw

一、中文摘要 我們考慮在四維緊緻里曼流型上楊-米爾 斯熱流方程弱解在有限時間奇點產生的問 題。我們證明了在有限時間內奇點產生所 有的能量損失可由吹有限多個『泡泡』的 能量所捕捉到。而這些『泡泡』是由在

_S

4 上的正能量楊-米爾斯連絡所組成。相似 的現象在定義域為二維緊緻里曼流型上的 調和映照熱流方程亦有發生。 關鍵詞：四維緊緻里曼流型、楊-米爾斯熱 流方程、弱解、

_S

4-泡。 Abstr act

We show that there is no unaccounted energy loss for blow-ups of a solution of Yang-Mills flow equation in 4 dimensional manifolds. Any energy loss corresponds precisely to the product of bubbles, that is, nontrivial Yang-Mills connections on

_S

4. Our result is similar to the evolution of harmonic maps of Riemannian surfaces into spheres.

Keywor ds: compact Riemannian manifold,

Yang-Mills heat equation, weak solution

1.Intr oduction

Let M be a closed, connected Riemannian

4-manifold and Π:η→M a smooth vector

bundle with fiber

Π

−1≅

_R

n, fiber metric

〉

〈

⋅⋅

x

, and compact structure group )

(n SO

G⊆ . There is the group ) ( ) (η

η

x M x Aut Aut G ∈ = = Υ of gauge

transformations, where Aut G

x)⊆

(

η

are automorphisms of the fiber

η

x. The group G acts on G by conjugation. We denote by

) ( : ) (η

η

x M x ad ad

Υ

∈

= the adjoint bundle whose sections s∈

Ω

0(ad(η)) may locally represented as maps s:

_U

_α→g, where g

is the Lie algebra of G . G acts on ad(η) via adjoint action. Denote

Ω

p(ad(η)) (or simply

Ω

p), the spaces of g -valued

p -forms and D , the affine space of

connections, D=

_D

_ref +A,A∈

Ω

1.

For D∈D,F(D)=

_F

_D= DοD∈

Ω

2 is a 0-order operator called the curvature of D .

The Yang-Mills action of a connection D

with curvature F is defined by

∫

= M dx F D YM |

|

2 2 1 ) (

where dx is the volume form of M ,

〉

〈 = g F F F , |

|

2 and 〈⋅⋅

〉

g , is the Cartan-Killing metric on g . The Euler Lagrange

equations is † _{八十六年度及以前的一般} 國科會專題計畫(不含產學 合作研究計畫)亦可選擇適 用，惟較特殊的計畫如國科 會規劃案等，請先洽得國科 會各學術處同意。

0

=

∗

F

D

where

_D

∗is the adjoint operator of D

with respect to the Riemannian metric of

M .

M.Struwe, in [Stru1] showed the existence and uniqueness of the weak Yang-Mills flow in four dimensions: . ) 0 , ( ) 2 . 1 ( , ) 1 . 1 ( 0

D

D

D D t = ⋅ − = ∂ ∂ ∗

. Concerning the blow-up analysis and the long-time behavior of the

Yang-Mills flow, Struwe claimed and proved in detail by A.Schlatter in [Sch].

In this paper, our purpose is to study what happens to the solutions near these singular points. The result we obtain as follows:

Main Theor em

Suppose D(t) is a solution of Yang-Mills flow equations and (x,t)∈Σ, the set of singular set of the D(t) at time t . Then there exists finitely many nontrivial Yang-Mills connections

{

}

1 p j j =

∆

over

_S

4 such that ) ( | ) , ( | 1 2 ) ( 2 ) (

|

|

lim

1

∑ ∆

∫

∫

= → + = ⋅ p j j x x D t t YM dx D B dx t BR

F

R

F

for any disk

_B

_R(x)=

_B

_R with φ = Σ\{( , )} ) ( 2 x x t

B

R Ι .

2. Solutions of the heat equations

In this section we study the local behavior of the solutions for the Yang-Mills equations near a singular point. In particular, we prove what is needed for the proof of our main theorem.

By using the local version of the monotonicity formula (see Lemma 2.2 of

[Sch]) we have

Pr oposition 2.1

Suppose that D(t) is a smooth solution of Yang-Mills flow equation, and that

Σ ∈

) ,

(x t and

_B

_R are given as in Main Theorem. Then dx t BR

F

D t t ) , ( |

|

lim

∫

2 ⋅ → exists.

Pr oof: A straightforward computation with

the help of the local monotonicity formula (see Lemma 2.2 of [Sch]) .

With regarding the blowing up sequence of rescaled connections, we have the following more general result:

Pr oposition 2.2

For any sequence

) , ( { 1,2 1 1

}

∞ ⊆

Ω

=

H

B

D

j _j j with ∞ < ) (

_D

_j YM and 0 → ∗

F

D

j _Dj in

_L

2_loc(

_R

4). Then there is a family of gauge

transformations { 3,2 ( 4, ) 1

}

_H

_R

G j j ε

σ

∞ − = ⊆ such that

D

D

D

_j j j ∞ ∗ _→ = ( )

ˆ

σ

weakly in

_H

2,2− (

_R

4,

Ω

1) loc ε .

_D

_∞ is a nontrivial smooth Yang-Mills connection over

_R

4Υ{∞}≅

_S

4 so that

ε

3 ) 0 ( 2 2

|

|

≥

∫

B

F

D∞ , a constant independent of j . Pr oof:

Using exactly the same arguments of [Sch] (See pp. 17--20 of that paper) with necessary notational modification, we have the

assertion.

3. Sketched pr oof of the main theor em

First of all, by the global monotonicity

we know that YM(D(t)) is a nonincreasing function of t , the limit

∫

⋅

BR

F

D t dx

|

2 ) , ( |

exist as t →t and it sufficies to show that we may blow the first bubble to compensate the loss of energy as time evolves, because we can keep blow bubbles within finite steps until the energy loss are captured by the bubble we blow near the singularity. We write , ) , ( |

|

|

|

lim

2 2 dx B D L dx t B R t R

F

F

D t t

∫

∫

+ = ⋅ → where

_B

R as defined in the main theorem

and L>0 is a positive constant. For any 0

↓

λ

j , there are

x

j→x,

t

j↑t such that

for some

_R

₀>0 we have

(3.1) dx B F t t t B dx B F N x t x x j j j j j R j

t

∫

∫

≤ ≤ − ∈ = = = ) ( 2 ), ( 2 0 1

|

|

) ( | sup ) ( | 0 2 λ λ τ

ε

ε

where N is a positive integer to be chosen

later and

ε

₀ is the positive constant stated in Theorem 1.1(9) of [Sch], ) ( 4 ₀ 2 0

D

R

NYM j j

ε

τ

= , ) ( 0 x R

B

x

j∈ ,

and that the bundle η is trivialized over )

( 0

2_R x

B

.

Now we do the scaling procedure by setting ) , ( ) , (

_x

2

_t

D

j xt =d+

λ

jA j+

λ

jx−

λ

jt+ j

We note that

_D

_j(t) is a classical solution of (1.1) on

_B

_R x

_t

_B

_j

_I

j 0 0,0] [ ) ( 0 × = × λ ,

where the ∗-operator is taken with respect to the rescaled metric and

) ( 4 ₀ 0 0

D

t

= _NYM−

ε

.

By the conformal invariance of the

_L

2 -norm of the curvature

_F

_j,

, ) 0 ( | ) ( | sup 1 ) 0 ( 2 2 ) ( , | | 1 1 0 0

|

|

ε

λ = ≤

∫

∫

∈ < B dx t B I R

F

F

j x j t x j Fix x∈

_R

4. Since

λ

↓0 j ,

B

1(x)⊂

B

j for

j

x j > , and

_D

_j(t)is defined on ) ( 1 x

B

for all

j

x j > . By choosing

sufficiently large N , by (3.1), we may apply

Theorem 2.1 of [U1] to

_D

_j(t),

j

x

j> ,

I

t∈ ₀ to get a sequence of gauge

transformation {

σ

_j,t}⊂

_C

∞(

_B

_R,G) such that the transformed connection

) ( )) ( ( ) (t _,

_D

t d

_A

t

D

j = jt j = + j ∗

σ

satisfy (i)

_d

∗

_A

_j(t)=0; (ii) || |

|

1,2 CYM( ) C

ε

₁ H

D

A

j ≤ j ≤ . We notice that

∫ ∫ ∂

∫ ∫ ∂

= − → t t A dxdt dxdt t B j j j j t

A

j τ M t 0 | |

|

|

2 0 2 0 there is a sequence {

}

j j

s

s

j∈

I

0,, such that (3.2) || ( )|

|

0 ) ( 2 →

∂

t

A

j

s

j _L K

as j→∞ for any compact K⊂

_R

4. We also notice that _, ( ₁(x),G)

sj

C

B

j ∞ ∈

σ

. Define ) ( )) ( ( : ) (

~

~

,_s t d t t

_D

_A

D

j j j j j + = =

σ

∗ which satisfies

F

D

D

j j j dt d

~

~

∗

~

− = with

_F

_F

Dj j ~

~

_{= . By (3.2)}

0 | ) ( || | || |

|

~

|

~

|

~

~

||

) ( ) ( ) ( 2 2 2 → = =

∂

∗ K j j t K j K j j L L dt d L

s

A

D

F

D

as j →∞ for any compact K ⊂

_R

4 . Now we may apply Proposition 2.2 to

_D

j

~

to obtain the first bubble

∆

₁=

_D

_∞ with

N B |

F

dx 2 0 ) 0 ( 2 2

|

≥

ε

∫

∞ . 4 Some r emar ks

In the case of the heat flow of harmonic maps from surfaces into spheres, much more is known about the structure of the bubbles (see [Q]. In our case, the space of connections is an affine space and they are invariant under gauge transformations, so we have to construct various time-independent gauge transformations to blow our

_S

4-bubbles. This difficulty make it more challenging to find similar convergence results as in the harmonic maps case.

5 Refer ences

[L] Lawson, B., The theory of gauge fields in four dimensions, CBMS Regional Conf. Series, Vol. 58 Providence, RI, AMS 1982.

[QJ] Qing, J., On the singularities of the heat flow for harmonic maps from surfaces into spheres, Commun. in Analysis and Geom. 3, no. 2, 297--315 (1995).

[Se] Sedlacek, S. A direct method for minimizing the Yang-Mills functional over 4-manifolds, Comm. Math. Phys. 86, 515--527 (1982).

[Sch] Schlatter, A., Long-time behaviour of the Yang-Mills flow in four dimensions, Annals of Global Analysis and Geom., 15, 1--25 (1997).

[Stru1] Struwe, M., The Yang-Mills flow in four dimensions, Cal. Var. 2 123-155 (1994).

[U1] Uhlenbeck, K., Connections with

L

p bounds on curvature, Comm. Math. Phys. 83 31--42 (1982).