m âe"V ilo = 1M4L0,. 1991 Mathematics Subject Classification. Primary 53C21; Secondary 58G03. *Research supported in part by NSC. Communicated by Prof.
J. M ath. K yoto Univ. (JMKYAZ) 39-3 (1999), 435-454
On the existence of extremal metrics for L -norm of scalar curvature on closed 3-manifolds 2
By
Shu-Cheng CHANG* a n d Jin-Tong Wu
Abstract In this paper, based o n Bochner formula, mass decay estimates and elliptic Moser iteration, we show the global existence and asymptotic convergence of a subsequence of solutions of Calabi flow on some closed 3-manifolds, and then the existence of extermal metrics of L -norm of scalar curvature functional on a fixed conformal class is claimed. In particular, we may re-solve p art o f th e Yamabe conjecture on closed 3-manifolds. 2
1. Introduction Let (M, go ) be a closed smooth n-manifold w ith a given conformal class [g o ] on M . Then the Euler-Lagrange equation of
Ss(g)—
I m R 2 dit (fMdf0
4'
g E Egd
1
is given by
(1.1)
AR
—
f3R2 + fir =0, f m R 2
where clti =eltig , A = Ag , R is the scalar curvature with respect to the metric g,r =
m cly n— 4 a n d fi = . N ow consider th e negative gradient flow o f S s(g) o n a given 4(n —1) conformal class [g o ] , t h a t is, we consider th e following initial value problem of fourth order parabolic equation:
= A R fiR + fir, 2
at (1.2)
g =e 2 Â g o ; 2(p,0)= .1 0 (p), m „e"V il o = 1M4L0,
1991 M athem atics Subject Classification. Primary 53C21; Secondary 58G03. *Research supported in p a rt by NSC Communicated by Prof. K . Fukaya, November 4, 1998
436
Shu-Cheng Chang and Jin-T ong Wu
where 2:M x [0,00)-4? is a smooth function and dtt o i s the volume element of g o . When n=2 1 , if the background metric g o h a s the constant Gaussian curvature, P. C hrukiel ([C hru]) proved the long tim e existence and asymptotic convergence of solutions of (1.3), and the first author generalized his results to any arbitrary background metric g o a n d then re-solve the uniformization theorem for surfaces ([C h 3 ]). Furthermore, we also proved some partial results for the long time existence of solutions of (1.3) when n =4 ([Ch2]). W hen n = 3, then 13 = 1 ,- and we will consider the following flow: 1
1
= AR + 8 01 (1.3)
g e2Ag.
0
- - r, 8
; 1(p,
fM3e3 2 ° 4 0
=
Adp),
M340 •
Although (1.3) is at heart a parabolic equation, due to equivariance under the group of diffeomorphisms, which m akes it highly degenerate. O n the other hand, Richard Hamilton's original proof o f sh o rt tim e existence of the Ricci flow was involved and used the Nash-Moser inverse function theorem. Soon after, D. DeTurck simplified short tim e existence proof by "breaking th e symmetry" (which causes difficulty in th e directly applying standard theory) to prove short tim e existence ([D e]). T h e n , by using the Deturck's trick, w hich w as done by the first author's previous work ([Ch5, Lemma 4]) in general case, short time existence of (1.3) follows easily. W e m a y a ls o compare this to [LT]. In this paper, we will show the long-time existence and asymptotic convergence of solutions of (1.3) on M x [0,00). 3
Theorem 1.1. L et (M, g o ) be a closed 3-manifold and 2 satisfy (1.3) on [0,7) with
2> — H f or the positiv e constant H w hich is independent o f t. ex ists on M x [0,00).
T hen the solution of (1.3)
Theorem 1.2. T he sam e assum ptions as in Theorem 1.1. T hen there ex ists a le ,2 ,1 (t)g 0 l subsequence of solutions o f (1.3) on M x [0,00) which converges smoothly to an extremal m etric g, i.e. its scalar curvature R = R (g op ) satisfying Aœ i?.+ A r
=0. Now consider the Yamabe constant which is conformal invariant
Q(M,g 0 )= inf
g °E( ( p ) 114 1 4 0 ) 9 6
(
02 'F o r n = 2 , we consider the so-called Calabi flow — =A R only.
at
3
Extremal metrics f o r L 2 -norm of scalar curvature where Eg o ((P)=SIV91 2 4
437
+k-SR o cedn o . As consequences of Theorem 1.1, we have 0
Theorem 1.3. If (M ,g 0 ) is a closed 3-manifold with Q 2#2.
A
P ')
Pi
0
B(p,p')
This leads to a c o n trad ic tio n . Then for small enough p again, one obtains
f
12 ( R 2
o
44 A-21\7212 ) 2 4
-
0
0
C(/3, A 0 ) f m p , p )
e
..
4 ..I.
C21110 .
B(P,P)
a(P ,P )
This completes the proof of Lemma 2.3. 3. A priori estimates and long time existence I n th is section, follow ing [C hi], [ G ] a n d Theorem 2.4, w e w ill have the C ° -bound a s in Lemma 3.2. Then, based o n [Ch2] a n d [Chru], one can get the bounds o n a ll W k 2 norm s a s in Lemma 3.3. A ll these together w ill im ply the long-time existence of solutions of (1.3). Let (M,g 0 ) be a closed 3-manifold with the background metric g o a t 1= 0 . One has ([G]) ,
Lemma 3.1. L et (M,g 0 ) be a closed 3-manifold w ith Q < O . For g e[g o ], say g = e 'g o . If a n d f , R dp -H .
(*
)
(*) holds also, up to the conformal group, for (M,g 0 ) is conformal equivalent to the standard sp h e re . T h at is, there exist conform al transform ations (p o f M such that, if e g t =e g 0 , then (*) holds for i
27
Now we are ready to prove the Theorems 1.1, 1.3 a n d 1.4. Since
R =e
o -2 A
R 0 - e N4A 0 A+21V21 ), -
2
and
1 -40eA 0, w e have (3.1)
Extremal metrics f o r L 2 -norm of scalar curvature
447
Then, from Corollary 2.5
beLq for some q > 4 . M ore precisely, 8
8
Wito C JIe22R r dt10 + ORO = Cf e lle d / I + < C (f (eF) 5 0/p) 5 (1(1Riss)idtlY + C(R 0 )
(3.2)
_C(fe 2 cli.t) (fIRI 2 dp) +C(Ro) C(go V , 13, T). But
(3.3)
f
24
0 =
nd ito .
e
C.
All together with (3.1), (3.2), (3.3), a n d Moser iteration ([Chi, Theorem 3.3], [G]), this leads
supeyt0, IV Al4
21 14
Extrema! m etrics f o r L 2 -norm of scalar curvature
451
on B . Otherwise d
—f e d p dt ,
C— 1 3"
0
f
e du o .
Bp
It follows that we have the uniformly bound of 1B , e 2 dit o , 5 < a < . This leads to a contradiction for E #0. Hence
I m
e
- À
o o elVAI4dito+ e l V 1 I dito
o 1V21 dpo =
4
4
M\ B p B
p
o
e - 1V21 dtto + C f e 3 1 dit o
< f
(4.2)
À
p
4
B
p
M \ B
0 e l v 2 i4 d ii0
