Scalar curvature on noncompact complete Riemannian manifolds

Nonlinear Analysis, Theory, Methods & Applications, Vol. 26, No. 12, pp. 1985-1993, 1996 Copyright © 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0362-546X/96 $15.00+0.00

Pergamon 0362-546X(95)00051-8

SCALAR

CURVATURE ON NONCOMPACT RIEMANNIAN MANIFOLDS

COMPLETE

S E O N G T A G KIM Department of Mathematics, Universityof Illinois, Urbana, IL 61801, U.S.A.

(Received 4 September 1994; receivedfor publication 21 March 1995) Key words and phrases: Scalar curvature, complete Riemannian manifolds, nonlinear elliptic partial differential equations, critical exponent.

1. INTRODUCTION Throughout this paper, we let (M, g) be a noncompact complete Riemannian manifold of dimension (n >_ 3) whose scalar curvature S(x) is positive for all x in M. Here we look for a conformal deformation of the given metric g, which makes the scalar curvature of the deformed metric a positive constant. Finding a suitable conformal transformation satisfying the above condition turns out to be equivalent to finding a positive smooth function u(x) satisfying the following partial differential equation

- A u + - n-- 2S u

(A)

4(n-

= qu (n+2)/(n-2),

1)

where q is a positive constant and Au = (1 / d,,/-d-~gi~) O~( g i J d ~ e Oi u) in a local coordinate system. We plan to study equation (A) by using variational methods. To do this we introduce some notation f

Q(M)-Inf,,~c~(M)

Bu 2 dVg

IVul z + - n - 2 m

4(n -

(f M u2n/(n-2)

1)

\(n-2)/n dVg)

and t

n - 2 1~ Su2 dVg, [Vu]Z + 4(n

Q(M) - lim Infu e C7tM-Br) M-B, r--~ oo

where r is the Riemannian distance from x to the fixed point Xo in M and Br is the ball of radius r with center at Xo. We also use the notation Q(M) for Q(M, g) and Q(S n) for Q(S n, go), where go is the standard metric. Now we state our theorem. THEOREM 1. Let (M, g) be a noncompact complete Riemannian manifold with positive scalar curvature. Assume that 0 < Q(M) < Q(M). Then there exists a positive solution u of the partial differential equation (A). Therefore, there exists a conformal metric (M, U4/(n-2)g) with positive constant scalar curvature. 1985

1986

SEONGTAG KIM

There are no restrictions on the boundedness of curvatures and the injective radius on given conditions of the theorem 1. There are examples of manifolds satisfying given conditions of theorem 1. Assume that M is not locally conformally flat and there exists a compact subset K such that M \ K has a conformal map to S ~, then Q(M) < Q(M) (see Schoen and Yau [1]). Let us consider the manifold M . . . . M3 # Mz # M1 # M2 # M3 "" with Q(Mk) = 1 + 1/k. We can easily construct a minimizing sequence which escapes to infinity. Therefore, we need Q(M) < Q(M) in general. The conformal change of metric of a given compact Riemannian manifold to constant scalar curvature was studied by Yamabe [2], Trudinger [3], Aubin [4, 5] and Schoen [6]. Schoen [6] finally showed that every compact Riemannian n-manifold M has a conformal metric of constant scalar curvature. For the noncompact case, Aviles and McOwen [7] showed that manifolds with nonpositive scalar curvature and strictly negative scalar curvature on the outside of a compact subset admit a complete conformal metric with constant negative scalar curvature. We divide our approach into two parts. The first part concerns the minimizing sequence on compact subsets and the second part concerns the minimizing sequence on the outside of a compact subset. For the first part, we use the variational method which was used by Yamabe [2] and later developed by Trudinger [3], Aubin [4, 5] and Schoen [6]. Unlike the Yamabe Problem in the compact case, we must worry about the behavior of the minimizing sequence on the outside of a compact subset. In the second part, using the Concentration Compactness Principle developed by Lions [8, 9] (see also Evans [10]), we control the minimizing sequence which is trying to escape to infinity (that is, the outside of a compact subset).

2. C O M P A C T N E S S

OF A M I N I M I Z I N G

S E Q U E N C E ON C O M P A C T S U B S E T S

Since we assume that 0 < Q(M) < Q(M) (
n-2 Slgi 4 ( n - 1)

J¢- -

= qi

U~n+2)((n-2)

on K i

(1)

u i = 0 o n OK i and ~ dVg = 1, where qi - Q(Ki) ~ Q(M) as i -~ oo. JK i ,,2n/(~-2) tci We extend the domain of ui by defining u i = 0 on the outside of K~. We use the same notation u~ for this extension. Note that the extension ui is in H~(M, g), the completion of C~'(M) with the norm IlulL -- f

M

Ivul 2 +

n-2 Su 2 dVg. 4(n - 1)

Let f~ be the set [x e M I ui(x) >- II. Then, [£)1 = Volume of ~) < 1, because

l

u~n/(n-2) dVg = 1. Ki

Complete Riemannian manifolds

1987

N o w let ui = 1 + wi. Then (1) is equivalent to n-2 -Awi + --S(1 4 ( n - 1)

+

Wi) = qi(1

+ Wi) (n+z)/(n-2).

(2)

By multiplying (2) by w) ÷2b where b > 0 and integrating over f~, we have qi l

~ , + 1)(n+2)/(n-2)dVg = l ~ (11 + + 2b b) 2 v- w i 1+b2 + C,,S(1 + wi)w~ +2t'd~, wi1 + 2~Wi

(3)

where Cn = (n - 2)/4(n - 1). By using H61der's inequality we have

lfl w2n/(n-2)+2bdVg<~(Sfl w'l+b)2n/(n-2'dVg~(n-2)/n(l/ \J~ w'2n/(n-2)+2b-(l+b)2)(n/2)dl/g)2/n W}l+b)2n/(n-2)dVg:

<~

•

(4)

Let a = (n + 2)/(n - 2) and x = l / w i. First consider when w i _ 2 (that is Ixl -< ½). (1 + wi) u -

w'~ = w'{ 1 +

- w'~

= w~'(1 + x) a - w~ ( a ( a - 1) 2 = w7 1 + ax + 2----T---. x . . . .

) 1

<_ c~ w:/(~-2)

(5)

for some constant cl (since the series converges by the comparison test). By the Sobolev Embedding T h e o r e m on Riemannian manifolds (see Aubin [11]) for wi ~ C~(f~), f~ C Ki C M (note Ki is compact) and (3), for any given e > 0, there exists C(e) with

_< (1 + ~) Q--~-~

-

-

1

a

(1 + b) 2 .f (qi Wil+2b (1 + Wi)(n+2)/(n-2) CnS(Wi + 1)w) +2b) dVg

< ( 1 + e) Q(Sn )(1 + 2 b ) _

+

c(~) i (w~+b)2 dV~. J

(6)

1988

SEONGTAGKIM

N o w l e t £21 ~- {x e ~lwi(x) >- 2} and DE - £2 - £21. Clearly, we have 1£211 < 1 and 1£221 < 1 because ID[ < 1. Using (5), the above integral (6) can be estimated by 1 (1 + b)Z ( l _< (1 + e)Q(S~ ) (1 + 2b) a,

+f

qiw2n/(n-2)+2b+Cwi(n+2)/(n-2)+2bdVg

qiw~+2b(1 + wi)(n+z)/(~-2)dVg - Cn f S(wi + 1)W'+2bdVg)

f12

fl

+ C(e) l (wil+b)2 dVg, where C is a general positive constant which does not depend on i. F r o m (4), we have

(1 + b) 2 q; Q(S ~) (1 + 2b)

_< (1 + e ) - -

+Cf

1

l+b

t~

(wi

)

2n/(n-2) dVg

I0(n+2)/(n-2)+zbdVg -i

(7)

+ C(e) f (w~+b)2 dVg. J Since qi < c <

Q(S") for some c, we can take e > 0 and 0 < b < 2/(n - 2) so that (1 + e )

q ~ ( 1+ b )

2
Q(S") (1 + 2b)

1.

(8)

Therefore, we have 1 -(1

+e)--

qi (1-F b)2X~/f w!l+b)2n/(n-2)dG)(n-2)/n

Q(s") (1 + 2b)/

. 1+2bdG + C S S(w~+ l~w~ ~ C(~) l~ (w~+b)2dVg+ C f~l w'n+2)/(n+2)+2bdVg q- C f qi w~+2b(1 + Wi)(n+2)/(n-2)dVg. ~2

(9)

Complete Riemannian manifolds

1989

Note that the last term is uniformly bounded by a constant c3 which does not depend on i by the definition of ~'~2" Finally, we have

(lf~ gl'l+b)2n/(n-2,dVg)(n-2)/n < Zl(C 3 +

(wi

12

) dig + C

QI

w~in+2)/("-2)+Zbd

,

(10)

where A1 is a positive constant. By using H61der's inequality, we have

<_ If~l 1-'1 < 1 and

f

w}"+z)/~"-z)+zb dVg <_ I~ w[ "/('-z)

~2

f~l

dlg <

1,

where tl = (1 + b)(n - 2)/n. Therefore, we have a uniform bound on the last two terms of the above inequality (10). Finally, we have

W}l+b)2n/(n-2) dVg

_< A2,

(11)

where A2 does not depend on i. Using the standard elliptic regularity theory (see Gilbarg and Trudinger [12] or Lee and Parker [13]) we now show that wi is C 2'~ bounded on each compact subset in the following way. Consider sufficiently large compact subsets K C Ko C K~ C K2 with smooth boundary satisfying Q(K) < Q(Sn). On K 2 w e have

IAwil = ICES(1 + wi) - qi(1 + Wi)(n+2)/("-2)[ <-- C(1 + wi) ("+2)/("-2), where C is a constant depending on_ K2 and max/~ x2 S(x). From the above, Ix2 wZ"/(n-2)+Z°dVg < Ao where b = b n / ( n - 2) and A 0 is a constant independent of i. Therefore, A w i ~ Lq(K2) when q = (2n + 2b(n - 2))/(n + 2). By the standard elliptic regularity theory (see Gilbarg and Trudinger [12]), we have wi ~ Hq(KO. From the Sobolev Embedding Theorem, we have w~ ~ LS(KO where s=

+2

,)

n -4b

> - - + n - 2

2b.

Continuing this procedure, we have wi e H~(Ko) for all t > 1. By the Sobolev Embedding Theorem, we have w~"/('-2) ~ C'~(Ko) for some a > 0. Using elliptic regularity theory again we have wi ~ C2'~(K). Therefore, we have a C2'~(K) bound. By the definition of f2 and u i = wi + 1 we have a uniform C 2'~ bound for u~ on each compact subset of M. Therefore, we can find a subsequence [uik] which converges to its limit u uniformly on each compact subset by the Arzela-Ascoli Theorem. F r o m the above, we have the following lemma.

1990

SEONGTAG KIM

LEMMA 1. If Q(M, g) < Q(M), then there exists a subsequence uniformly on each compact subset of M.

[ui~l which converges to u

Remark. We do not know whether u is a positive solution or not because u can be identically zero. This case happens when the support of u i escapes to infinity.

3. COMPACTNESS OF A MINIMIZING SEQUENCE AT INFINITY Next we consider behaviors of a minimizing sequence [nil at infinity (the outside of the compact subset). We use the Concentration Compactness Principle to control the minimizing sequence which is trying to escape to infinity.

CLAIM 1. Q(M) is well defined.

roof. Let n-2 tM-Br Ivul 2 + 4 ~ -- 1)

f(r) -

2

Su dVg

Inf u e C~(M-Br)

(lu_su2n/< ~-2) dVg) ~"-2)/~

f(r) is nondecreasing as a function of r by definition. Since f(r) is bounded above by Q(Sn), limr-.~ f(r) exists.

Then

For the convenience of the notation, we let

El(u) ~

l

Vk--Uk-- U Note that vk goes to 0 weakly in

n-2

Ivul z + - M 4(nand

1)

Suz dV~,

c~-

n+2 n-2"

H2(M, g).

By using the generalization of Fatou's Lemma by Brezis-Lieb [14], we have the following claim.

CLAIM 2. As k goes to infinity, lul °÷1

CompleteRiemannianmanifolds

1991

Proof. lu - u~l ~+' dVg

=

i ~fo ~l

a

lu,~ - tul ~÷' dtdVg

=(ff+l) f '1.~/0 j u(u n u(u

-

-

tu)lu k -

tu)lu

-

tul ~-1

tu[ ~-1

= l ~ [u[~+l dVg. CLAIM 3. As k goes to infinity,

El(uk) ~ EI(u ) + Ex(vk). Proof. El(Uk) = El(t/ + Ok) =EI(U)+EI(vk)+21(--AUM + --* E1(u) + El(Vk). CLAIM 4. AS k goes to infinity, El(Vk)>--

Q(MkBR)(t"

Ivkl ~÷l

dVg/2/(~+l)+ 0(1)

M\R R

for any fixed BR.

Proof. EI(UR)

=

->

M

IVUkl2 + 4 ~ -

MkB R

i) Svz dVg

Vvk[2 + 4(n -

(since v, ~ 0 uniformly on Bn by lemma 1)

> Q(MkBR)

[vii ~+t dVg M\B R

+ 0(1).

dtdVg

dtdVg

1992

SEONGTAG KIM

u weakly in H~(M, g) and JM luV ÷1 dig = 2. Then we have the following Assume that u~ facts f r o m the homogeneity and invariance of the conformal change. If :. > 0, then EI(u ) = 22/(4+ 1)E1() , - 1/(c~+1)u )

>__,~2/(~+I)Q(M). Furthermore if ;t < 1, we have

El(v,) = (1 - 2)2/~+1)E1((1 - ~.)-l/~'-l)V k) _> (1 -- )~)2/('~+I)Q(M)

+ 0(1).

(12)

Proof o f compactness of a minimizing sequence Q(M) = Ex(u,) + O(1) = El(u) + El(vk) + O(1) _> J . z / t ~ + I ) Q ( M ) + (1 - A)z/(~'+I)Q(M) + O ( 1 )

_> (2 2/~+1) + (1 - 2)2/t~+l))Q(M) + O(1).

(13)

Case 1 . 0 < )~ < 1. (13) > Q(M) because

(1 - ) t ) t + X t > (1 - ; t + A ) ' = 1

when 0 < ;t < 1 and 0 < t < 1.

Case 2. :. = 0. (12) implies El(uk) = El(vk) + O(1). In this case, we have

Q(M) = InfEl(Uk) = InfEl(Vk) --* Q(M) as k goes to infinity.

Case 3. A = 1. That is JM lU['~+1 dig = 1. In this case, we have compactness. By considering the above three cases, we have J~t [ul ~+1 dig = 1 only, because the given condition Q(M) < Q(M) precludes case 2. Finally, we have a HZ(M, g) solution by the above argument. The regularity, which is a local property, comes f r o m the result of Trudinger [3]. THEOREM (Trudinger). Any HE(M, g) solution of the Y a m a b e equation (A) is smooth. The positivity of the solution comes f r o m the M a x i m u m Principle. Finally, we have a positive smooth solution of the Y a m a b e equation (A). Acknowledgments--This paper is a part of a Ph.D. work directed by thesis advisor Patricio Aviles at the University of Illinois at Urbana. The author would like to express gratitude to Patricio Aviles for his advice and encouragement. The author is partially supported by G.A.R.C.

Complete Riemannian manifolds

t993

REFERENCES 1. SCHOEN R. & YAU S. T., Conformally flat manifolds, Kleinian groups and scalar curvature, Invent. Math. 92, 47-71 (1988). 2. YAMABE H., On a deformation of Riemannian structures on compact manifolds, Osaka math. J. 12, 21-37 (1960). 3. TRUDINGER N. S., Remarks concerning the conformal deformation of Riemannian structures on compact manifold, Annali Scu. norm. sup. Pisa 3, 265-274 (1968). 4. AUBIN T., M~triques riemanniennes et courbure, J. diff. Geom. 4, 383-424 (1970). 5. AUBIN T., Equations diff6rentielles non lin~aires et probl~me de Yamabe concernant la courbure scalaire, J. Math. pures appl. 55, 269-296 (1976). 6. SCHOEN R., Conformal deformation of a Riemannian metric to constant scalar curvature, J. diff. Geom. 20, 479-495 (1984). 7. AVILES P. & MCOWEN R., Conformal deformation to constant negative scalar curvature on noncompact Riemannian manifolds, J. diff. Geom. 27, 225-239 (1988). 8. LIONS P.-L., The concentration-compactness principle in the calculus of variations, the locally compact case. I, II, Ann. Inst. H. Poincar~ Analyse non Lindaire 1 109-145,223-283 (1984). 9. LIONS P.-L., The concentration-compactness principle in the calculus of variations. The limit case. I, II, Rev. Mat. Ibero. 1(1), 145-201, (2), 45-121 (1985). 10. EVANS L. C., Weak Convergence Methods f o r Nonlinear Partial Differential Equations. CBMS Regional Conference Series in Mathematics, Vol. 74. American Mathematical Society, Providence (1990). 11. AUBIN T., Nonlinear Analysis on Manifolds, Monge-Ampere Equations. Springer, Berlin (1982). 12. GILBARG D. & TRUDINGER N. S., Elliptic Partial Differential Equations o f Second Order. Springer, Berlin (1983). 13. LEE L. M. & PARKER T. K., The Yamabe problem, Bull. Am. math. Soc. 17, 37-91 (1987). 14. BREZIS H. & LIEB E., A relation between pointwise convergence of functions and convergence of functionals, Proc. Am. math. Soc. 88, 486-490 (1983).