Math. Ann. 295, 75-80 (1993) Malltematische Annalen �9 Springer-Verlag 1993

Conformal deformat ions on a noncompact R iemannian manifo ld

Ma Li

Department of Applied Mathematics, Qinghua University, Beijing 100084, People's Republic of China

Received September 26, 1991; in revised form May 12, 1992

Mathematics Subject Classification (1991): 58G30, 35D05, 35J60

1 Introduction

This paper is about the Yamabe problem of noncompact type. The method we will use is to minimize a corresponding functional in a suitable class.

More precisely, we consider the following question: Given a complete noncompact Riemannian manifold (M, g) of dimension n > 3, which conditions are suitable for us to find a complete metric of scalar curvature - 1 conformal to g. It is equivalent to finding a smooth positive function u satisfying the following equation:

(1.1) - - c ~ A u + s ( x ) u + u p = 0 in M

with the condition 4

Un 2g

n - I n + 2 is a complete metric. Here c,~ = 4 2 ' p -- and s(x) is the scalar curvature

n - n - 2 ' of (M, 9). In their interesting paper Aviles and McOwen [1] found the following result

Theorem 1.2 [ 1, Theorem C]. Suppose that there exists a nonnegative smooth function ~5 with compact support such that

(1.3) f (CnlV~)12 ~- 8(X)~ 2) dv < O. M

Then (1.1) admits a positive solution u. I f we further assume that 3 M o �9 M such that s(x) <= - c o < O for all x E M \ M0; or s(x) <_ - -c l r (x ) -d for all x C M \ M o and Ric(u, u) >= - c z r ( x ) -2e for all x E M (here 0 <= e < 1, 2e <= d < 1 + e, r(x)

0 is the geodesic distance from a fixed point x o of M o, and u = Or is the outnormal

4 (whenever being defined) at x) then u n - Z g is a complete metric.

76 L. Ma

They used the sub and super solution method. Here we consider a case where (1.3) is not satisfied. Now we wish to fix some notations. Lq(M) , 1 =< q < ec and H i ( M ) are the usual Lebesgue L~-space and the Sobolev space on (M, g) respectively. L = Lg = - c A + s(x) is the conformal Laplacian operator. For u E C2(M) we write F(u) = - L u - u p.

Our main result is

Theorem 1.4. Suppose that there exists a positive constant A such that

(1.5) f ( c n l V ~ [ 2 Av 8 ( X ) ~ 2) dv ~ A f ~f)2 dv M M

holds for all q5 c C~(M) , and there exists another positive constant s o such that s (x) >= - s o f o r all x E M . Then, i f ~ a nonnegative smooth function such that (either

2n r ~ LZ(M) or r ({ Lp+I(M)), and (either F( r E LZ(M) or F ( r E L n+2 (M)) , (1.1) has an unique positive solution u with u - r c H i ( M ) .

Remark 1.6. 1) On the standard simply-connected hyperbolic space form H ~ ( - 1 ) , the condition (1.5) is satisfied. In fact, By McKean's inequality

AI (_A, H n ( _ I ) ) > ( n - 1) 2 = 4

we know that

AI(L, H n ( - 1 ) ) > cn(n - 1)2 n(n - 1) _ _ _ ( n - 1) 3 n(n - 1) > 0. = 4 n - 2

So a small perturbation of this metric will still satisfy (1.5). 2) It could be directly verified that if 3% > 0 such that u(x) > Cot(X) (2-n)/2 for

r(x) __> 1, then the metric U4/(n-Z)g is complete [1, Lemma 5.2]. We believe that if r is complete, then u4/n-2g is also complete. But we left it open.

3) ~b could be considered as a gauge or boundary condition for (1.1).

Remark 1.7. As an application of Theorem 1.4, we have the following perturbation result.

Theorem 1.8. Let 90 be the standard metric on M = H n ( - 1 ) . Let h be a radial symmetric 2-tensor with compact support on M . Then there exists c o > 0 such that f o r 0 < ~ < ~o, 9 = 9o + eh is conformal to a complete metric with scalar curvature - 1.

1 Proof Obviously, Vol(M,g) = +ec , so r = ( n ( n - l))p -~ meets our request. By Taylor's expansion for .~I(Lg) we know that if e is small, it is still positive. By

4 Theorem 1.4 we get a radial positive solution u of (1.1). The completeness of U T~-zg follows easily from u - r E H i ( M ) since we need only to verify that a finite radial geodesic could be extended to infinity. []

We wish to point out that the hypothesis r ~ L2(M) in Theorem 1.4 is natural. The reason may be seen in the following

Conformal deformations on a noncompact Riemannian manifold 77

Proposition 1.9. Assume (1.5) holds on (M, g) and u is a positive solution of (1.1). Then u ~ L 2.

Proof. If not, choose a cut-off function ~ with ~ =-- 1 on B R. Multiplying both sides of (4.1) by u~ 2 and integrating over M we obtain that

(Cn(VU V(U('2)) q-8(x)~l,2~2)dv ~= O.

111

By this, we have

A 1 / u2~2 dv <= C n / u21V~12 dv.

AI M

Letting R ---+ oc, u = 0, it is absurd. []

Remark 1.10. Some results of this paper were announced in [5]. One could weaken the condition on s(x) in Theorem 1.4.

2 The proof of Theorem 1.4

First, we recall a well-known fact about solutions of the differential inequality

(2.1) c~Au > s(x)u + u p

in a bounded smooth domain Y2 C (M, g).

Lemma 2.2 [1, Theorem 1.1]. For any compact subset X C Y2, 3c I > 0 depending only on p, n, and the elliptic constant of A in Y2 such that for every nonnegative H 1 (S2)-solution u of (2.1)

(2.3) max u(x) <= q . a:CX

Since (M, 9) is complete, there exists a bounded smooth domain sequence {D 3 ) in (M, g) such that

M = O D j j=l

and

D 3 G Dj+ l ~ M j = 1 , 2 , . . . .

We recall here that AI(M) = l imA(D 3) and Al(Dj) is the first eigenvalue of the Laplacian operator with zero Dirichlet boundary condition.

Then, we come to a key ingredient.

78 L. Ma

Lemrna 2.4. Suppose ~ is a nonnegative smooth function on M with r ~ 0 on 0~?. Then, the problem

u = r on 0S? (2.5) F ( u ) = 0 in ~2 and

admits an unique smooth positive solution.

Proof a) Existence. Clearly the even functional

on the set

1/ I(~t) = ~ (Cn[V~t l 2 -~- 8(x)~t2)dv -~-

f2

1 /lul p+l dv p + l .

s

A = { u e H I U ? ) ; ~ = q 5 on 0~2}

is coercive and weakly lower semi-continuous. So by a direct method we get that 3z~ r A s.t. u >__ 0 and I (u ) = infA(. ). Since for any r/ r H~ (f2), t C R, ~1 + t'r! c A,

d it necessitates ~ l ( u + trl) It=0--- 0. By Trudinger 's regularity theorem [3, Chap. 5]

we know that u is smooth. The strong maximum principle [1, Proposition 3.75] tells us that ~z is positive in D. b) Uniqueness. It is an elementary fact that there exists an absolute positive constant c 3 such that

(2.6) (a - b)(a p - b p) > c3la - blp+l

for any a, b ~ 1R. Suppose u l, u 2 are two solutions of (2.5). Let w = ~u I - ~2, then w satisfies

L w + ( u P - u ~ ) = O in g2,

w = 0 on 0X?.

By (2.6) we g e t / ' ( L w , w)dv <= O. From (1.5) we know that w 0. []

~Q

We end this part by giving the proof of Theorem 1.4: From Lemma 2.2 we get a positive solution u.j of (2.5) with g2 = Dj. By Lemma 2.1 and the standard elliptic estimates [4] we know that, for any compact set X C M , there exists an absolute constant c 4 such that

II'II~jllC3(X) ~ C 4"

By a diagonal argument we get a subsequence { ~ } such that

(2.7) u i -~ u in C2oc(M)

and u is a nonnegative solution of (1.1).

Confonnal deformations on a noncompact Riemannian manifold 79

S ince u, - r E H~(D~), we ob ta in by (1.5) and (2.6) tha t

A / lu~ - r < / (L(u~ - r u,~ - r dv

D z D ,

f<, = - u , - Lr u~ - r dv

D~

t ]

D~

< f (F(r uz - O} - c3 f f lu~ - Olp+l dv. D~ D z

By H o l d e r ' s inequal i ty we k n o w that

I I ~ - r + [[U, -- r162 ~ C(~)IIF(r

Taking i --+ oc, we get on one h a n d that

u - q) e L 2 ~ L P + l ( M )

by (2.7). On the o ther hand , for J2 = D~,

= / ( - u P - LO, u z - @ d v - / s ( x ) ( u ~ - r

f2

< f (F(r ui - 6) dv + so f ,% - d),2 dv s ~2

< c2(M) < +oc.

So, by (2.7) again we get that

f l v ( u - r < +~. M

Hence u - r E H i (M) . If u - 0, then r E L 2 N Lp+I(M), a cont rad ic t ion . By the

s trong m a x i m u m pr inc ip le aga in we k n o w that u > 0 in M . The u n i q u e n e s s fo l lows f rom (1.5). [ ]

Acknowledgment The author would like to thank Prof. K,C. Chang for his interest on this paper. The author thanks the referee for his remarks on this paper.

80 L. Ma

References

1. Aubin, T.: Nonlinear analysis on manifolds, Monge-Ampere equations. Berlin Heidelberg New York: Springer 1982

2. Aviles, P., McOwen, R.: J. Differ. Geom. 27, 225-239 (1988) 3. Aviles, P., McOwen, R.: J. Differ. Geom. 21,269-281 (1985) 4. Schoen, R., S.T. Yau: Differential Geometry, Beijing, Academic Press, 1988 (In Chinese) 5. Morrey, C.B. Jr.: Multiple integrals in the calculus of variations. Berlin Heidelberg New York:

Springer 1966 6. Ma Li: On the positive Solutions of the Yamabe Equation on an open manifold. Collected papers

of Chinese Postdoctor in Sciences, Beijing: Peking Univ. Press 1991 (In Chinese)