differential equations on riemannian manifolds and their geometric applications

COMMUNICATIONS ON PURE AND APPLIED MATHEMATICS, VOL. XXVIII, 333-354 (1975)

Differential Equations on Riemannian Manifolds and their Geometric Applications”

S. Y. CHENG AND S. T. YAU Courant Institute Stanford University

Most of the problems in differential geometry can be reduced to problems in differential equations on Riemannian manifolds. Our main purpose here is to study these equations and their applications in geometry.

In Section 1, we prove that if the volume of the geodesic balls of a complete Riemannian manifold grows at most like a quadratic polynomial, then there is no non-trivial positive superharmonic function defined on this manifold. In fact, a more precise quantitative result is obtained. We apply this to prove a Bernstein type theorem: If H is a function of the same sign defined on R 3 with aH/aX,=aHfaXz=O and aH/aX,SO, then there is no graph defined on R 2 whose mean curvature at each point is given by H.

In Section 2, we simplify a proof of the “maximal principle” in [9] and give some applications. For example, if M is a complete Riemannian manifold such that outside a compact set the Ricci curvature is bounded from below by (dim M + & ) / r 2 , where r is the distance from some fixed point and E > O is a fixed constant, then M does not admit any non-constant positive superharmonic function.

In Section 3, we study the first eigenvalue of a complete (non-compact) Riemannian manifold M. This is defined to be the infinimum of the first eigenvalues (for the Dirichlet problem) of all compact subdomains of M. We prove that if f is any positive function defined on M, the first eigenvalue of M is bounded from below by inf (-Aflf). We then observe that if the volume of the geodesic balls of M grows polynomially, the first eigenvalue of M has to be zero. These two facts together give strong restrictions on positive functions defined on M. One purpose of studying these facts is to give a complete proof of the following theorem studied in [8]: Any complete convex hypersurface with constant mean curvature in euclidean space is a generalized cylinder. (The two-dimensional version was obtained by Osserman and Klotz [4] by a completely different method.)

In Section 4, we apply the method of [9] to prove that if the first eigenvalue A,(M) of a complete Riemannian manifold M is positive, then a

*This research was supported in part by the National Science Foundation, NSF-GP-32460X. Reproduction in whole or in part is permitted for any purpose of the United States Government.

@ 1975 by John Wiley & Sons, Inc. 333

334 S. Y. CHENG AND S. T. YAU

positive solution of Af =-A1(M)f exists. In the course of the proof, a generalization of a theorem of [9] is also obtained.

In the last section, we discuss differential inequalities of the form Au 2 f (u) on a complete Riemannian manifold with Ricci curvature bounded from below by a constant (or on a properly immersed submanifold of the euclidean space with bounded mean curvature). We prove that if f > O and lim inf,,, (f (t)/g(t)) > 0 for some positive continuous function g non-

decreasing on an interval [a, w) with I( la' g(7) di)-"*dt <w for some b, then

no Cz-function u can satisfy Au 2 f(u). In case the manifold is the euclidean space and f = g, this was proved by Keller [3] and Osserman [6 ] . Calabi [2] also proved the theorem when the manifold has non-negative Ricci curvature and f = g. The theorem in this general form will be applied to study affine hyperspheres in the next paper.

1. The Parabolicity of Certain Complete Riemannian Manifolds

It is well known that there is no non-constant negative harmonic function defined on the euclidean space. Among the euclidean spaces, R 2 enjoys a very special property. Namely there is no non-constant negative subharmonic function on RZ. It turns out that this is not true for higher-dimensional euclidean space. The purpose of this section is to explain this phenomenon.

Let M be a complete Riemannian manifold with dimension n. Let r be the distance function of M from a fixed point p. Let V(r) be the volume of the geodesic ball of radius r. Then we define the order of M to be the smallest number O ( M ) such that lirn inf,,,(VCr)/r"'M))<w. It turns out that this number has an intimate relation with various function-theoretic proper- ties of M. We list first some elementary propositions.

PROPOSITION 1. Let s be a proper function defined on M such that, for some constant c > 0, s(x) 5 cr(x) + c. Suppose, for some constant a,

Vol{x I s ( x ) 5 b } < w b" lim inf

b-rn

Then O ( M ) S a .

PROPOSITION 2 (Milnor [5]). Suppose M covers isometrically another compact Riemannian manifold N. If the growth function r(t) of the fundamental group m,(N) is dominated by t", then O ( M ) S m .

PROPOSITION 3. If M is a complete Riemannian manifold with non- negative Ricci curvature, then O(M) 5 dim M.

DIFFERENTIAL EQUATIONS ON RIEMANNIAN MANIFOLDS 335

The main theorem we want to prove in this section is the following

THEOREM 1. Let M be a Riemannian manifold. Let f be a negative function defined on a compact geodesic ball B(R) of radius R around some

fixed point p. Suppose, for some a > 0, (Aflf) is a decreasing function for

r La and, for some sequence a = r o c r , < r2 < . * < r,, = R, Vol ( r , ) / r t i c. Then Lr)

Proof: Let f be a negative subharmonic function. Then we have

For all r > O , let B ( r ) be the geodesic ball of radius r around a fixed point p. Let 9 6 be a Lipschitz function defined on M such that cpS(X)=O for x~ B(r) and q ( x ) = 1 for x E B(r- 6). Then, integrating (1.2), we obtain

Since we can assume that IVtplS 1/6, we see that

Before we go on, we observe the following fact: If h is any bounded

measurable function defined on M , then the function h(r) = J h is a locally

Lipschitz function and the derivative 6'(r) exists almost everywhere. Hence if we define

B(r)


then it follows from (1.4) that

(1.5) F ( r) 5 (V’( r))”*(F’( r))”’ + ( G ( r))

for almost all r.

r L a , we see that F ( r ) > G ( r ) for r L a and In order to prove (l.l), we can assume F ( a ) > G ( a ) . Since G’( r )SO for

(1.7)

F’(r) - G’(r) 1 ( F - G)2(r) zm ’

1 dx - 1 F(r ) - G(r ) F ( P) - G ( i ) 1- v’(x) ’

for i > r > a. By the Schwartz inequality,

Combining (1.7) and (1.8)’ we have

(1.9) 1 - 1 ( i - r)’

F ( r ) -G( r ) F(F)-G(T)LV(P)-V(r ) ‘

It follows easily from (1.9) that

(1.10)

Inequality (1.1) is an immediate consequence of (1 .lo).

COROLLARY 1. Let M be a complete Riemannian manifold with O ( M ) 5 2. Then M does not admit any negative subharmonic function.

Proof: By assumption, we can find. a constant c > O and a sequence rl < r2 < r3 < 00 such that Vol B(ri)/r? 5 c. By passing to a subsequence, we can assume that ri/r,+, <$. The corollary then follows immediately from the theorem.


COROLLARY 2. Let M be a complete Riemannian manifold with O ( M ) 5 2. Let f be a C2-function defined on M. If sup f <m, then we have sup f =

Let C = sup{f(x) I Af(x)<O}. Let A be a function defined on the real line such that A ( x ) = 0 for x S C, A'(x) > 0 and A " ( x ) 2 0 for x > C. Then direct computation shows that the function A 0 f is subharmonic and bounded from above on M. By Corollary 1, this implies that A o f is a constant function and Corollary 2 follows.

Let us now apply Theorem 1 to prove a Bernstein type theorem. To do this, we shall use notations of [l]. Let u be any smooth function defined in an open set of R". Then one can define

sup {f(x) I Af(x) < 01. Proof:

au du siu =- - c vivj- ax, j = l axj

in this open set where

is the unit normal to the level surface of u. The following proposition is well known (cf. [l]).

PROPOSITION 4. Let S be a hypersurface in R" and let u be a smooth function in R". Then, if u I S has compact support,

where H is the mean curvature of S.

PROPOSITION 5 . Let S be a graph in R" given by xn = U ( X I , * * * , ~ " - 1 ) .

Then

satisfies the equation

Av, + IlN(12 vn = (n - 1) S.H,

338 S. Y.,CHENG AND S. T. YAU

where A is the Laplace-Bettrami operator of S and JIN112 is the length of the second fundamental form of S.

In order to apply Theorem 1, we shall find a condition for 6,H S O . Since

we can simply require that H can be extended to a function H in R " so that H depends only on x,, and aH/ax, 50.

THEOREM 2. Let H be a function defined in R 3 such that aHlaxl= aHlaxz=O and aH/ax,SO. Let M be a piece of a surface defined over a domain of RZ. Suppose H has the same sign and M contains a compact geodesic ball B(aZ) of radius a 2 > 1 around some fixed point p. If the mean curvature of M is given by H, then

(1.12)

where /IN112 i s the length of the second fundamental form.

Proof: First of all, let us prove that if H has the same sign, the volume of the balls B(r) grows like r2. In fact, if M is defined by some function u, its

mean curvature H is given by t (av,/axi). Integrating it over a ball D(r) of

radius r in R2, we get

2

i = l

where ni is the i-th component of the outward unit normal to aD(r). Let A be a function defined on the real line such that A(x) = 0 for x 5 -r,

A'(x) = 1 for -r S x 5 r and A(x) = r for x 2 r. Let p be a function defined on R 2 such that p(x) = 0 for 1x1 2 r, p(x) = 1 for 1x1 S i r and l V p l 5 112r. Then the function pA(x3) has compact support on M and, according to Proposition 2 ,


Since S3[pA(x3)] = 63(p)A(x3)+pA’(x3)(1- u;) , we have

(1.15)

Hence

which proves our claim. According to Proposition 5 and the hypothesis, we have

Therefore, ( l . l ) , (1.16) and (1.17) imply that, for any sequence a =roc r l < ’ < r,, = a’,

Inequality (1.12) follows quite easily from (1.18).

COROLLARY 1. Let H be a function defined on R 3 with aHlax, = aH/ax, = 0 and a H / a x 3 s 0 . If H has the same sign and M is a graph defined over R 2 whose mean curvature i s given by H, then M is a plane.


If we drop the first restriction on H , we still have the following corollary of Theorem 1.

COROLLARY 2. Let M be a graph of some positive function defined over R’.

This follows from the well-known equation Ax3 = Hu3 and the fact

Then if the mean curvature of M is non-positiue, M is a plane.

Proof: that u3 > 0.

Remark. One can use Lebesgue’s method to derive a Harnack type inequality from Theorem 1. We may state the result as follows:

PROPOSITION 6 . Let M be a two-dimensional Riemannian manifold such that all the geodesic circles of M around some fixed point p E M are connected. Let u be a positiue harmonic function defined on a compact geodesic ball of radius 2R on M. Then there is a constant depending only on S U ~ ~ ~ ~ ~ ~ ~ (Vol B(r)/r2) such that rnaxeLn, u S c mine(a, U.

2. A Generalized Maximal Principle

We shall give a simplification and a generalization of the “maximal principle” of [9]. Let p be any fixed point of a complete Riemannian manifold M with dimension n. Let u : [0, l ] + M be any unit speed geodesic joining a point x to p. Let u’(t) be the tangent vector of u and Ric(u’(t)) the Ricci curvature of M in the direction u’(t). Define

‘( t - k)‘ Ric (u’(t)) dt . 1 n - 1

If x is not on the cut locus of p, then we can take cr to be the unique shortest geodesic from p to x and define K(x) = K,(x). Otherwise, we define K(x) = min, K , ( x ) , where u ranges over all the minimal geodesics from p to x. Let r(x) be the distance function from p. Then clearly if the Ricci curvature of M is bounded from below, K(x) is bounded from above by a constant when r(x) 2 1.

THEOREM 3. Let M be a complete Riemannian manifold. Let f be any Cz-function bounded from above on M. Then, for any p e M , there is a


sequence of points {xk}c M such that

lim f(xk) = sup f , k-m

Proof: For all k > 0, let

Then since

we see that g must attain its supremum at some point xk. If X k is not on the cut locus of p, then we can differentiate u at XI, to obtain

(2.4)

Using (2.3) and Lemma 1 of [9], one can then simplify (2.4) to obtain the last two inequalities of (2.1).

Now consider the case where Xk is on the cut locus of p. We shall use a method of Calabi [2]. Let (+ be a minimal geodesic joining p and xk. Then,

342 S. Y. CHENG AND S . T. YAU

for any point q in the interior of u, q is not conjugate to xk. Fix such a point q and let Nq be a conical neighborhood of the geodesic segment of u joining q and xk such that, for any x E N,, there is at most one minimizing geodesic joining q and x.

Let Fq be the distance function from q taken in the manifold Nq. Clearly, Fq(x)2rq(x) and Fq is smooth in a neighborhood of x k . We claim that the function

also attains a local maximum at the point xk. In fact, for any point x E N,, we have

Therefore the claim is proved and we can take the gradient and the Laplacian of the function

The same argument as before then shows that the last two inequalities of (2.1) are valid, by taking q + p.

The first inequality of (2.1) can be proved in the same way as in [9].

COROLLARY 1. Let M be a complete Riemannian manifold. Suppose for some point p E M there exist E > O , c > O such that, for all minimal geodesic IT : [ O , l ] + M, Ric ( ~ ’ ( t ) ) 2 (n + E)/t2. Then every positive superharmonic function defined on M is a constant function.

Proof: Let f be a positive superharmonic function defined on M. Then clearly we can assume f(p) = 1 and inf f > 0. Furthermore, direct computation shows that Af2 S IVf 1’.


Applying (2.1), we conclude that there is a sequence {Xk}CM such that

lim f '(Xk) = inf f ', k--

Under the assumption on the Ricci curvature, one can prove that, when r(Xk) is very large, r(Xk)K(Xk) + 1 5 -fe. From (2.6), we conclude immediately that lim SUpk,, r(xk) < rn and f ' attains its minimum somewhere. By the minimum principle, f must be a constant.

COROLLARY 2. Let M be a complete Riemannian manifold with Ricci curvature bounded from below. Let F be a function defined on the product space of R and the tangent bundle of M such that, for all sequences t,+t, {(x,, v , ) } ~ T ( M ) with u, -+ 0; we have lim inf,,, F(t,, x,, u,)>O. Then, for all C2-functions u defined on M with A u ( x ) Z F ( u , x , V u ( x ) ) for all XEM, u cannot be bounded from above.

3. A Cylinder Theorem

One of the main purposes of this section is to continue the work of [8] and prove the following theorem: Any complete convex hypersurface with constant mean curvature in the euclidean space R"" is a generalized cylinder. (When n = 2 , this is due to T. Klotz and R. Osserman [4].)

Before we do this, we shall introduce some terminology. Given a compact domain D with C' boundary in a Riemannian manifold M, the first eigenvalue of D is defined to be the smallest positive number h l ( D ) such that, for some non-zero function f, f I dD = 0 and A f = -hl(D)f. The first eigenvalue of M is then defined to be inf h l ( D ) , where D runs through all the compact subdomains with C' boundary.

PROPOSITION 7.

344 S . Y. CHENG AND S . T. YAU

where D runs though compact subdomains of the form {x 1 g(x)<O, g is Lipschitz}.

Proof: It is well known that if aD is C', then

Therefore, if D , c D2 are domains with C' boundaries, then X1(Dz) 2 X1(Dl). The proposition follows easily from this.

In this section, wc need conditions for h l ( M ) = 0. We shall see that they follow essentially from O ( M ) < a . Let B ( a ) be a geodesic ball around some fixed point p. Let r be the distance function from the point p. Then a - r is a Lipschitz function which is zero on the boundary of B(a) . Therefore,

for any b < a .

PROPOSITION 8. Let M be a complete Riemannian manifold. Then, if

Proof: By assumption, there exists a sequence (a,}+m such that Vol (B(a,)) 5 Cq:, where C is independent of i. By choosing a subsequence if necessary, we may assume that aLtl - a, 2 $ai.

If Vol (M) <to, Proposition 8 is clear from (3.1). If Vol ( M ) = m, (3.1) shows that

O(M) 5 2 , h i ( M ) = 0.

which decreases to zero.

PROPOSITION 9. Let M be a complete Riemannian manifold. Suppose there are numbers a and c such that, for all geodesic balls B(r) of radius r around


some point p, Vol B(r) 5 cra. Then lim infi,, 4'X1(B(2')) is bounded. In particular, lim inf,,, r2hl(B(r)) is bounded and Al(M) = 0.

Proof: By (3.1),

(3.2) 4 Vol ( B ( 2 ' ) ) lim inf 4'hl(B(2')) s l i m inf

I'm 1-m Vol (B(2'-'))

Therefore it suffices to prove that the right-hand limit is bounded. This follows from the hypothesis because if

=to, Vol (B(2')) lim inf

I+= Vol (B(2'-'))

then for any constant d>O

when j is large enough.

THEOREM 4. Let f and g be two smooth functions defined on a compact Riemannian manifold M (with boundary). Suppose f > O , g 2 0 on M and g(8M) = 0. Then either

Proof: Consider the function h = g / f defined on M. It satisfies the following equation:

(3.3)

The Hopf maximal principle implies that if Ag/g-Af/fZO on M, then h

Theorem 5 follows easily from this principle because h(8M) = 0. cannot attain its maximum in the interior of M unless h is constant.

COROLLARY 1. Let f be a positive function defined on a Riemannian manifold M. Then h l ( M ) Zinf (-Af/f).

COROLLARY 2. Let M be a complete Riemannian manifold. Suppose there are numbers a and c such that, for all geodesic balls B(r) of radius r around


some point p E M, Vol (B(r)) 5 cra. Then, for any positive function f defined on M,

is bounded from above.

In order to state the next corollary, we recall that a minimal hypersurface is called stable if and only if the second variation of its volume function is positive for all compactly supported deformations.

COROLLARY 3. Let M be a compact hypersurface (with boundary) in euclidean space. Then if the Gauss image of M lies in a n open hemisphere, M i s stable.

Proof: It is well known that if M is not stable, then we can find a subdomain MI of M such that dM1 is the first conjugate boundary. Therefore there exists a non-negative function g such that g(dM1) = 0, g > 0 in MI and

where llNlr is the scalar curvature. On the other hand, if the normal v lies in an open hemisphere, then we

can find a vector a such that (v, a ) > O on M. Since (v, a ) is also a solution of (3.4), Corollary 3 follows.

Remark 1.

Remark 2. Since Bernstein's conjecture is false for dimension greater than or equal to 7, Corollary 1 of Theorem 2 is not valid for higher- dimensional euclidean space. However, under the same assumption, one can conclude that lim inf,,, r2 infe(,) llN112 is bounded from above, where llN1r is the length of the second fundamental form and B(r) are geodesic balls of radius r.

This corollary is also known to L. Simon.

THEOREM 5. Let M be a complete convex hypersurface with constant mean curvature in the euclidean space. Then M is a generalized cylinder, i.e., the product immersion of a linear subspace and a sphere.

Proof: Let u be the normal vector of M and the length of the second fundamental form of M. Then we have the well-known formula

Since the Gauss image of a complete convex hypersurface lies in a closed hemisphere, there exists a unit vector a such that (v, a ) l O . By Proposition


3, Theorem 5 and ( 3 3 , we conclude that (v, a) has to be identically zero unless llN112 tends to zero. If the mean curvature of M is a non-zero constant, (IN((Z is also bounded from below by a positive constant. Hence either the mean curvature is identically zero or (v, a) is identically zero.

In the former case, the convex hypersurface M is clearly a plane. In the latter case, M is the product immersion of a line and another manifold. (Differentiating the equation (v, a) = 0, one obtains A,(e,, a) = 0 for all principal curvatures A, and principal direction e , . Projecting a onto M, we get the unit vector field 1 (a, e,)e, which, by computation, is parallel on M.) An

induction will then show the validity of the theorem. L

4. Gradient Estimate and the Existence of the First Eigenfunction

In Section 3, we introduced the number Al(M) for a general Riemannian manifold M. In this section, we shall apply the method of Section 2 and the arguments of [9] to prove the existence of a positive function f on M such that A f = -h , (M)f . In fact, results of [9] will be generalized.

Let B(a) be a geodesic ball of radius a and center p in some complete Riemannian manifold M. Let f be a non-negative non-constant differentiable function defined on B(a). Then, for any c >0, we can consider the function

where r, is the distance function from p (when lVf l= 0, F, is defined to be a). This function has to attain its minimum at some point q in the interior of

We assert that we may assume the function F, is differentiable at q. If q is not on the cut locus of p, this is clear. Otherwise, we take a shortest geodesic u joining p and q. Let i j be any point in the interior of u. Then we can consider the function

B ( a ) .

in a conical neighborhood of u and the same argument as in Section 3 shows that the claim is valid. Hence, at the point q,

Denoting r, by r and the covariant derivatives of f with respect to some

348 S . Y. CHENG AND S. T. YAU

orthonormal frame field {el , . . . , en} by f,, fi,, etc., we obtain the following equations at q :

(4.2) - = 0 , 1 [f, + 2rrt(f+ c) (a2- r2) IVf 1 (aZ- r2) IVf l2

4r2(f+ c) c r? 2r c firi +A

A f + a 2 - r 2 a*- r2 (a2 - r2)' a 2 - r2 2(c r')(ft ') + 2r Ar(f + c) +

From equation (2.11) of [9], we see that

where R,i is the Ricci tensor of M. By choosing the frame field suitably at q, we may assume that

(4.5) fi = 0

for i > 1. Then (4.2) yields

4r2(f+ c ) c r' - 2rfli-1 + --

a'--' (a'-r2)*

DIFFERENTIAL EQUATIONS ON RIEMANNIAN MANIFOLDS

Putting (4.4), (4.5) and (4.6) into (4.3), we have

349

2(f + c)(c ”) 2r Ar(f + c) + a’-r2 A f + a 2 - r 2 ’ +

(4.7)

-(@$(c ft--Xf;,)- ~ ( A f ) l - ( f + c ) R 1 l Z O . f l i,i f l

Let us assume

(4.8)

where mi are constants. Then, following [9], we have

(4.9)


Putting (4.9), (4.10) into (4.7) and multiplying by (a2-r2)’/(f+c), we obtain

1

1 (a2-rZ)’f: ( f + c ) (a2- r2)Rl lz - + m3(a2- r2)’ -- 4(n-1) (f+C)’ . fl

Finally, assume

(4.12) m+ m4 IVf I + m,(f+ c ) . Af 5 4(2n + 1) f + c

Then

Let K be the lower bound of the Ricci curvature of B ( a ) . Then, by Lemma 1 of [9], r Ar is bounded from above by some multiple of JKJ + 1. It follows easily from (4.13) that there is a constant an depending only on n such that

Since q is the minimum point of F,, we have proved the following:

THEOREM 6. Let M be an n-dimensional complete Riemannian manifold. Let f be a non-negative differential function defined on a geodesic ball B(a) of radius a. Suppose, for some c > O , there are constants m l , m2, , ms such that


Then we can find a constant an depending only on n such that

(4.15)

where r is the distance from x to the center of B ( a ) and K is the lower bound of the Ricci curvature of B ( a ) .

Now let fa be the first eigenfunction of B ( a ) such that f a ( p ) = 1, fa 1 a B ( a ) = 0 and Afa = -Al(B(a))f , . It is well known that fa is non-negative so that we can apply Theorem 6 . Hence we have a one-parameter family of functions {fa} defined on the balls B ( a ) for which (4.15) is valid. More precisely, for any fixed ball B ( c ) , 1V log (fa + l)/ is uniformly bound for a > 2c. Since fa@) = 1 for all p, we conclude that {fa} has a subsequence converging uniformly on compact sets to a function f with f(p) = 1. Standard estimates then show that Af = -Al(M)f.

THEOREM 7. Let M be a complete Riemannian manifold. Then there exists a positive function f defined on M such that A f = -A,(M)f, where Al(M) is the first eigenvalue of M defined in Section 3.

Remark. Note that according to Corollary 1 of Theorem 4, there is no positive solution of A f = -Af for A > Al(M).

5. Differential Inequalities on Complete Riemannian Manifolds

Throughout this section, we shall assume that our Riemannian manifold M

Let u be a C2-function defined on M such that is complete and its Ricci curvature is bounded from below by a constant.

where f is a function to be determined later. Let F be a positive Cz-function defined on the real line. Then we can consider the function 1/F(u) which is bounded from below on M. According to Theorem 3, there is a sequence { x t } c M such that

(5.3)


and

Multiplying the last equation by F’(u)’/F(u)’ I F”(u)l, we see that, at x k ,

F”(u) F’(u)’ JVuI2 F’ (u )~ Au (F”(u)( F4 F4 IF‘’(u)[

_-

2F’(u)’ F’(u)’ J V U ( ~ , -F‘(u)’ (5 .5 )

- F ( u ) IF”(u)l F ( u ) ~ - kF(u)’ IF”(u)I * +

If we could choose the positive function F such that

and

(5.7)

IF’(t)l” <m, lim sup t-m F(t) IF’’(t)I

F’(tI3f(t) < 0 lim inf t-m F(t)4 JF”(t)( ’

then (5.1), (5.3) and (5.5) would imply that

lim sup u ( x k ) < 00 . k--

To guarantee the conditions (5.6) and (5.7), we define, for s 2 b,

(5.8)

where g is a positive continuous function defined on [ a , m ) with -112 jb= (ja‘9(7) dT) dt <m. Then we extend F to be a positive monotonic

increasing C’-function defined on the real line. In this case, (5.6) and (5.7) are equivalent to

(5.9)


and

(5.10) lim inf A,, t+- F(t)4g(t) *

-112

Since F ( t ) 2 ( t - b) (I' g(T) dr) for t large, condition (5.9) will follow from

the following relation:

(5.11)

With this choice of F, we have proved

THEOREM 8. Let u be a Cz-function defined on a complete Riemannian manifold M (with Ricci curvature bounded from below) such that Au Zf(u). A sufficient condition for u to be bounded from above on M is that there is a continuous function g positive on some interval [a,m) such that

~ l i Z

( j a ' g ( r ) dr) dt < m for sorile b 2 a,

Furthermore, if f is lower semi-continuous, f(sup u ) 5 0.

Proof: Theorem 3 that f(sup u) 5 limk,, Au(xk) 5 0.

If g is non-decreasing, the condition

We have only to check the last statements. But it follows from

is automatic and we have the following

COROLLARY. Let M be a complete Riemannian manifold with Ricci curvature bounded from below. Let u be a C2-solution of the inequality Au Zf(u), where f is a non-negative lower semi-continuous function and


lim inf,,, ( f ( t ) / g ( t ) ) > 0 for some positive continuous function g non-decreasing

on some interval [a ,m) with Ibm ( I a 1 g ( r ) dr)-"* dt <m for some b. Then sup u

exists and is a zero of the function f .

Remark 1. In case M is the euclidean space and f = g is positive continuous, this was proved by Keller [3] and Osserman [6]. In this special case, Redheffer [7] generalized their result by replacing monotonicity of f by

the condition tf(t) 2 c f ( ~ ) dr for some constant c. This last condition is also

a special case of Theorem 8. However, in case the manifold is the euclidean plane, he can drop the condition of monotonicity completely.

In Theorem 8, we may replace the condition that the Ricci curvature be bounded from below by the condition that M be a properly immersed submanifold of the euclidean space with bounded mean curvature.

I"[ Remark 2 .

Remark 3 . The most general form of the theorem can be stated as follows: If Au 2 f ( u ) is such that, for some C2-function F, the equalities (5 .6 ) , (5.7) are valid with

sup F( t) < lim sup F( t) I1IC-r 1-m

for all r >0, then sup u <m. Furthermore, if f is lower semi-continuous, f(sup u ) 5 0.

Bibliography

[l] Bombieri, E., Theory of Minimal Surfaces and a Counter-Example to the Bernstein Conjecture

[2] Calabi, E., An extension of E. Hopf's maximum principle with an application to Riemannian

[3] Keller, J. 13., On colutions of Au = f ( u ) , Comm. Pure Appl. Math., 10, 1957, pp. 503-510. [4] Klotz, T. and Osserman, R., On complete surfaces in E' with constant mean curvature,

[5] Milnor, J . , A note on curuature and fundamental group, J. Diff. Geom., 2, 1968, pp. 1-7. [6] Osserman, R., On the inequality Au>f(u) , Pacific J. Math., 7, 1957, pp. 1641-1647. [7] Redheffer, R., On the inequality AuZf(u ,grad u) , J. Math. Anal. Appl., 1, 1960, pp.

[ X I Yau, S. T., Complete Hypersurfaces with Constant Scalar Curvature, Lccture Notes, A.M.S.

[9] Yau, S. T., Harmonic functions on Riemannian manifolds, to appear.

Received October, 1974.

in High Dimension, Lectures at the Courant Institute, New York University, 1970.

geometry, Duke Math. J., 25, 1958, pp. 45-56.

Comment. Math. Helv., 41, 1966-67, pp. 313-318.

277-299.

Symposium, Stanford 1973.

differential equations on riemannian manifolds and their geometric applications

Documents