Some characterizations of the euclidean sphere

NonlinearAnalysis, TheoryMethods& Applications.Vol. 1 No. 1. pp. 37-48. PergamonPress, 1976. Printedin Great Britain.

SOME

CHARACTERIZATIONS

OF

THE

EUCLIDEAN

SPHERE*

PI-HLIP H A R T M A N Mathematics Department, The Johns Hopkins University, Baltimore, M D 21218, U.S.A.

(Received 26 January 1976) Key words: Riemann manifold, curvatures (Riemann, Kronecker-Gauss, principal, sectional, mean, scalar), covariant derivative, Codazzi condition, second fundamental form, umbilic, Laplacian 1. I N T R O D U C T I O N

THE MAINtheorems of this paper are "local" in nature, i.e. deal with a neighborhood of a point on a Riemann manifold. The results however have consequences "in the large"; see, e.g. Theorem 1.1. Definition (Pl). Let ~ + be the set of class C 3 pieces ( = open patches) M of Riemann manifolds of positive sectional curvature and dimension n. Let hij(x), x e M, be a symmetric covariant tensor of class C 2 satisfying the Codazzi condition Vkhi~ = V/li~, where V denotes covariant differentiation on M, with its ordered eigenvalues 21(x) =< ... =< 2(x) satisfying (2i(x) . . . . . 2(x)) Do, a fixed subset of R". Let ~go be a set of pairs (M, Izij), M ~ Jr' + and tzi~as above. Let A0. ) = A(21 . . . . . 2 ) and B(2) = B ( 2 t , : . . , 2,) be real-valued functions on D o. We say that the ordered pair (A, B) has the property Pl(Jgo, Do) if, for every pair (M, hit ) ff "14/0' a point x ° ~ M is an umbilic (i.e. 21(x °) . . . . . 2 (x°)) whenever A(2(x)) has a local minimum and B(2(x)) a local maximum at x = x °. We write Pl(Jgo) for /~l(Jgo, D) if D = {2 = (21. . . . . 2 ) e R " : (0 <)21 __<... __<2}, and fit for Pl(~go) if~g o consists ofaU pairs (M, hij), arbitrary M e ~g+ and arbitrary hi~ as above. Definition (PI). We shall write Pl(~//o) in place of/51(Jr'o) and P~ in place of Pl if, in the above definition, d¢+ is the set of class C a pieces M of hypersurfaces of R" ÷ 1 of positive sectional curvature, J / o is a subset of ~¢t'+, and the only tensor hij(x ) considered on M is its second fundamental tensor (so that (0 <)21(x) < ... < 2(x) are its principal curvatures). Remark. In Definition (P1), we could replace R "+1 by a Riemann manifold N "+ 1 of constant sectional curvatures c and require M ~ ~t'+ to have positive sectional curvatures exceeding c. Thus, for example, R "+ 1 in Proposition 1.1 and Theorem 1.1 can be replaced by S "+ ~ if the sectional curvatures of M exceed 1. The meaning of J/t'+ or J / o in any situation will be clear from the context. Our main interest is in the property Pl(Jt'o) or P r The reasons for introducting Pl(~//o) or/3j are twofold: first, the sufficient conditions to be obtained for Pt(~t'o) are actually sufficient for Pl(~t'o) and, second, if (A(2), B(2)) have property P~, then both (A(2), B(2)) and (A(1/2), B(I/2)), where 1/4 = (1/4 . . . . . . 1/21), have property PI" This follows by a remark of Simon [8] on replacing M by the sphere S" with the metric given by the third fundamental form of M and h o is the second fundamental form of M. Definition (P2)- Let A(2), B(2) be real-valued functions on D ~ R". We say that the ordered pair (A, B) has the property P2 if A(2) < B(2) for all 2 ~ D and A(2 °) = B(2 °) if and only if 40 = (2°, .. . ,4 °.) satisfies 2 o = . .. = 4 °. * This research was supported in part by NSF Grant No. MPS75-15733 and was undertaken at C o u r a n t Institute. 37

38

P. HARTMAN

Proposition 1.1. Let x° ~ M ~ ~t[o. Let (A, B) have the properties P~(J(o), P2 and let A(,~(x)) have a local minimum and B(2(x)) a local maximum at x = x °. Then every point in a neighborhood of x ° is an umbilic (thus a neighborhood o f x ° on M is part of a sphere).

This is clear from A(2(x°)) __
For example, if ~b(u, v) is strictly increasing with respect to u, let B(2(x)) assume its maximum value B ° at x ° and let A(2(x°)) = A °. Then ~(A °, B °) = 0 = ~(A(2(x)), B(2(x)) __<~(A(2(x)), B °) for x e M. Thus A ° _<_A(2(x)), so that A(2(x)) has a minimum at x °. For generalizations in the case n = 2, see Hartman [5] and references there to Alexandrov, Hartman &Wintner, H. Hopf, and Pogorelov. For some cases with n __>2, see Simon [8] and references there; cf. Corollary 1.2 and (1.22) below which, together with Theorem 1.2, generalize some results of [8]. A result of Weyl [9] implies that if n = 2, A(2) = 2122 and B(2) = (21 + 22)/2 , where K(x) = A(2(x)) is the Gauss curvature and H(x) = B(2(x)) is the mean curvature of M a t x, then (A, B) has property P1 (and so, A = (21.~.2)1/2 and B = (A1 + 42)/2 have properties P1 and P2); cf. Chern [2]. Correspondingly, a variant of a result of Hilbert [6] shows that A = At and B = 42 have properties P1 and P2; el., e.g., Chern's [2] proof of Weyl's theorem. As remarked in Sacksteder [7], Hilbert's result implies Weyrs. Sacksteder also gives examples to show that Hilbert's result is false if it is only assumed that M ~ C a (rather than M ~ C 4) and that Weyrs result is false under the assumption M e C 2. The situation concerning M ~ C a in Weyrs result was left undecided. For n > 2, Yau [10] (cf. Simon [8]) has generalized Weyl's result and has shown that if A=ZZ2i2#= i-~j

2,

-

i=1

22

and

B=

/=1

2In,

(1.1)

j=l

where A(2(x))/n(n - 1) is the scalar curvature and B(2(x)) the mean curvature of M, then (A, B) has the property P1, so that ([A/n(n - 1)] 1/2, B) has the properties P1 and P2. Our main results give sufficient conditions for a pair of functions A, B : D ~ R to have property Pl(J/o). Partial differentiation of A, B is denoted by subscripts; for example, Aj = OA/O2j, Ajk = ~2A/~2j~2k. Theorem 1.2. Let A, B E C2(D) be real-valued functions. Then (A, B) has property P1 if A 1 >= A 2 ~ . . . >= A n >= 0, 0 < l 1 ~ 1 2 ~< . . . ~ B ; AIB .-AB

1 >0

j=l k=l

if 21 < 2 n ;

j=1

(1.2) (1.3)

j=l

(e.g. if A is a concave function); ~, Bjk(2)~j~ k ~ 0 j = l k=l

whenever

~ As(2)~j= ~ Bs(2)~j=0 j= l

j~ l

(1.5)

Some characterizations of the Euclidean sphere

39

(e.g. if B is a convex function) ;finally if, f o r any partition of the set { 1,..., n} = It w . . . u I into nonempty sets, where i ~ 1 , j ~ I a and ~ < fl imply i < j, Iz~ = #{I }, and A = E2~for i E 1 , the functions

A*(A, . . . . . A ) = A(A,II~,

.....

AJflv),

A J i l t , A2/112 . . . . .

B*(A t . . . . . A ) = e(A1/lz t . . . . . A1//tl, A2//~ 2. . . . , A J / t ),

(1.6)

satisfy

A(2~ . . . . ,2,) < A*(AI . . . . . A )

and

B(2~ . . . . . 2,) > B*(A~ . . . . . A).

(1.7)

(In (1.6), AJ#~ occurs l~ times as an argument of A, B on the right.) Remark 1. It is clear that if f(t), g(t) e C o are increasing functions of t e R ~, then (A, B) has p r o p e r t y / 5 if and only i f ( f (A), #(B)) has p r o p e r t y / 5 . Also, if f, g ~ C 2 have positive first derivatives, then A, B satisfy the conditions of T h e o r e m 1.2 if and only if f (A), #(B) do. Remark 2. If A, B are defined on the set 2~ > 0 for i = 1. . . . , n and are symmetric (i.e., A(T2) = A(2), B(z2) = B(2) for all permutations z2 = (2,1 ~. . . . . 21~,~)of 2 = ( 2 1 , . . . , 2)) and A is concave and B is convex, then (1.7) holds. Remark 3. N o t e that (1.2) implies that

Aj__>0

and

Bj___0

for

d(A, B)/d(2; 2k) = A j B k - a k B j >= 0

1 ~j
(1.8)

1 < j < k < n,

(1.9)

and AyB k - AkB j ~ AiB,, - AraB i if 1 < i < j < k < m < n. Consequently (1.2) implies that (A,Bj - AjB,)(2j - 2,) 2 + (AiB k - AkB ) (2 k - 2 ) 2 < (A,B k - AkB,)(2 i - 2k)2

(1.10)

for 1 < i _< j _< k < n. A sufficient condition for (1.10) is that (1.11)

(A,Bj - AjB,) + (Aj.B k - AkB ) < A,B k - AkB ,.

In particular, if A(2) = X2j [or if B(2) = X2j], then (1.11), hence (1.10), is trivial. Corollary 1.1. Theorem 1.2 remains valid if condition (1.2) is replaced by (1.8)--(1.10),.and conditions (1.4)--(1.5) by ~

[Ajk(2)Bi(2 ) - A i ( 2 ) B j k ( 2 ) ] ~ j ~ k < O

whenever

j=lk=J

~ Aj(2)~= j=l

~ Bj(2)~ = 0(1.12) j=l

and i = 1. . . . , n, while conditions (1.3) and (1.7) are retained. In order to state the next result, put Atp)(2) =

2 n

for

- - ~ =< p =< 0o,

j=l

where, as usual, A c_ oo7(2) = 21, Ato)(2) = (1-I2j)l/", and Atooj(2) = 2,. N o t e that A
40

P. HARTMAN

Theorem 1.2 contains the case (1.1) treated by Yau [10]. In fact, we have the following generalizations: Corollary 1.3. Let (A, B) and (A, Z) be pairs satisfying the conditions of Theorem 1.2. (i) If, in addition, A s - Z j > _ - 0 and Z B 1 - ZIB >0, (1.13) ( A j - Z s ) ~ j = ~ BS~s = O j=l

~

~ AS¢s=O,

j=l

then (A - Z, B) satisfies tile conditions of Theorem 1.2. (ii) If, in place of(1.13)-(1.14), > O, Z B ~ - Z~B > A > O,Z > O, ZA s - ZjA = = 0, ( Z A s - Z s A ) ~ i = ~ BS~S= 0 j=l

(1.14)

j=l

=*"

~ Aj~i= 0,

j=l

(1.15)

(1.16)

j=l

then (A/Z, B) satisfies the conditions of Ttleorem 1.2. Under analogous conditions, (A, B - Z) or (A, B/Z) satisfy tile hypotheses of Theorem 1.2/f(A, B), (Z, B) do. As applications, it follows, for example, that the pairs •

A=~Xi2

s and B = ( n - 1 )

22~,

A= ~

(1.17)

j=l

i¢:j

412j/(n - 1) ~ 4k and B =

igj

k=l

r-[~ 4if~nil~2 ,'-t

(1.18)

j=l n

4,4/(n-

A = EX

1 ) ~ 4k and B = ~ X2/2 4k

i:/:j

k=l

j=l

(1.19)

k=l

have properties P~ and P2- This is a consequence of Corollary 1.3 and the case (A~ v A~2) of Corollary 1.2, since EEX~4i = nZA~l} - nA~2) by (1.1). It follows from (1.17) that the pair (cf. (1.1)) I-

/ X X 4 , 2 / n ( n - 1)] ',2 and B = ~ 4/n, j=l L i~j and from (1.19) that the pairs A

A=ELX,2/(ni~j

1) ~ 2, and B = ~ 2/n, k=l

A = ~ 4/n j=l

and B =

(1.20)

(1.21)

j=l

t

4~/~ 2k

j=l

(1.22)

k=l

have properties P1 and P2. For example, if (A, B) is the pair in (1.20) and A(4(x)) has a local minimum and B(4(x)) a local maximum at x °, then the same is true of the pair (A, B) in (1.17). The pair (1.22) is suggested by results of Simon [8]. We can also formulate analogues of Theorem 1.2, where P1 is replaced by P1(~1¢o) for suitable sets -//0- To get an example, let

AX(4, n) = ~...E2,12, ...4,,:,BX(2, n) = n!

4/n

/(n - K)t,

(1.23)

Some characterizations o f the Euclidean sphere

41

where 1 ~ ij ~ n, ij ~ i k ifj v~ k, and K < n. (Note that any given term occurs K ! times and there are n !/(n - K)! terms in AX(2, n).) Put A°(2, n) = 1 and AX(2, n) = 0 if K > n. Corollary 1.4. Let 3 < K < n. Then (At(2, n), BK(2, n)) has properties Pl(~IK) and P2, where

JCr is the set of pairs (M, hu), arbitrary M ~ ~ + with arbitrary h~j subject to the "pinching" property 0 < (K - 2) 2 /~r-2, =< (n -- K + 3)~, K - 2

(1.24)

or, more 9enerally, 0 < (K - 1)(K - 2)2Ar-3(2, n - 1) < AX-2(2, n).

(1.25)

If K = 3, the conditions (1.24) and (1.25) reduce to 2 < (n - 1) 21 and 2 2 < 21 + ... + 2 . Corollary 1.4 remains correct if K = n but a sharper result is contained in Corollary 1.2 with p = 0, q = 1. I f K = 2, then (1.23) in Corollary 1.4 reduces to the case (1.1) which is contained in Corollary 1.3. Our proof of Theorem 1.2 depends on Lemma 2.1 below which is a generalization of an identity of Weyl in the case n = 2 involving the second covariant derivatives of the principal curvatures at a non-umbilic; cf. Chern [2]. The proof involves a "generalized (not necessarily self-adjoint) Laplacian" adapted to the problem at hand; see (3.6) below. (See the Remark at the end of Section 3 for an analogue of Weyl's inequality; cf. Chern [2]). We wish to acknowledge with thanks useful conversations with Professor Yau, also the availability of preprints of the papers Yau [10], Cheng & Yau [1], and Simon [8]. 2. G E N E R A L I Z A T I O N

OF AN IDENTITY

OF WEYL

Let M be a sufficiently small open neighborhood of a point x ° of a Rieman manifold of class C 3, and dimension n. There are no assumptions about the curvatures of M or the eigenvalues of (hu) in this section. Let {e91. . . . , o9} be a C 2 orthonormal frame field on M satisfying the usual structure equations

dot = - ~ OimO~m'Oim + Omi= 0,

(2.1)

m

d°)ij ~ - E

f~ij = ~1 ~

R

O')imO')mj +

f~ifl

Ukme~kOgm,Rijkm + Rumk = 0;

(2.2)

(2.3)

kra

cf. [3]. Here and below, products of exterior forms are understood to be exterior products. Sums are over the range 1 to n unless otherwise indicated. If a function f ~ CZ(M), its gradient or covariant derivative is d f = Z f~Om

(2.4)

tn

and its second covariant derivative is Z f ,. m

Similarly, if ~

ij

= dy, - Z ra

hozoi ® o~j e C2(M) is a symmetric tensor, its first covariant derivative is

(2.5)

42

P. HARTMAN

(2.6)

2 hijmogra = d h u - ~ ]lm~mi - ~ himogmj

and its second is ff~ hijkmogm = dhijk -- E hijmogmk -- 2 himkfDraj -- ~ hmjkogmi" m m m m

(2.7)

There is a partition of {1,..., n} = I x u ... u I into non-empty sets such that the ordered eigenvalues 21(x ) < ... < 2(x) of (hl) satisfy 2i(x ° ) = 2 j ( x °)

if i , j ~ I , 2 i ( x ° ) < ) , l ( x °)

if i ~ I , j ~ I a

and

a
(2.8)

A superscript 0 on a function indicates its value at x °. Put 2°~ = 2 0 =21(x °)

for

ieI.

(2.9)

We shall use the following abbreviation for sums means i

~. iel=

Put (~)

A(x)=~2i(x)

for

~ = 1. . . . . v.

(2.10)

i

It can be supposed that the C 2 orthogonal frame field (o9x. . . . . o9) is chosen so that i f H = (h0, then H ° . diag(2 . . . °, . 20,), H ( x ) = diag(Hl(X) . . . . . H (x)) for x ~ M , where Ha(x ) = ( h i ) f o r i, j E 1, so that hij(x ) - 0 if i • 1, j ~ I~ and a ~ fl, Am(x) = trace H (x) = ~ h.(x). i

From now on, the orthogonal frame field (o91. . . . . o9) is fixed. I f f ~ C2(M), we define the a-th partial Laplacian o f f by the sum A j = Z f . over i e I , where f . is given by (2.5). Similarly, the a-th partial Laplacian of the tensor Z X h i ~ ~ ® o93is defined by the sum A hlj -----~,hi3kk over k e I , w h e r e hijkm is given by (2.7). Lemma 2.1. Let 1 < a, fl < v. Let the symmetric tensor hi j satisfy the Codaz zi condition h ikj = h ~jk. Then ° -

('LA) ° =

-

2

R,j,j +

+

(2.11)

l

where, in terms of the coefficients of ogmi = ~" amiko9k' amlk + aimk = O,

(2.12)

k

P~lJ = P ~ and T ~ = Tij~ are given by (~) (#) (~)

(~) (#) (#)

P,¢ = 2 ~, 2 ak~i + ~ X 2 f jk

i jk

a2

kifl

(2.13)

Some characterizations of the Euclidean sphere

43

TO =0/fv<2and (~) (#) (r)

To =

Z

ZEE

7~t, fl i j

a~k( 2° -- 2~)2/(2~- 2°)( 2° -- 2°)

(2.14)

k

/fv > 2. N o t e that some terms of the sum T ,# O can be negative. This fact is the reason for the imposition of condition (1.10) in Corollary 1.1. The formulas (2.13) and (2.14) at x = x ° can also be expressed in terms of h.°..; cf. (2.16) and (2.17). ZJg Proof It is . clear from the definition ofdh.l j ,where f = h~.J in (2.5), and from (2.6) that h i. /.k = h.ik . . ) " The Codazzl condltmns are equivalent to h~jk = h~k; Hence hij k = h~kj = h u ; and, by (2.7), hljk~ = h~ik~ = hik~. An exterior differentiation of (2.6) leads, by (2.7), to hikpq -- hikqp = Z him Rmkpq "q- 2 hkmRmipq, m m

cf. [3]. Thus hikjj = hijkj = (hijkj -- hijjk ) -~ hijjk or

hlkjj = ~ him R,,jk j + ~ h ~ R,,ik j + hj~ik. rtl

ra

This gives h°"ltjj= 2°Ri~i~ + 2°R°"'j j,~ + h°~ji~ at x = x °, or huj~o _ h ojj. = (20 - 2%i,R°iiij" Summing over i e I and j e Ia gives (~)

(#)

(~) (#)

Z (A#hii) 0 - E (A~hjj) 0 = (20 - 2~) 2 E R°ij" i

j

(2.15)

i j

Since hij = 0 if i e I a n d j ¢ I , (2.6) gives Z hijmO)m ~

- E him(°mj - 2 hmj(Omi,

so that

hij k = - ~himamj k - ~hmflmi k for m

Hence, at x = x °, a.,jk =

-

aim k

iel,j¢I~.

m

implies that

h ° ~ = - 2°a ° = ( 2 ~ - 2 ° ~ aa I ijk ° i °ijk - 2°a j jik By the Codazzi equations, hog hikj,

if

iel~,,jel#,cx~ [3.

(2.16)

=

h ijk ° = h °ikj = ( 2 ~ -

o aiko j 2a)

if

ieI~,keI#,~

~ ft.

Consequently, the last two equations give o ~ aikj o aijk Let j = i in (2.6) and sum over i e I , (~)

if

i ~ I a and (~)

j, k e l # , ~ v~ ft. (~)

~ humOgm = ~, dh u - 2 ~ ~ hi ogi , m i

i

m i

(2.17)

44

P. HARTMAN

or since him =--.0 unless m ~ I , the last sum is 0 since /(~)

)

hml = him,coi

=

(¢0

-

o91~n.

Thus, it follows that

(~)

d A ~ ' = d [ ~ hii=~'~'h"''ogm'. ,

i'e" A'm = ~ h u " i

(2.18)

L e t j = i in (2.7) and sum over i s I to get 1:0

(~)

m

i

(~)

EZhiimo%- 2 E E h.,~co.,.

Edh.j-

Z E hiij~o~m =

i

m

m

i

T h e definition (2.5) of the second covariant derivative of a function, say f = A , and (2.18) show that (~)

(:0

Z Z hiijm°9. = E A~jm°gra - 2 Z ~, h.@%r m i m m i Thus, we have (~)

(~)

huj k = A j k - 2 ~, ~, h,.oa,.ik, i

m

i

where the contribution of m ~ I to the last sum is 0. Put k = j and sum over j ~ I~, ~ # fl, (~)

(~) (~) (P)

2 Aahu = A a A - 2 E E ~ ~ i

~,@a m

i

h,,,,flmij"

j

Consequently /(~)

(P)

ApA - A Ap = ~

",~

Aphu - ~,j A h j j ) + S p,

(2.19)

where

i j

~

kelp

,ia4

The contribution from k E I comes from the last sum and is (~) (P) (a)

(~) (P) (~)

S O = - 2 ~ E E (20 - 20''Ok) (akji)x2 i

j

2(20 _ 2~) ~, E E (a0k~,)2

=

k

i

j

k

at x = x °, by (2.16). Similarly, the contribution from k ~ Ip at x = x ° is (~) (#) (P)

s o = 2(~° - ~ ) E E E ( aO~ , )2 i

T o obtain the contribution from

j

k

k¢I~ u Ip at x = x o, note that, by (2.16),

h°,,.a°i -, - h°-,a°~ ;., = (h~,)° ~/(,~,o _ ,~o~)-,O'°~ji,~/~'~°,,;

_ ;~o) = (hJ

(,~o _ ,~o)/(,~o _ ;o)(,~o

_ ,~o).

(2.20) Thus the contribution of k E I for 7 # a, fl to S p at x -- x ° is S O = 2(2~ - 2o) T °~p,where TOp is 0 if v = 2 or is given by (2.14) if v > 2; cf. (2.16). This completes the proof, by virtue of (2.19) and (2.15).

Some characterizations of the Euclidean sphere

3.

PROOF

OF

THEOREM

1.2

AND

45

COROLLARY

1.1

We assume that A, B satisfy the conditions of Corollary 1.1. Let x ° ~ M e ~ + . We suppose that A(2(x)) has a local minimum and B(A(x)) a local maximum at x = x °. We continue the notation of the last section. Introduce the functions (1.6) of A = (A 1. . . . . Av) where 0 < A1/#1 < ... < Av//~~. The analogues of conditions (1.3)--(1.12) and (1.8)-(1.9) hold if A, B, Ai, Bj, Ajk , Bjk , 2, n are replaced by A*, B*, OA*/t3Ai, aB*/c~A; 02A*/t3AjOAk, 02B*/t3AjOAk, Ai/Pi , v. Also if 1 < ct _< fl < y < v and (1.10) is divided by #~#,#r and summed over i t I , j ~ I~, and k e I , we obtain the following analogue of (1.10):

(A*B~ - A~B*)(AJ#~ - Aa//~a)2 + ,IA*B*t~ ~ - A*B'~)(Aa/#a - AJ/~) 2 < (Aat B - A ' B * ) ( i J # ~

- iJ#at) 2,

(3.1)

when 21 = AJ#at, ~.j = A~/#a, 2 k = AJ#~. (Of course, the derivation of (3.1) and the analogues of (1.3)-(1.5) and (1.8)-(1.9) depend on (at)

(~) (#)

A'at = Z Ai/#at, A 5 = Z ~, Ai/patlza i

i

j

for 1 < a, fl < v.) We do not require an analogue of (1.7). Since (1.7) implies that A*(A(x°)) = A(A(x°)) _6_
a(x) = a(A(x)) and

b(x) = B(A(x))

and let a k, b k and ajk, bjk denote the components of the first and second covariant derivatives of these functions. Then, at x = x °, v

a k° =

v

X A°A° ~t atk = 0 , bk°

~---

at=l

X B°A° at ak = 0 ,

(3.2)

at=l

BaAa. fl=l

.~ ,i ~i = q = l 1:=1

Multiply by A ° _>_0 and sum over i ~ I and a to obtain (#) at

at=l

~

/~=1

" " fl ~ o t "

fl=l

" t r k " "~k"

(3.4)

k q=l r=l

Also, we have o

akk

~

~

~ ' . A °~ rA °arkA °",,k + ~ A°A ° >0. at ~kk

a=l 7=1

~=1

Multiply by Bao > = 0 and sum k E Ia and fl to obtain a generalized Laplacian of a(x), B~(Apa)° = E ~. Bt~Q ko + /~=1

~=1

k

A oat at=l

#=1

B~(apAat)° => 0,

(3.5)

46

P. HARTMAN

where

Qk is the double sum in (3.5). By the lemma of the last section, we can write

B~(A~a)° = E E BaQRO + fl=l

. B~(A A~)O

A o~

,a=l k

at=l

fl=l

+

A°B°(2°, a' • - 2 a ) [ ~

R°ij + 2PL + 2 T L

a=lfl=l

]

> 0. =

(3.6)

By (3.4), the sum of the first two terms on the right does not exceed o

: 1 k

o

~ I ,= 1 [A°'Ba

~ (rrd ak ,kJ'

which is non-positive by (3.2) and the analogue of (1.12). Consequently, the last sum in (3.6) must be non-negative. Since the factor in brackets is symmetric in (~, fl), the result can be written

R°ij +

t A ° B ° - A ~o oB ) ( 2 o - 2~)

E ~', l
2P°# + 2 T

> 0.

(3.7)

t..

But the analogue of (1.9) and 2 °~ - 2~ =< 0 give r(,) (a) o

(A~Ba°° _ AOB%a , , "(20~, - 2 ~ ) / X X RO*j + 2 "

X X l_-<~
Li

oa]

< 0.

(3.8)

3

j

We shall verify that E~',

Q-

2°-2~)T O <0.

tA°B°-A°B%(

(3.9)

15~
If this is granted for a moment, then every term in (3.7) is 0, and so the analogue of (1.3) implies that 2 0 = 2°., i.e. that x ° is an imbilic, and the proof will be complete. If v < 2, then T a = 0 and (3.9) holds. If v > 2, then, by (2.16), Q can be written in the form (~) (#) (~')

Q=

2E

Z EEX(h°0:

l_~at
j

( , Lo B~ o -

A~)(2

° -

2 °, ,~, 2, ,/ ( 2 ° -

2~)(2~

-

2 °) , . . ,(2 0 -

2°).

k

The sum Q can be rearranged as (~) (#) (r)

Q =

E2Z

Z 2 2(hL)2 {r~, + r~,~+ r~},

1-
j

k

where r~,

=

,I A O ~B ~o

_

A O~B %~ , ,(20 = - 2 ~ ) / ( 202

o - 2~)(2~- 2°1(20 ~--,-

2o).

Thus, (3.9) will be verified if it is shown that {... } < 0 for arbitrary a < fl < ?. If the denominators inF~,FmF~arewrittenas +(2~-°)(2°2~) o (2~o _ 2 o ~), then it is seen that {...} < = 0 is equivalent to ,(AOB ~ po _

A pon ) (o, B

+ (AOAO -o _ 20)2 • . , ~ ~

-

AOB ~ ~o ) ( 2 ~o

--

2~)~

-

( A oB

o -

A o B )o( 2

o -

2 0~.) 2 < =

O.

But this is (3.1). Hence Corollary 1.1 follows. Remark. If, in the above proof, it is not supposed that a(x) = A(2(x)) has a minimum at x = x ° but it is supposed that the proviso EAj(2)~j = 0 can be omitted in (1.4)-0.5) or in (1.12), then the

Some characterizations o f the Euclidean sphere

47

arguments lead to the following inequality at x = x °, E~

B~(Aaa)° <= j=l

(Ao B ~ o -- A aoB )o( 2 ° - 2~) ~, ~ R ° O.

l<~t<#
(3.10)

i j

(It should be pointed out that A, B in this last inequality are not the functions occurring in the statement of Theorem 1.2 but are the functions A*, B*.) The inequality (3.10) can be considered a generalization of Weyl's main inequality for estimating the mean curvature in terms of intrinsic quantities; cf. Chern [2]. For an analogous use of such an inequality, see Cheng & Yau [1]. 4. P R O O F

OF

COROLLARY

1.2

We shall consider the cases p ~ - oe, 0 and q :# ~ . These excluded cases are simple (and also follow by limit processes). In view of Remark 1 following Theorem 1.2, it suffices to show that A =

2

,B=

j

2q, where

-oo
1
o%0#p
(4.1)

j=l

satisfy the conditions of Theorem 1.2. From Aj = (A/2j) l-p, B k = q2~- 1,

~,, AjkCjCk =

(1

-

p)A 1-

2p

¢J2; -1

j=lk=l

j=lk=l

j

2 --

1

2~" j=l

22p-2~]

< 0,

k

j=l

it is clear that conditions (1.2)--(1.5) hold. Finally, the inequality (1.7) is a consequence of Hflder's inequality. 5. P R O O F

OF

COROLLARY

1.4

When convenient, we write AK(21. . . . . 2) for AK(2, n). Thus if K __>j, the significance of AK(21. . . . , 2 ) is clear. The functions (1.14) have property P 2 ' i.e. At(21 . . . . . 2,,) <

2/n nl/(n - K)I (5.1) J with equality only if21 . . . . . 2 , by an inequality of MacLaurin; cf. [4], p. 52. Thus it is sufficient to show that AK(2, n) and B(2) = X2i have property/~x(~t'x). It is clear from the proof of Theorem 1.2 that it suffices to verify the conditions (1.2)-(1.7) when 2 = (21. . . . . 4,) is subject to (1.25). We shall not give all details but concern ourselves only with the apparent difficulties, namely, conditions (1.4) and (1.7). First, we deal with (1.7) which will not involve (1.16). The second inequality in (1.7) is trivial, so that we only have to examine the first. If v = 1, so that 11 = {1. . . . . n}, then the desired inequality is (5.1). If v > 1, write K

-j "! - j ) l , AK(2, n) = ~, K! . A j A K _~,I/j.(K

j=0

where Ai_,l = A'(2 1+1. . . . . 4.) and Aj = AJ(21. . . . . 2~1). By (5.1), A j < (A1//~ly/~ 1 !/(/~1 - j ) ! . The desired inequality now follows by an induetion on v (for all K, n in AK(2, n)).

48

P. HARTMAN

Next, we consider condition (1.4). The quadratic form on the left of (1.4) becomes

Q = Z..- Z4,...

2,~_,

¢,._, ¢,~,

where 1 =< ij =< n, ij # i k if j ¢ k. We shall consider ~ subject to the condition ~ j = 0 (i.e. YBj~j = 0). We write Q in the form

= Z

. .

22,2,, . . . .

2,~ _ 2

~j-

~,t

)2

-

k=l

j=l

¢~ + Z ¢ik2 k=l

or, since Y~j = 0 and (Z~i~) 2 __<(K - 2) Z~2t,

Q=<~,

2i,,-2 (K- 1)

. . . "22,1 ..

~2it_ j=l

it~j

j=l

In the factor Xj = 2 j Z . . . E2i ... 21~_~, the sum is over all indices i~. . . . . iK_3,

w h e r e iv ~ iq

if

p # q, 1 <= i k <= n, a n d i k # j. It is e a s y t o see t h a t

x j - x,, = (2j - 2,.) Z . . . Z 2 , . . . 2,,,_3, ik C~j,m

so t h a t X I < X

2<...
andX

=2.AK-3(2, n - 1 ) . T h u s

Q < ~ ~ [ ( K - 1)(K - 2 ) 2 , a K - 3 ( 2 , n -- 1) - AK-2(2, n)], j=l

and so, Q <- 0 when (1.25) holds. Since we have A K - 2 ( 2 , n) = ( K - 2 ) 2 , A X - S ( 2 , n -- 1 ) + A X - 2 ( 2 , n -- 1),

condition (1.25) is equivalent to (K

-

2) 2 2" A K-

3(4, n -- 1) _--
-

-

1).

Hence (1.24) and 41 =< ... < 2 imply (1.25). REFERENCES 1. CHENG S. Y. & YAU S. T., Hypersurfaees with constant scalar curvature, to appear. 2. CHERN S. S., Some new characterizations of the Euclidean sphere, Duke Math. J. 12, 279-290 (1945). 3. CI~RN S. S., DO CARMO M. & KOBAYASI~S., Minimal submanifolds o f a sphere with second fundamental form of constant length, Functional Analysis and Related Fields (ed. F. Browder), pp. 59-75. Springer, Berlin (1970). 4. HARDY G. H., LITII.EWOOD J. E. & P6LYA G. Inequalities, Cambridge (1934). 5. HARTMANP., On elliptic partial differential equations and uniqueness theorems for closed surfaces, J. Math. Mech. 7, 377-392 (1958). 6. HILa~.RT D., Grundlaoen der Geometrie, p. 238. Teubner, 7th ed. (1930). 7. SACKSTEDERR., On hypcrsurfaces with no negative sectional curvatures, Am. J. Math. 82, 609-630 (1960). 8. SIMONU., A further method in global differential geometry, preprint. 9. WEYL H., Uebcr die Bestimmung einer gesehlossen konvexen Fl~che durch iher Linenelement, Viertelijahrsschrift der Naturforschenden Gessellschaft in Zi~rich, 61, 40-72 (1916). 10. YAU S. T., Hypersuffaces with constant scalar curvature, preprint.