Curvature pinching and eigenvalue rigidity for minimal submanifolds

Math. Z. 191, 537-548 (1986) Mathematische Zeitschrift

�9 Springer-Verlag 1986

Curvature Pinching and Eigenvalue Rigidity for Minimal Submanifolds

Sebastian Montiel, Antonio Ros and Francisco Urbano

Departamento de Geometrla y Topologia, Facultad de Ciencias, Universidad de Granada, E-18071 Granada, Spain

1. Introduction

Simon's formula ([1, 9]) is a basic and useful tool in the study of some problems of global rigidity for minimal submanifolds immersed in kind Rie- mannian manifolds. This formula is related to a special sort of submanifolds, those that have parallel second fundamental form, and allows us to character- ize some submanifolds of this family as extreme cases in different geometric situations (generally about the variation rank of the curvature). If in the cases of surfaces and hypersurfaces Simons' formula gives the best result, in the general case the fine use of this formula states some problems, although they are of a tensorial-algebraic nature, they are surprisingly intrincate.

In this paper, developing some simple ideas used by the authors in [-7] and [11], a modified version of Simons' formula is obtained, that even though this essentially coincides with the classic version, it allows a more controlled treatment and new view points.

We give sharp applications in two different contexts. First we obtain a complete solution to a pinching problem for the Ricci curvature of n-dimensional compact totally real minimal submanifolds in the n-dimensional compact projective space (Theorem 1). The result gives a characterization of nice submanifolds of this family. The corresponding pinching problem for the sectional curvature has been solved by Urbano in [-10] and [11].

The second application is the study of eigenvalue rigidity for compact Riemannian manifolds which can be minimally immersed in a sphere by the first eigenfunctions. This family includes the compact irreducible homogeneous spaces and the strongly harmonic manifolds.

In the paper we continue the study started by Ros in [-6]. In that paper Ros obtained two inequalities for the first eigenvalues of these manifolds. Here we give a new inequality involving the first eigenvalue and its multiplicity, and using the obtained Simons' formula, we proved that the equality in all the cases characterizes the compact rank one symmetric spaces (Theorem 2).

538 S, Montie[ et aL

2. Preliminaries

Let M be an n-dimensional compact Riemannian manifold. We denote by UM the unit tangent bundle over M and by UMp its fiber over p~M. If dp, dv and dvp denote the canonical measures on M, UM and UMv respectively, then for any cont inuous function f : UM~R, we have:

fdv= ~ { ~ fdv~}dp. UM M UMp

If T is a k-covariant tensor on M and VT is its covariant derivative, then, we have (E7]): ,,

S I ~ (VT)(ei'ei'v ..... v)}dv=O (2,1) UM t i= 1

where el . . . . . e, is an o r thonormal basis of TpM, peM. Now, we suppose that M is isometrically immersed in an (n+p)-

dimensional Riemannian manifold M. We denote by ( , ) the metric of )~r as well as that induced on M. If a is the second fundamental form of the immersion and A t the Weingar ten endomorph ism associated to a normal vector 3, we define

L: T,M~TpM and T: T•MxTvaM--+R

by the expressions

Lv= ~ A~I .... )e~ and T(r i = i

where Tv~M is the normal space to M at p and e 1 . . . . . e, is an o r thonormal basis of TpM. Then, L is a self-adjoint linear map and T a symmetr ic bilinear map.

F r o m now on, we suppose that M is a curvature-invariant submanifold of M, i.e., _R(X, Y)ZeTpM for all X, Y, Ze TpM, be ing /~ the curvature opera tor of 2~. Then, if 170 and V2a denote the first and second covariant derivatives of a respectively, one has that Va is symmetr ic and V2a satisfies the following relation:

(v 2 ~,)(x, Y, z , w) = (v ~ ~)(r, x , z , w) + R• Y) ~(Z, W)

-a(R(X, Y)Z, W)-a(Z, R(X, Y)W) (2.2)

where R ~ and R are the curvature operators of the normal and tangent bundles over M respectively.

If S is the Ricci tensor of M and M is minimally immersed in M, from the Gauss equat ion we have:

S(v, w)= ~ R(v, ei, el, w) -(Lv, w). (2,3) i = 1

Now, we prove the following version of Simons ' formula:

Curvature Pinching and Eigenvalue Rigidity for Minimal Submanifolds 539

Proposition 1. Let M be an n-dimensional compact minimal curvature-invariant submanifold isometrically immersed in an (n + p)-dimensional Riemannian manifold M. Then

n + 4 0 = ~ - JM [(Vcr)(v, v, v)]2dv + (n + 4)v~ IA~(v'~)vl2dv

- 4 ~ (Lv, A.(~,~)v)dv-2 ~ T(o(v,v),a(v,v))dv UM UM

i : 1

where e l , . . . , e, is an orthonormal basis of TvM, peM.

Proof Using (2.1) for the 5-covariant tensor T on M defined by

we have: T(vl, v2, v3, v4, vs)= ((Va)(vl, v2, v3), o'(v 4, vs))

Now, from (2.2) and the Gauss and Ricci equations, it follows:

~, ((VZ~)(el, ei, v, v), ~7(v, v)) = S(v, A~(v,v)v ) i = 1

+ ~ R• v,a(v, ei),a(v,v)) - Z R(ei, v,v,A~(v,~,ei) i = 1 i = 1

= S(v, Aot~,v)v) - (Lv, A~t~,~)v) - T(cr(v, v), o(v, v))

+2 ~ (A~tv,v)ei,A~(v, eOV) i = 1

+ ~, {R(ei, v, o(v, e3, a(v, v)) -R(e i , v, v, A~(~,,)e3}. (2.5) i=1

Let e be the 1-form on UM v given by

%(e) = (A~(~,v)e, A~(~,~)v)

with v e U M v and eeT~UM v. If e I . . . . . en_ 1 is an orthonormal basis of T, UMv, then it is easy to see that the codifferential of c~ is

n - - 1

(cS~)(v) = ~ ei.%(ei)= -(n+4)lA~(~,~)v] a +2(Lv , A~(~,~)v) i = 1

+ T(ff(v, v), or(v, v))+2 ~ (A~(~,v)ei,A~(,,eoV), i = 1

540 s. Montiel et al.

where e 1 . . . . . e ,_l , e,=v is an orthonormal basis of TvM. Now, over UM v and using that ~ (6a)(v)dvp=O, we have:

UM v

=(n+4) ~ [A,(v,v)vl2dvp-2 ~ (Lv, A,(v,~v)dv p UMp UMp

- ~ T(a(v,v),a(v,v))dvv. (2.6) UMv

In a similar way, if fi is the 1-form on UMp defined by

fi~(e) = ( ( Va)(e, v, v), ( Va)(v, v, v)5

with veUMp and eeT~UMp, we get

{ ~=1 ,(Va)(e~, v,v), 2} dvp - n + 4 3 ~ I(Va)(v,v,v)12dvp. (2.7) U p i UMp

Now, using (2.5), (2.6) and (2.7) in (2.4) we have:

n + 4 0 = ~ ~ I(Va)(v,v,v)]2dv

UM

+ ~ S(v,A~,(v.~)v)dv+(n+4) ~ [Aa(~,~)vl2dv UM UM

- 3 ~ (Lv, Ao(~,~v)dv-2 ~ T(a(v,v),a(v,v))dv UM UM

+v~ M ~= {R(e~,v,a(v,e~),a(v,v))-K(e~,v,v,A~r dr. (2.8)

Finally, from (2.3) it follows

integrating

S(v, Ao(v,v~ v) = ~, R(v, ei, ei, A~(~, ~)v) - <Lv, A~(~, v)v>. i = 1

Again, integrating o v e r UMp the codifferential of the 1-form 7 given by

we have 7v(e) =/~(e, v, v, A~(v,v)v)

2 S R(ei, v,v,A~(~,eov) dvv UMp i

Now, the Proposition follows substituting (2.9) and (2.10) in (2.8).

Remark 1. Notice ambient space is

(2.9)

(2.10)

that the hypothesis used in Simons' formula ([1, 9]) "the locally symmetric", has been changed in Proposition 1 by


"the submanifold is curvature-invariant". In practice, this change is not out- standing because in almost all the cases in which Simons' formula has been used (submanifolds in real space forms, Kaehler, and totally real submanifolds in complex space forms, see [1, 2, 5, 9, 12]) both hypothesis are satisfied simultaneously.

We prove a Lemma which will be used later.

Lemma 1. Let M be an n-dimensional compact minimal submanifold isometrically immersed in a Riemannian manifold f/l. Then, for all p6M, we have:

2 a) ~ (Lv, A~(~,v)v)dv p- ~ ]Lv[2dvp

VMp n + 2 VMp 2

b) ~ I ~ ( v , v ) l ~ d v , - S (Lv, v)dvp UMv n + 2 UMv

c) ~ (Lv, v)dvv= 1 ~ [al2dvp. UMp UMp

Proof Let a 1 be the 1-form on UMp defined by

~(e) = (Lv, A~(v,~)e)

with veUMp and eeT~UMp. For each veUMp, let e 1 . . . . . en_l, e.=v be an orthonornal basis of TpM. Then

(5al)(v) = - (n + 2) (Lv, A~(~,,)v) + 2lLvl 2 § (a(v, v), 4)

where 4 = ~ a(e~,Lei). Integrating o v e r UMp we have: i=1

(n+2) ~ (Lv, A~(v,v)v)dv.=2 ~ ILvl2dvv+ ~ @(v,v),~)dv.. UMp UMp UMp

As M is a minimal submanifold, it is easy to see

~ (6(v,v),~)dvp=~ ~ ( ~ (a(el,ei),~)}dvp=O. UMp UMp ~i= 1

Thus, we obtain a). b) and c) are obtained using the same way for the 1-forms cd and ~3 on

UMp defined by %Z(e) = (a(v, v), a(v, e)) @(e) = (Lv, e).

3. A Pinching Problem for Totally Real Submanifolds

Let CP"(c) be the n-dimensional complex projective space with the Fubini- Study metric of constant holomorphic sectional curvature c. Suppose that M is an n-dimensional compact totally real minimal submanifold isometrically immersed in CP"(c). Then, M is a curvature-invariant submanifold, and it is known that Asvw=-Ja(v,w) for vectors v,w tangent to M, where J denotes

542 S. Montiel et al.

the complex structure of CPn(c). From this, it follows that the tensors L and T defined in w 2 are related by

r(Jv, Jw) = (Lv, w). (3.1)

In this section we prove the following:

Theorem 1. Let M be an n-dimensional compact totally real minimal submanifold isometrically immersed in CW(c). Let S be the Ricci tensor of M. Then

3(n - 2 ) S_> c

- 16

if and only if one of the following conditions is satisfied:

n - 1 a) S= T c and M is totally geodesic,

b) S=0 , n = 2 and M is a finite Riemannian covering of the unique fiat torus minimally embedded in Cp2(c) with parallel second fundamental form,

c) S 3 (n -2 ) 16 c, n > 2 and M is an embedded submanifold congruent to the

standard embedding of." SU(3)/S0(3), n = 5 ; SU(6)/Sp(3), n=14 ; SU(3), n = 8 ; or E6/Fr n = 26.

Remark 2. The uniqueness in b) is proved in [3]. Also, the standard embedd- ings of the examples in c) are described in [3].

Remark 3. When n = 2 the condition about the Ricci tensor is equivalent to: " M has non-negative Gauss curvature". In this case Theorem 1 has been proved by Yau in [12].

Remark 4. When n>2 , the minimality of M can be weakened to " M has parallel mean curvature vector". In fact, if H denotes the mean curvature vector, then JH is a parallel tangent vector field on M and, hence, the Ricci curvature of JH vanishes. Since the Ricci curvatures of M are positive, we have that H = 0 , i.e., M is a minimal submanifold.

Proof If e 1 . . . . . e, is an or thonormal basis of TpM, p~M, we have

C C 2 { R(ei, v, a(v, el), a(v, v)) + 2 R(e i, v, v, A~(v,e~) v) } = ~ ( Lv, v) - ~ La(v, v)[ .

i = 1

So, using (3.1) and Lemma 1, Proposition 1 gives

n + 4 (n+ 1)c 0 = ~ ~ ](Va)(v,v,v)[2dV+ 2n(n+2~ ) ~ ]a]2dv

U M U M

+ ( n + 4 ) ~ ]A~(v,~)vl2dv-4 S (Lv, A~(~,~v)dv U M U M

- 2 ~ (LAjvv, Aj~v)dv. (3.2) U M


Now, if peM, let f : UMp~TpM be the vectorial function given by

f(v)=A~(~,~)v.

As f is odd, we have ~ f(v)dvp=O, and so UMp

- ~ (Af, f)dvp>(n-1) ~ [f[2dvp UMp UMp

where A is the Laplac ian of UMp and n - 1 the first non-zero eigenvalue of A. By a direct computa t ion , we have

(Af)(v) = - 3(n + 1 ) f (v) + 4Lv.

Then, the above relat ion gives

2 S [A~w,~v[2dVv>n+2 ~ (Lv, A~(~,,)v)dvv. (3.3)

UMp UMp

Using (3.3) and L e m m a 1 in (3.2) we have

O> n + 4 " V ( n + l ) c = ~ - )M I( ~)(v'v'v)lZdv-~ 2 n ( n + 2 ) Jv lal2dv

4n (n+2) 2 VM5 ILv] 2 dv- 2 UMf (LAser, Ajar)dr. (3.4)

Suppose that S > 3 ( n - 2 ) n + 2 = 16 c. Then f rom (2.3) one has 0__< (Lv, v) <~-~- c for

all veUM. Then we obta in

n + 2 (LAs,v, Ajvv ) <=~(- c la(v, v)l 2. (3.5)

Also, if e l , . . . , e , is an o r t h o n o r m a l basis of TpM, peM, such that Le~=eiei, we have

tha t is

ILv] 2= ~ o~2(v, ei) 2<n+2 i= a = 16 c ~ ~ifv, ei) 2,

i = 1

]Lvlz <n + 2 c ( Lv, v ). = 16

(3.6)

Using (3.5), (3.6) and L e m m a 1 in (3.4), we get

0 > n + 4 = 3 ~ I(V~r)(v'v'v)[Zdv" UM

So, V~=0, that is, M is a submanifo ld of CP"(c) with parallel second funda- menta l form.

544 S. Mont ie l et al.

Now, we use the classification of the totally real submanifolds of CP~(c) with parallel second fundamental form obtained by Naitoh and Takeuchi in [3] and [4]. Suppose that M is not totally geodesic. Then, if n=2, b) follows from this classification. If n > 2, the universal Riemannian covering 2~/ of M is a symmetric space with positive Ricci curvature. So, ~/ has no Euclidean factor and thus is irreducible ([3]). From the classification we have that /V/ is one of the examples of c). As these examples are embedded in CP"(c) and satisfy S=3(n -2 )c /16 , we obtain c).

4. Eigenvalue Rigidity for Minimal Submanifolds of a Sphere

Let M be a compact Riemannian manifold and 0 = , t 0 < 2 1 < . . . the eigenvalue sequence of its Laplacian A. Let qS: M ~ E '~ be an isometric immersion in the Euclidean space. Then, we have the L2-decomposition qS= ~ q~, ueN, where

u > 0 d~: M-*E" is a differentiable mapping satisfying Aq~u=-2uq5 .. Note that q~0 (the center of gravity of the immersion ~b) is a constant map.

Let ul, u2~N, l <ul <u 2. We say that the immersion 4) is by the {ul,u2}-th eigenfunctions if q~,=0 for all ueN, u~=O, u 1 or u 2. If u 1 =u z we say that the immersion is by the ul-th eigenfunctions.

We say that ~b is a standard immersion by the u-th eigenfunctions if q~ = k( f 1 .. . . ,f,,~) where k is a real constant and f l . . . . ,f,,~ is an orthonormal basis (for the L2-metric) of the ;t,-eigenspace.

Now, let Sin(c) be the sphere of constant curvature c. Consider on the space SM(m)= {P~gI(m,R)/P=P ~} of the symmetric matrices of order m the metric ( , ) given by

(P,Q)=(c/2)traceP.Q for all P, Q eSM(m).

Then, the mapping f: Sm(c)~SM(m+I) given by f ( x )=x tx is an isometric immersion by the second eigenfunctions.

Let 0: M~Sn+P(c) be a full and minimal isometric immersion of an n- dimensional Riemannian manifold M. If ~ is considered as an immersion in E "+p+I, then ~ is by the u-th eigenfunctions for some u~N with 2~=nc. Now, we consider the isometric immersion q~=foO: M~SM(n+p+I ) . If M is a compact rank one symmetric space and ~ is the first standard immersion, then ~ b = f o ~ is an immersion by the {1,2}-nd eigenfunctions. In [6], the minimal immersions ~ for which the associated immersions q5 = f o ~ are by the {ul, u2}- th eigenfunctions were studied. We will use the following results:

Lemma 2 ([6]). Let 0: M~Sn+P(c) be a full and minimal isometric immersion, and (o=foO: M-*SM(n+p+I) the associated immersion. Then, we have the following relations

(4,~)=~/2c (4,I)=1/2 -(q~,A~)=~,

(AO, A4)=2n(n+l )c - (A24 , Aq))=4n(n+l)2c2+4clcrl 2

where a is the second fundamental form of M in S"+P(c).


Lemma 3 ([6]). With the same assumptions as in Lemma 2, c~ is a immersion by the {ul,u2}-th eigenfunctions if and only if M is Einstein and T = k ( , ), where T is the tensor defined in w on the normal bundle of the immersion ~ and k is a real number. Moreover, in this case ~b o is proportional to the identity matrix in SM(n + p + 1).

Lemma 4. Let ~: M~S"+P(c) be a full and minimal isometric immersion by the first eigenfunctions, i.e., )h =nc. I f

xlxjd p = o~(~ij (4.1) M

where ~ is a real number and x o . . . . . x,+p are the coordinate functions of ~, then

(n + 2)p 21 lal2dp> - - vol(M). (4.2)

~t = 2 ( n + p + 2 )

The equality holds if and only if the immersion ~b=foO is by the {1,u}-th eigenfunctions for some ueN.

Proof The assumption (4.1) means that the center of gravity of ~b is proportional to the identity matrix in SM(n+p+l ) . From Lemma 2, at every

I 1 point of M, we have ( q ~ , ) = ~ . So, we have the same relation for the center of gravity. Hence, 1

r I

where I is the identity matrix in SM(n+p+ 1). For each real number 2, let Fx: M-~SM(n +p + 1) be the mapping given by

( ) F x = - A d p - 2 (o ( n + p + l ) c I .

Then, ~ Fxdp=O, and using the minimum principle for 22, we have M

- ~ (AFz, Vz)dp>21 ~ IF~j2dp. (4.3) M M

As this inequality is polynomicat of second degree in 2, then it is equivalent to

>(~ {[A4)12 + 21(A(~,O)}dp) 2. M

(4.4)

Moreover, the equality in (4.4) is equivalent to the equality in (4.3) for some 2~R.

Now, using Lemma 2, (4.4) gives (4.2). If the equality in (4.2) holds then the equality in (4.3) holds for some 2~R and, so, AFa= - 2 1 F x. If ~b= ~ (bu, then

u > 0

the decomposition of F~ is given by F~= ~,, (2u-2)~b u and thus A F z = - 2 1 F ~ is u > l

546 S. MontieI et al.

equivalent to 05=050+~bl +05, for some u~N, with 2=2 , . This means that 05 is an immersion by the {1, u}-th eigenfunctions.

Lemma 5. Let M be an n-dimensional minimal submanifold isometrically immersed in Sn+P(c). Suppose that M is Einstein and T = k ( , ). Then

np(n + 2) lal2>2(n+p+2) c

and the equality holds if and only if M is isotropic and has parallel second fundamental form.

Proof. Let e 1 . . . . ,G be an orthonormal basis of TpM, pEM. Tt~en

i {R(ei, v, a(v, ei), a(v, v)) + 2R(ei, v, v, Acr(v, ei)V)} i - - 1

= 2e(Lv, v) - 2ela(v, v)[ 2.

As T = k ( , ) , taking the trace, we have k=la12/p. So, using Lemma 1, Proposi- tion 1 gives

n + 4 2c 0 = - 5 - + +

4 pn(n+2) vM ~ [al4 dv+(n+4)vMf IAa(,,o)v[ 2dr

- 4 ~ (Lv, A~,~)v)dv. (4.5) UM

Now, given paM, let co be the 1-form on UMp defined by

cov(e) = ( a(v, v), a(v, e) ) [a(v, v)l z

for all veUMp, eeT~UM~. Integrating over UM v the codifferential of co, we have

(n+6) f I~(v,v)14dvv UMp

= 4 f ]A~,~,)v[Zdvp+2 ~ (Lv, v)la(v,O[2dv;. (4.6) UMp UMp

By the Schwarz inequality fa(v, v)]4< IAr z and, so, (4.6) gives

2 IAo(v,olvl2dvv>=n+2 ~ (Lv, v)j~r(v,v)12dvp. (4.7)

UMp UMp

The equality in (4.7) holds if and only if M is isotropic at p. As M is Einstein, Lv =t iv for all ve UM, with # = [al2/n. So, using (4.7) and Lemma 1, (4.5) gives

21r 2 ( n + p + 2 ) } 0>= ~ ~ j c np(n+2) [~r12 dr,

UM


and the equality holds if and only if M is isotropic and Va =0. So, the Lemma is proved.

Remark 5. Notice that the bounds for ~ IA~(~,avl2dvp obtained in (3.3) and UMp

(4.7) are different. In fact, the equality in (3.3) has not imply the isotropy of M at p. Moreover, using the bound of (4.7) instead of that of (3.3), we cannot prove Theorem 1.

The main result in this section is the following

Theorem 2. Let M be an n-dimensional compact minimal submanifold isometrically immersed in Sn+V(c) by the first eigenfunctions. Then, we have

S rdp < (n/4) {2(n + 5)21 - ( n + 2)22} vol(m) (4.8) M

where r is the scalar curvature of M. I f the immersion is standard, then we have

n + 2 2m 1 21__>22 (4.9) n m 1 +1

and

, r d p < ( 2 1 / 2 ) { n - 4 + ~ } v o l ( M ) (4.10) M

where m 1 is the dimension of the 21-eigenspace. The equality in (4.8), (4.9) or (4.10) holds if and only if M is a compact rank one symmetric space.

Proof Notice that 21=nc and that in the cases (4.9) and (4.10) the immersion is standard, and, so, it is full. In the case (4.8) we can suppose that the immersion is also full.

The inequalities (4.8) and (4.9) were proved in I-6, Corollaries 3.2 and 3.3]. Now, since every standard immersion satisfies (4.1), we have from Lemma 4 that M satisfies (4.2). Using in (4.2) that m l = n + p + l and r = ( n - 1 ) 2 1 - l a ] 2, we obtain the inequality (4.10).

We suppose that equality in (4.8), (4.9) or (4.10) holds. If 0: M~S"+P(c) denotes the immersion, we consider the associated immersion ~b=fo0 : M ~ S M ( n + p + 1). From Lemma 4, the equality in (4.10) holds if and only if ~b is by the {1,u}-th eigenfunctions for some ueN. Also, from [6], the equality in (4.8) or (4.9) holds if and only if ~b is by the {1,2}-nd eigenfunctions. So, in the three cases the equality implies that ~b is by the {1,u}-th eigenfunctions. But in this case the Assumption (4.1) is satisfied. So, from Lemma 4, we have that the equality in (4.2) holds.

On the other hand, from Lemma 3 it follows that if 4) is by the {1,u}-th eigenfunctions, then M is Einstein and T = k < , > . So, in our case the assumptions in Lemma 5 are satisfied.

Finally, the equality in (4.2) implies the equality in Lemma 5 and, so, M is isotropic and has parallel second fundamental form. Using [8] we have that M is a compact rank one symmetric space, and ~ is the first standard embedding.

548 S. Montiel et al.

References

1. Chern, S.S., D o Carmo, M., Kobayashi, S.: Minimal submanifotds of a sphere with second fundamental form of constant length. Functional Analysis and Related Fields, Proc. Conf. honor of M. Stone, 59-75. Berlin Heidelberg New York: Springer 1970

2. Itoh, T.: Adendum to my paper On Veronese manifolds. J. Math. Soc. Japan 30, 73-74 (1978) 3. Naitoh, H.: Totally real parallel submanifolds of Pn(C). Tokyo : J. Math. 4, 279-306 (1981) 4. Naitoh, H., Takeuchi, M.: Totally real submanifolds and symmetric bounded domains. Osaka :

J. Math. 19, 717-731 (1982) 5. Ogiue, K.: Differential geometry of Kaehler submanifolds. Adv. Math. 13, 73-114 (1974) 6. Ros, A.: Eigenvalue inequalities for minimal submanifolds and P-manifolds. Math. Z. 187, 393-

404 (1984) 7. Ros, A.: A characterization of seven compact Kaehler submanifolds by holomorphic pinching.

Ann. Math. 121, 377-382 (1985) 8. Sakamoto, K.: Planar geodesic immersions. T6hoku: Math. J. 29, 25-56 (1977) 9. Simons, J.: Minimal varieties in Riemannian manifolds. Ann. Math. 88, 62-105 (1968)

10. Urbano, F.: Totally real minimal submanifolds in a complex projective space. Proc. Am. Math. Soc. 93, 332-334 (1985)

11. Urbano, F.: Non-negatively curved totally real submanifolds. Math. Ann. 273, 345-348 (19861 12. Yau, S.T.: Submanifolds with constant mean curvature I, II. Am. J. Math. 96, 346-366 (19741:

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Received May 21, 1985; received in final form September 30, 1985

Curvature pinching and eigenvalue rigidity for minimal submanifolds

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