Harmonic mappings and minimal submanifolds

lnventiones math. 62, 2 6 9 - 298 (1980) Il l l2 ell tiolle$ mathematicae �9 by Springer-Verlag t980

Harmonic Mappings and Minimal Submanifolds

S. Hildebrandt 1, J. Jost I, and K.-O. Widman 2

1 Mathematisches Institut, Universit~it Bonn, Wegelerstral3e 10, 5300 Bonn 2 Mathematisches Institut der Universitiit, Link/Sping

1. Introduction

By a classical theorem of Liouville, a bounded entire harmonic function on •" is necessarily a constant. In this paper, we shall prove an analogous result for harmonic mappings of Riemannian manifolds:

Theorem 1. Let U be a harmonic map ol ~ a simple or compact Riemannian manifold 37 of class C 1 into a complete Riemannian manifold Jl[ of class C 3, the sectional curvature of which is bounded from above by a constant ~c>=O. Denote by BM(Q) a geodesic ball in Jig with radius M < ~/(2 IlK) which does not meet the cut locus of its center Q. Assume also that the range U (Y{') of the map U is contained in BM(Q). Then U is a constant map.

Here, a Riemannian manifold ~r is said to be simple, if it is described by coordinates x on ~," and by a metric

da2 =7~t~(x) dx~ dx ~

for which there exist positive numbers 2 and # such that

for all x, ~ in IR". In other words, ~+r is topologically IR" furnished with a metric for which the associated Laplace-Beltrami operator is uniformly elliptic on ~".

Consider a minimal n-dimensional submanifold M in the (n +p)-dimensional Euclidean space E "+p which is represented non-parametrically on IR" by

z i= f i ( x i , x2 . . . . ,xn), n + l < _ i < n + p ,

where the p functions f i ( x ) are of class C 2 on IR" and satisfy the minimal surface system

Dlj{] /77~D=fi}=O, i = n + l . . . . . . n+p, on IR" (1.1)

270 S. Hildebrandt et al.

with

7~t~(x) = 6~ + fi~(x) fj~(x). (1.2)

It is well-known that the functions J'i are then even real analytic on IR". If, in addition, we suppose that ]Vf(x)l={f~=(x)fj,(x)} 1/2 is uniformly bounded on IR", the manifold ~={IR", do'Z}, daZ=?~dx~dx ~, is simple. Moreover, by virtue of a remarkable theorem due to Ruh and Vilms, the Gauss map ~: 3 f~G(n ,p) associated with the immersion F: Yf--*E "+p, F(x)=(x,j '(x)), is a harmonic map of the simple Riemannian manifold 5f" into the Grassmannian manifold G(n.p) which is identified with the symmetric Riemannian space SO(n+p)/SO(n) x SO(p) which carries a canonical metric.

Applying Theorem 1 to the Gauss map N, we obtain the following "Bernstein theorem":

Theorem 2. L e t z i = f i ( x ) , i = n + l . . . . . n+p, x = ( x l . . . . . x")elR', be a non-parametric C2-solution of the minimal surface system (l.1) and (1.2) on IR". Suppose that there exists a number flo with

flO < COS- m (~/(2 l/~m)),

such that

where we have set

m:,} 113, m=min{n ,p} , K= if m > 2

As(x)<-ri o jor all xelR", (1.4)

A z(x)= {det(b~t~ + f j , (x) fjr 1/2. (1.5)

Then, f l ( x ) , f2(x) . . . . , f " (x) are linear Junctions on IR" representing an affine n- plane in E "+p.

If p = 1, we have m=~c= 1. Therefore, (1.3) yields no restriction for ~0, and condition (1.4) becomes I Vf(x)l < const. Thus, Theorem 2 furnishes for p = 1 the following result: An entire solution f ( x ) of the minimal surface equation

vf div { ~ } = 0

with sup I Vf(x)l < oQ is necessarily a linear function. This is nothing but Moser's

weak Bernstein theorem cf. [33], pp. 590-591. In the classical Bernstein theorem which has been proved up to n__<7 by Bernstein (n=2), De Giorgi (n=3), Almgren (n =4), Simons (n=5, 6, 7), no uniform boundedness of lVf(x)l on IR" is required. This is equivalent to replacing < in (1.3) by __<, or to replacing the condition M < 7t/(21/7 ) in Theorem 1 by M__< rr/(21~. We are unable to handle this situation. In fact, many examples for analogous situations show that, in general, the Liouville property breaks down if the equality sign is admitted in the decisive criterion, cf. [18], [-19], [29]. Also, it is known that Bernstein's theorem does not hold for p = 1, n > 8, because of the example in [3].

Thus, our Theorem2 can be considered as a generalization of Moser's theorem to higher codimension p. It is not clear in which cases our quantitative

Harmonic Mappings and Minimal Submanifolds 271

assumption is sharp. But it is very likely that quantitative conditions of the kind (1.3), (1.4) are necessary for a Bernstein theorem to hold for C2-solutions on IR", since Lawson and Osserman have found Lipschitz solutions on IR" of (1.1) which are not linear, provided that n>4. For instance, the Lipschitz function f : IR4~IR 3 given by

where

f(x)= Ixl q for x#O,

~l(z~, z2)=(Iz~ lZ-lz21 ~, 2z, ~-2)

denotes the Hopf map v/: $3---*S 2, t u r n s out to be a solution of (1.1), (1.2), cf. [27], p. 15. On the other hand, using an idea of Barbosa as well as results from geometric measure theory, Fischer-Colbrie [11], Theorem 5.4, has sketched a proof that each entire solution f : IR3~IR p of the minimal surface system has to be linear if it satisfies sup lVfl < vc. For n=2, Chern and Osserman found an

~,3

even better result. In [7], they proved that a complete, two-dimensional regular minimal surface in E 2+p has to be a 2-plane if its image under the Gauss map does not intersect a dense set of hyperplanes.

During the last years, various authors have proved Liouville theorems for solutions of non-linear systems. To our knowledge, the first paper dealing with systems has been written by Frehse [12], Hildebrandt and Widman [19] have dealt with a class of non-linear systems which contain the harmonic maps as subclass. Ivert ]-22] has pointed out that several of the results of [19] can be derived from the a priori estimates established in [40] and [16]. Meier [29], [30] has extended the results of 1-19]. In addition, Sect. 3 of his paper [29] contains surprising examples. Cheng [5] has found the following Liouville theorem: Suppose that S is a complete Riemannian manifold of non-negative Ricci curvature and J / i s a simply connected manifold of nonpositive sectional curvature. Then, a harmonic map U: Y'~,//[ is constant if its range U(,~) lies in a compact subset of ,//g.

This result partially overlaps with our result specialized to the case ~c=0. Another Liouville theorem for harmonic maps U: o~'~,#, with sectional curvature of J [ <0, has been found by Karp [26] which follows from a remarkable "dispersion formula" for harmonic maps, cf. Theorem C, [26], p. 9. Further- more, Garber et al. [13] proved, that each harmonic map U: 1Rn~S N with finite Dirichlet integral on IR" has to be a constant, if n>2, on account of a surprisingly simple scaling argument. Eells has pointed out to us that one can replace S N by an arbitrary N-dimensional Riemannian manifold.1 However, the proof of [13] fails if one tries to replace IR" by a general n-dimensional manifold Y', n > 2. Finally, we mention a Liouville theorem due to Schoen and Yau [38] which has been used by these authors for interesting topological applications.

Our paper is organized as follows: In Sect. 2, we establish interior a priori estimates for harmonic maps U: Y'--,Jr (cf. Theorems 3 and 4). We have taken care to ensure that the estimates depend only on certain geometric quantities of

l Letter dated Nov. 7, 1978


Y', J/r and U. Therefore, these estimates might also be useful for other applications, for instance, deformation theorems. A proof of Theorem 1 follows directly from these estimates. In Sect. 3, we have collected information on the Grassmannian manifolds G(n, p) which will be needed in Sect. 4 where we apply the Ruh-Vilms theorem to derive Theorem 2 as well as further Bernstein type theorems for minimal submanifolds.

Finally we wish to thank Shiu-Yuen Cheng, Doris Fischer-Colbrie, Leon Karp and Robert Osserman for acquainting us with their unpublished work. Moreover, we are grateful to James Eells, Paul Ehrlich, Blaine Lawson, Ernst Ruh, and Richard Schoen for valuable information concerning the literature.

The first author has enjoyed the hospitality and the support of the University of California at Berkeley 2 and of the Institute for Advanced Study in Princeton 3 during the academic year 1979-80 which has been a great help for the preparation of this paper.

The referee has pointed out to us that Theorem 2 can also be deduced from the results of Fischer-Colbrie [11] for minimal cones and the regularity estimates of Allard, Annals of Mathe- matics 95, pp. 417-491 (1972). We are grateful for this reference as well as for a correction of our original s tatement of Theorem 7.

2. Harmonic Mappings

Let Y" and Jg be Riemannian manifolds of dimension n and N, and of class C x and C 3, respectively. Furthermore we assume that ~t' is complete.

For every C1-map U: ~ we can define the energy

E(U)= ~ e(U)dR" ~7

where dR n denotes the volume element on X, and

e(U)= ltr~c (U, , U , ) ~ ,

the energy density of U, is the trace with respect to the metric tensor of Y" of the pull-back of the metric tensor of Jr the mapping U.

A mapping U: Y ' ~ J I is said to be harmonic if it is of class C g and satisfies the Euler equations of the energy functional. In local coordinates, these can be written in the form

where

A~ul+Fi~(u)~,~#D~uiD#uk=O, 1 < I G N ,

d _ ~,-1/2 ~r- ~ D~,{Yl/27~Dt~}

(2.1)

(2.2)

is the Laplace-Beltrami operator on Y'. Here 7,~ are the components of the metric tensor of Y" with respect to some local coordinate system, (7 "~) is the inverse of (7~), and y = det (7,p). The coefficients of the metric tensor of Jr are denoted by gig, and (gig) is the inverse of (gik), while

~,=g~g~k, 2 NSF grant MCS77-23579 3 NSF grant MCS77-18723(02)

1 ffijk = 2 (g jk, i - - gik , j + g i j , k)


denote the Christoffel symbols for (gig). Moreover, we write u=u(x) for a representation of the map U: Y'-~,/// in local coordinates x = ( x 1, ..., x") and u =(u 1 . . . . . u N) on 5/" and J{, respectively. We use the Einstein summation convention: Repeated Greek indices e, fl, ... are to be summed from 1 to n, Latin indices i,j . . . . from 1 to N.

In the following, we need to recall some differential geometric estimates which follows from Rauch's comparison theorem and from Gauss's lemma (cf. [14] and [17]). We use the following functions in our estimates:

a ~ ( t ) = ~ t l / ~ c t g ( t l f ~ ) if v>O, O<t<~z/1/v

" --(t]/-vctgh(t V - v ) if v<0, 0 < t < o o

[sin(t ] / ~ if v<=0, 0_<_

b,,(t) = ] sin h(t 1/77) i if,, o, k

Lemma 1. Let u=(u l ,u z . . . . ,u N) be normal coordinates on #/~ associated with an arbitrarily chosen normal chart ,#'(Q) around some point Q of J//[ such that Q has the coordinates (0, 0 . . . . . 0). Assume that the sectional curvature K of Jg satisfies

o.~ <-- K <- ~c on A/'(Q)

with constants m and ~c, - ~ < to <_0<_ ~c < co. Then, for all u ~Jr satisfying lul =(uiui) 1/2 <l t / l f~ and for all ~ elR u we have

{a~([uD--1}gik(U)~i~k<Fikt(U)Ut~i~k<={ao)(lU[) --1}gik(U)~i~ k, (2.3)

{aik_a,o(lul)gik(U)} ~i {~.<=Fii~(U) Ut ~i {k<= {C~ik__a,~(lul)gik(U)} ~i ~k, (2.4)

g ~ <gik(u)~i{k<b2,(lu[){i{ k. b2(tu[)-i i (2.5)

Now we investigate when it is possible to introduce a normal chart of large radius around an arbitrary point Q E J//. In [17] the authors have used the somewhat unwieldy notion of "normal range" of Q. Fortunately, these con- siderations can now be simplified because of the following result due to Jost [25]:

Lemma 2. Suppose that the sectional curvature K of the Riemannian manifold vlr is bounded from above on By(Q) by some nonnegative number ~, where

BM(Q)-- {P~,///: dist(P, Q)<_M}

denotes the geodesic ball in .~lg with center Q and radius M . / f M < ~/(2]/~), and if the intersection of BM(Q) with the cut locus of its center Q is empty, then there exists no geodesic two-angle in BM(Q), and there is no pair of conjugate points on an 3, geodesic contained in BM(Q). Therefore, we may introduce normal coordinates BM(Q) around each point P of BM(Q) as center, and we can apply the estimates of Lemma 1 to all u in this coordinate patch.


By virtue of the Lemmata 1 and 2, we could carry out the proof of the Liouville theorem stated in Sect. 1 by a method which combines the arguments of [-19], pp. 51-55, and of [17], pp. 8-13. However, this procedure is quite tedious and yet not very enlightening. Therefore, following an idea due to lvert [22], we shall replace it by a scaling device based on a priori estimates. Unfortunately, the technique used in [17], Sect. 4, provides only for estimates of the H61der seminorm which depend on a certain way on the particular solution. Thus we shall start by deriving a priori estimates following the approach used in [16], pp. 157-168, which employs ideas taken from the important paper [-40] by Wiegner; cf. also [24].

Theorem 3. Suppose that the sectional curvature K of ~ is bounded on the ball BM(Q) by co <- K <_ ~c where co and K denote two constants satisfying - oo<co_<0_<~c<oo. Assume also that the cut locus of Q lies outside of BM(Q), and that

M <

Then, for any ~ ~', there are numbers a, 0<or< 1, and c>0 , depending only on n, N, M, (2, on the metric of Ys and on co and • such that the estimate

IUIc=la)<=c

holds Jbr each harmonic map U of Y{" into JC/ which maps ~" into the ball BM(Q). Moreover, if YE is homogeneously regular (in the sense of Morrey) with constants 2 and I~, then cr depends only on n, N, M, 2, I~, co, and K, while c depends, apart from these parameters, also on O.

We use here a slightly altered definition for homogeneously regular manifolds (cf. [32], p. 363):

A Cl-manifold s is said to be "homogeneously regular" if there exist numbers 2 and #, 0<2<=#, such that each point Po of 5" is the center of a coordinate patch {x: Ixl < 1} for which

21 ~12_-<?=e(x)~ ff_-<~ I ~12

holds for all ~ I R " and for I x l < l . The estimate, stated in Theorem 3, is interesting in its own right. It follows

immediately from a local version which will be formulated as

Theorem 4. Let BM(Q) be a geodesic ball in ~ which does not intersect the cut locus of Q. Suppose that the sectional curvature K of Jig satisfies co < K < ~c on BM(Q) , co<_O<_tr and that

M < rc/(21~).

Let u=(u 1, . . . ,u N) be normal coordinates of Jr on BM(Q), and assume that

JJ2d= {XElRn: lxl <2d }

is a coordinate patch for ~ such that the components 7~e(x) of the metric tensor of Yf on ~2d satisfy

for all ~ I R " and all XE~2d.


Assume that U: ~ ' ~ [ is a harmonic map with range U(~2a ) contained in BM(Q), and let u=u(x) be the representation of U with respect to the normal coordinates u 1, . . . ,u N. Then there exist numbers a, O < a < l , and c>O depending only on n, N, M, )~, Iz, co, tc but not on d>O such that the a-Hdlder seminorm of u on ~d is estimated by

[U]co~ o < c d-~. (2.7)

Proof of Theorem 4. We restrict ourselves to the case n > 3, since one can easily reduce n = 2 to n = 3 by considering the product manifold J4" • IR. In fact, there is even a simpler proof for n = 2 ; cf. [15] and [24].

Denote by G(x,y) the Green function belonging to the Laplace-Beltrami operator A~ on ~2d' It is well known that there are numbers K~ and K 2 depending only on n, 2,/~ (in fact: only on n and t~/2) but not on d such that

o < a ( x , y ) < = K 1 I x - y l 2-" (2.8) and

K 2 1 x - y l Z - " ~ G ( x , y ) if I x - y l < ( 3 / 4 ) ( 2 d - l y [ ) . (2.9)

Let us define the mollified Green function G~ ,) by

6P(x ,y)= ~ G(x , z )dz ~pl) ')

where, as usual, 1

=meas and Nl,(y) = {zelR": I z - Yl < P}- Moreover, we have

O<GP(x,y)< K3 I x - y [ 2-",

lira GP(x,y)=G(x,y) for x + y , (2.10) p ~ + 0

GP(. ,y)~t: l~nU'(~ed) if p < 2 d - l y l .

If xo6~d, T2R(Xo)=,~2R(Xo)--~R(Xo), R <d/4 , and [) ' -Xo[ <R/2, then

5 ]Vx Gp(x,y)I2dx<=K'*R2-"" (2.11) T2R(XO)

Again, the numbers K 3 and K4 depend only on n and/, /2. Finally, we shall also employ the relation

~, lf~7~'t~D~_DlsG~'(x,y)dx = y ~dx (2.12) P,~2a ~o(Y)

for all ~/4~(~2d)" Let v=v(x) be a representation of the harmonic map U with respect to

normal coordinates v on B~(Q) with arbitrary center P in BM(Q). Set

e(v) = 12gik(U ) Z 'a~ D~ v i D~ u k and

q (v) = 7 ~e [D~ v i Dr, v i - J Fik (v)D, v i Dt~ vk].

Clearly, e(v) is invariantly defined and agrees with the energy density e(U).

276 S. Hitdebrandt et al.

Note that Iv] < 2 M < rt/]/~.. Thus, in virtue of (2.4),

q(t,)> 2a~(Ivl)e(v). (2.13)

If t<M<r~/(21/~), we have in part icular

a~(t)>a~(M)>O. (2.14)

Let u = u(x) be the representat ion of U by normal coordinates centered at Q, and denote by w the solution of

Ae-w=O in "~2d (2.15)

which agrees on ~ 2 d with ]ul 2. F r o m (2.1), we derive the integral relation

0 = ~ (y~t~O, uiO~pi-~(u)y~O~uiO~uk~o'}l~,'dx (2.16) "~2 a

which holds for all q~e/~nE~176 NN). Inserting r we see that

q(u)G"(.,y)]/-Tdx=-�89 ~ lfYT~aD~ lul2DaGO(.,y)dx. (2.17) ~2a ~O2a

On the other hand, multiplying (2.15) by GP(.,y) and performing an integration by parts, we obtain

�89 ~ ]//77~aO, wOaG~ ~2 a

Adding this identity to (2.17), we infer from (2.12) that

2 ~ q(u)GP(.,y)]f7dx = ~ {w-lul2}dx. (2.18) ~2a ~o(Y)

Employing Fatou's lemma, we find

2 f q(u)G(.,y)l/-Tdx<__w(y)-Iu(Y)l 2 (2.19) "~2d

for all Y ~ 2 d " Moreover, the maximum principle yields

max w ____ max lul ~ ~2a c~2a

whence w(y)-lu(y)l z < w(y)- max w + max Jut z -I u(y)[ 2

~2d r

< o s c l u l 2. t~2d

Taking lul_-<M and (2.13), (2.14), and (2.19) into account, we arrive at

max{M2, osc lul 2 }

e(U) G(x, y)l/~ d x < ~ (2.20) ~ 2a~(M)


for all yC~2d. Returning to (2.18), we may now replace (2.19) by the representation formula

lu(y){ 2 = w ( y ) - - 2 5 G(x,y)q(u(x))]/y(x) dx (2.21) ,~2d

for yE~zd. Moreover ,

LA2 ~" lul2=utAxul+7~r ut

whence > )'~1~ {D~ u t D a u t - u I Fi~ (u) D, u i DI~ u k }

A~.lul2>0 on ~2d, (2.22)

on account of (2.4). Thus, we may carry over the Lemmata 4.2 and 4.3 of [16] to v= lu l 2, f2=~2d , b*=0 , and

L= - l fTAe-= -Dt~{a~(x)D~}, a ~: =]l-y3, ~.

Moreover , we can write (2.1) in the form

Lu =f(x, u, Vu) where

f = ( f l , f 2 . . . . . fu), fl(x,u,p)=a~r Set

,~=aK(m)b~(m).

Then, 6 >0, and (2.4) implies that

d . f ' ( x , u, p) < (1 - 6) a sp(x)p~p~ (2.23)

where u*=ut(x). An inspection of the proof of Lemma 4.4 in [16] shows that it remains valid if we replace the assumption 2* (M)< 2 of this lemma by condit ion (2.23). Thus we obtain

Lemma 3. Suppose that the assumptions of Theorem 4 are fulfilled. Then, for every e > 0 , there exist an integer m and a number R ~ > 0 with the following property:

For every ke(0,1), every Ro with O<Ro <min{Ro, d/8}, and every Xo6Nd, there is a number R with

(k/8)"-fro < 4R </~o such that

e(U) G(x, x o ) l ~ x ) d x < e. �9 ~2R(xo) - - ~ k R (XO)

Next we set h o = rain {n/(2 M 1/~) - 1, 1 }.

Note that 0 < h o < 1 since M < rt/(2]/-~). Then there exists a number e', 0 < e '< 1, such that

h '= re/(2 ] / ~ ) - ((1 - h0) 2 M 2 + e,}l/2 > 0.

Let h=min{2h']/~/rc, ho}, 0 < h < h o < l .


Choose an arbitrary number ~ > 0, and then an e" with 0 < e " < r2. Set

e 0 = 2 - 1 h(2 - h) rain {e', e"/8 },

and let S be the smallest positive integer such that

(1 - h) 2s < ~;"/(8 m2).

We shall determine a number p > 0 depending on the parameters as chosen before, but not on U, such that the oscillation of U on ~p(x0) satisfies

osc U < r for all XOE~d . ~p(Xo)

This yields a precise estimate for the modulus of continuity of U. In a further step, this estimate will lead us to an a priori bound for a H61der no rm of U.

In order to carry out the first step of our construction, we fix some X o e ~ e. Let (2 R be the open ball ~R(Xo), and T2R=Y22R--Q R. Consider now the analogue to (2.16) for a representation v=v(x) of U in normal coordinates v on BM(Q) with an arbitrary center in BM(Q):

f {2=PD=viDeq)i-Fi~(v)7"eD~viOevkq)'}]/Tdx=O (2.24) ~2 d

for all q0E/]lc~L~(~Zd,]RN ). Clearly, the Christoffel symbols are now to be computed with respect to the

coordinates of the fundamental tensor of J / in the v-coordinates. Inserting the test function q)=G~ where IXo-y[<R/2 , and q is the usual friend on ~2R' i.e., 0 < q < 1, q = 1 on ~'~R, q ~ 0 outside of ~2R, 117/,/I <c/R, c independent of R, 0 < p < R/2, then

~'7~PD~[Iv[2rl]D~GP("Y)]/-Tdx-~7~PD~rID~G~ (2.25)

+ 2 ~ q(v) G~ + Z ~7~ G~ y)D~v~ D~rlv~lf Tdx=O

where all integrals are to be extended over f22R. We write (2.25) as

I + I I + I I I + I V = O .

Then we shall investigate the integrals I - I V in two cases:

(~) v= Vho, R o, where Vho,R o denotes a representat ion of U in normal coordinates the center of which has the u-coordinates h ogRo (recall that the u-coordinates have Q as center), where

a~o= ~ u(~)dx. T Z R 0

(fl) V=Wh,R. Here, w(x) is the representat ion of U with respect to an arbitrarily chosen system of normal coordinates with center in BM(Q) , and W,R(X ) denotes the representation of U with respect to normal coordinates the center Q~ of which has the w-coordinates t #R, where

WR = T!RW(x) d x.


Since ~Jp(y) c ~Mg(Xo) = f2g, we get

I= ~ Ivl2 dx. "q~o (Y)

Now we shall est imate II, 111, and I V in case (fl). In case (~), one has to replace w by u, h by ho, wh. g by Vho.R o, R by R 0, etc. Since

I Wh, R (X)] 2 = d i s t 2 (Q 1, Q0 + [dist z (U (x), QO - d i s t 2 (Q 1, Q h ) ]

and I dist 2 (U (x), Qh) -- dist2 (Q 1, Q0[

< 4 M [dist(U(x), 0h)-- dist (Ol, On) l

< 4 m dist(U (x), Q O=4 m IWI,R(X)[

<4Mb~o(M) Iw(x)--#R], we obtain

II <(1-h) 21%12+c,R -1 j" Iw(x)-%l lVGqx, y)l~,'dx. T2n

Here and in the following, cl, c2, r . . . . denote numbers depending on the differential geometr ic pa ramete rs M, 2, p, ~o, tc . . . . of the problem, but which are independent of U and R.

F r o m Young 's inequali ty we infer that, for every ~> 0, the integral

V=c,R -~ ~ [ W - ~ g l I V G P ( ' , Y ) l l ~ d x r2N

can be es t imated by

V<cZg-IR , ~ [W-WR[ZTdx+gR "-2 ~ [VGP(x,Y)I2dx. TZR TZN

Employing Poincar6 's inequality

[W-Wal2dx<=c2 R2 ~ [Vw[ 2dx 7'2R 1"2R

together with the est imates (2.6) and (2.9) we get

e -" [. Iw-%12dx<=c3 f e(C)a( ' ,Y)l~dx, T2R T2R

and also R n - 2 ~ [ v a P ( x , Y ) l g d x ~ K 4 �9

T2R Therefore,

V <gK4 +g-' c2 c3 ~ e(U)a(',y)~Tdx. T2 R

Let ~=eo/(4K,,), and c4=g-,,c2c3=4go I c12 C3 K 4. Then we obtain

II<(1-h)2suplwl2+%/4+c,,(%) ~ e(U)G(',Y)l/Tdx. ~2R T2 R


Moreover , for any 51 >0,

lV<=c5 R-1 S IVvIGP(',Y)lf~ dx T2R

<=51R-2 I GO(x,y)dx+c~?'~IWl2G"(',Y)~ dx T2R

<qc6+5;ic7 [. e(U)G~ T2R

Let 51 =5o/(4c6) , and c8=5i -1 c 7 = 4 e o I c6c v. Then

IV<eo/4+Cs(%) ~ e(U)G~ 7'21 l

Thus, letting p tend to zero, we obtain from (2.25) the estimate

I v(y)l 2 <(1 - h ) 2 sup Iwl 2 +50/2 ~2R

+c9(5o) ~ e(U)G(.,y)l/Tdx (2.26) T2R

--2 ~ q(v)G(',y)q]fTdx

for all ysf2n/2, where c9=c4+c 8. Moreover ,

2 ~ q(v)G(.,y)rl]/-Tdx=2 ~ . . . + 2 ~ ...

>4a~(sup lv l ) j" e(U)G(',y)rll/Tdx -QR/2 f2R/2

+4a~(2M) ~ e(U)G(.,y)ql/~dx. ~ 2 a - -~R/2

Then, for yGOR/4, w e infer from (2.8) and (2.9) that

e(U)G(x,y)]fTdX<Cao ~ e(U)G(x, xo)lfTdx.

Setting c 11 = cl 1 (5o) = c9 ( 5 0 ) - - 4 min {0, a~ (2 M)} c l o, we thus infer from (2.26) that, for all yef2R/4,

I v(Y)l 2 < (1 - h) 2 sup I wl 2 + %/2 ~2tR

+q~(5o) ~ e(U)G(',Xo)l/Tdx-4a,,(suplvD ~ e(U)G(',y)lf~dx ~ 2 R -- ~'~R/2 ~R/2 ~'~R/2

(2.27)

In order to apply L e m m a 3, we choose 5 = 2 -150C111(50), and let m = m ( 0 and R~ =Rb(5) be the corresponding numbers from L e m m a 3.

Then the following statements hold:

Lemma 4

(ct) Let R o =mm{Ro, d/8 }. Then there exist numbers Po and Ro, O<Po <Ro <R*, Po = Po(eo, d) such that


sup I Vho, ao 12 < (1 -- ho) 2 sup l u 12 + ~-0. f2oo .Oa~

(fl) Suppose that, for some R* e(0, min {Ro, d/8}],

sup Iwl 2 <(1 - h J M2 + Y. f2R*

Then there exist numbers R and p=p(eo, R*), O < p < R <R*, such that

sup IWh, R] 2 <(1 - h ) 2 sup Iwl z + e o. ~p Y2R*

(2.28)

(2.29)

(2.30)

Proof of Lemma 4

(~) In virtue of U(M2d)CBM(Q), we have [u(x)l<=M on M2~, and [ffRol< M for each R o < R*. Moreover , we note that

Since

therefore

[ Vho,Ro I = dist (U (x), Qho)

< dist (U(x), Q) + dist (Q, Qho)

= lu(x)l +ho [aRol < M + h o M .

7C 0 < ho < 2 ~ - ~ - 1,

we obtain sup I Vho,Ro(X)l < ~/(2]f~). ~2d

Thus G(sup Iv[)>0 for V=Vho,~ ~ and for each Re(0,d] , whence we may drop the ~R/2

second integral in (2.27). In order to estimate the first integral, we apply Lemma 3, with the previously chosen e>0 , and with k=�89 and /~o=R~. Then there is a number R o with

(k/8)"R~ <4Ro <=R* such that

e(U)G(' ,Xo)]/Tdx <e. ~2R 0 -- ~Ro/2

Then we infer from (2.27) that

sup I Z)ho,Ro [2 ~_~(1 - - ho) 2 s u p ]ul 2 -I- e 0. ~RO/4 ~2R 0

Setting p o = 1 6 - m - l R *, R o = m m { R o , d/8},

we obtain (2.28), since po<Ro/4<2Ro<R~/2 .

(fl) By (2.29), the representat ion w(x) of U in w-normal coordinates with center (2o satisfies

iw(x) l<=l~_ho)2 M2 +e, = 7z -h '

282 S. H i ldeb rand t et al.

for every XE~'~R,. Therefore, also

7r ]V~R] < ~ - h ' for O<R<R*/2.

21/~c

Since U~Wh, R is a representation of U in normal coordinates with center Qh whose w-coordinates are h #R, we obtain that

]Wh. R(X)] = dist (U (x), Qh) < dist (U (x), Qo) + dist (Qo, Qh)

__<]w(x)] + h I%1=(1 +h)[rt/(2]~c)-h']

for Xef2R, and O<R<R*/2. Since

O<h <h ,21/~, Ti

it follows that

sup I w..R(x)l < =/(21~) f~R*

if O<R<R*/2.

Thus, in particular, a~(suplv])>0 for R<R*/2 and for v=%,R, so that we can f2R/2

drop the second integral in (2.27). Applying Lemma 3 to the first integral in (2.27), with the previously chosen g>0, with k= 1/2 a n d / ~ 0 = R *, we deduce the existence of a number R with

(k/8) m/~o --< 4R __</~o such that

(. e(U)G(',Xo)]/-Tdx <e.

This together with (2.27) implies that

sup IWn, RI 2 <(1 - h ) 2 sup Iwl 2 +%. DR/4 ~'~2R

Choose p = 16 - " -1R* . Clearly,

p <R/4 < 2 R < R*/2

and (2.30) follows at once from the previous inequality. Thus, Lemma 4 is proved. Now we define wi, Pi, R.*, for i=0, 1 . . . . . S by the following iteration pro-

cedure: Let Po, Ro, R* be chosen as in part (c 0 of Lemma4, and w 0 be the formerly defined representation v0.Ro of U. By virtue of (2.28),

sup Iwol 2 <(1 -ho) 2 sup lul 2 +~o ~PO aQR~

N (1 -- ho) 2 M 2 + eo, (2.3 l, 0) and % < d.

Suppose now that wi, Pi, and R* are already defined for 0 < i < j - l, such that O<pi<R* <R~, and

sup ] wi] 2 < (1 - h o ) 2 M 2 + ~' (2.31, i) L~p~


as well as sup I w,I 2 _<_ (1 - h) z sup I w i_ 112 + e,o, (2.32, i) "Qpt ~2P~ 1

where w i denotes a representation of U in a system of normal coordinates with center PieBM(Q). Let w l=U, P - l = R o .

In the j- th step we choose R*=pj_. l, and w=wa_~. Because of (2 .31 , j -1) , assumption (2.29) is satisfied. Thus, by Lemma 4, (fl), there are numbers R and p =p(%,R*) , O<p<R<R* , such that

sup Iwh.•{ 2 <(1 - -h) 2 sup Iwl 2 + %. ~20 f~R*

Set R~=R, R*=R*, pj=p=p(R*,co), and Wj=Wh.R =wh. R. Then, (2.32,j) is satisfied. Iterating (2.32, i) for i = 0 , 1, . . . , j , we get

sup [wj[ 2 <(1 -h )2J sup Iw012 + % [ 1 - ( 1 - h ) 2 ] -1 , -Qo~ f2oo

For j > 0, the inequality

[ l - - ( 1 - -h)2] -1 -1-(1 -h)2J<=2h-l(2-h) -1

holds. Invoking (2.31,0), it follows that

sup Iwj] 2 <(1 - h)2J(1 - h0)2 m 2 + rain {~', e"/8}, g2o t

i.e., (2.3 l j) holds. Moreover , our iteration procedure yields that

(OSC U) 2 ~ 4 sup ]wj] 2 ~ 4(1 - ]1) 2j M 2 -F e"/2 ~pj (2p /

whence, in particular,

osc U < {4(1 - h) 2s m 2 + e/'/2 } 1/2 < ]/~7 < ~. J2pS

The number p = Ps can be computed. It depends only on the geometrical data of the problem and on the choice of the parameters of our construction, but it is independent of U and of the choice of X o e ~ a.

Thus we have found: For every r > 0, there is a p > 0 depending on r and on the geometrical data such that

osc U < T for all Xoe~ a. (2.33) ~'.(xo)

Finally, we shall derive a uniform HSlder condit ion for U on ~a. For this purpose, we choose some p, 0 < p < d, such that

osc U < M for all Xoe~,l. (2.34) d.Op (xo)

Then we pick an arbitrary x 0 E ~ a and an arbitrary number R with 0 < 2 R < p . Set f 2 ~ = ~ ( x 0 ) for r > 0 , Tz~=f22 , - f2 ~, T*~=f2~/,~-f25~/4. Moreover , let t /be an


even bet ter friend than betore o n ~C22R; that is, we assume that rI6Cc'~'(f22R, IR), 0 < t /< 1, [ V~/I < c/R for some c independent of R, t / (x)= 1 for I x - x o I < 5 R/4, and ~/(x)-=0 for IX-Xol>7R/4. Let u(x) be the representat ion of U in normal coordinates on BM(Q) centered at Q, and let

aR= S u(x)dx. T2R

Then we choose normal coordinates v on BM(Q) with center P the u-coordinates of which are fiR, and we denote the representa t ion of U in the v-coordinates by v(x). Since

[ v (x)] = dist (U (x), P) _-< bo, (m) ]u (x) - ffRI

for xef2p, Poincar6 's inequali ty yields

Iv(x)12 dx<c~2R 2 j IVu(x)12 dx (2.35) T2R T2R

where, as usual, c~2 and the for thcoming numbers c 13, c~4 . . . . are independent of x o and R.

We test the "weak Euler equat ion" (2.24) once more with qo=v~/G~(.,Xo), 0 < e < R, thus obta ining

+ 2(~ q(c) G~(',Xo)rIV~dx + 2(~/~BG~(',Xo)CiD~viD~rl]f~ dx

= I + I I + I I I + I V .

Since ~ e ( X O ) ~ R , w e find that

I =

Secondly, we note that

Iv l2dx>O. ~ (xo)

q(v) > 2 a,,([ v l) e(U) > 2 a,,(M) e(U)

on No(Xo), by virtue of (2.34), whence

III>=a~(M) ~ rle(U)G~(.,Xo)V/Tdx. I22R

Thirdly, by Young 's inequali ty and by (2.10),

IIV1<=c13 S lv l 'R-11vl T2R

<=c14R2-n{ ~ IVvl2dx+R -2 ~ ]viZdx} . T2R T2R

Invok ing (2.35), we arr ive at

]IVI<cxsR 2-" f {[VuIZ +[VvIg}dx. T2R


For sufficiently small e>0, say, 0 < e < R / 2 , the mollified Green function =G~(',Xo) is a solution of Laplace's equation A~O=O in TzR. Thus, by a well- known reasoning due to Caccioppoli/Moser (cf. [33]), we obtain that

IvlZlVG~'(',xo)lZdx<cx6{R -2 ~ Iv[ZlG~(',Xo)lZdx+ ~ IG~(',Xo)12lVv[Zdx}. T~R TZR TZR

Employing (2.35) and (2.10), we obtain

[v}2l[TGe(',Xo)]Zdx~clvR2{Z-n) 5 {]Vttl2+l[Tvl2}dx. (2.36) T~R TZR

Since

1II1<c~8[R-2(5 -1 [. [vl2dx+(5 ~ Ivl21VG(',Xo)lZdx] T2R TIR

for each (5 > 0, we may choose (5 = R"-2, whence

III1<=C19 R2-n ~ {[Vul2 +IVt)[2} dX, T2R

on account of (2.35) and (2.36). Collecting these estimates, we can derive the inequality

rle(U)G~(',Xo)l/Tdx<=c20 R2-" ~ {[Vul2 +lVv[2}dx. ~2R I"2n

Letting e tend to zero, the properties of the friend r/also yield

~ e(g)a(',Xo)]/?dx<c2o R2-" ~ {IVul2 +lVvl2}dx. (2.37) f2n Tzn

Since [u(x)l<M, {v(x)I<M on ~22, we infer from (2.6), (2.9), (2.37), and Lemma 1 that

~ e(U)G(',Xo)]/-Cdx<=c21 ~ e(U)G(',Xo)]/Tdx. ~R ~2R - f2R

Adding C21 S " ' ' to both sides of the inequality and dividing the result by 1 -t-C21 ,

we obtain aR

(b (x0, R) < 0. q) (x0, 2R) (2.38)

for all XoeN a and all Re(O,p/2], where we have set

0 = C 2 1 / ( 1 + C 2 1 ) , 0 < 0 < 1 ,

and q)(xo, R)= ~ e(U)G(',Xo)]//-r dx.

,.~R (xo)

By a well-known iteration argument, cf. [33], there is a number o, 0 < o r < l , depending solely on 0, such that

cb(xo, R)<c22 cb(xo, P) (2.39)

for all Xo~YJ a and all Re(O,p/2].


In virtue of a well-known imbedding theorem due Morrey, (2.9), (2.20), and (2.39) imply that

[U]c~l~d ) < c* (2.40)

where c* depends on the geometrical data including d but does not depend on U. Now we shall investigate how c* depends on d. Clearly, we could derive an estimate of this kind from (2.39), by a close look at the dependence of the numbers cl, ..., Clz on d. But it turns out that a simple scaling argument gives the dependence of c* on d immediately. For this purpose, let c* =c*(d), and

c=c*( l ) .

Let ~ea be the part of 2' which is described by the coordinate patch N2e and by the representation ~&~(x)dx~dy ~ of its line element dcr 2 on ~2d. Denote by .~'2*a a part of an n-dimensional Riemannian manifold ~* which is described by the coordinate patch ~2 and by the representation 7*~(y)dy~dy ~ of its line element dr* .2 on ~2 where the components ?,*~ are defined by

7~ (y )=y~(d .y), lyl<2.

Condition (2.6) implies also that

for all ~elR" and all Ye~2 . Let u=u(x) be the representation of a harmonic map U: Y'zd--*B~t(Q) with

respect to u-coordinates centered at Q. Then, u*(y)=u(d .y), ]y]<2, yields the representation of a harmonic map U*: .~ with respect to the u- coordinates, since on ~'2

A~.,u*t+Fi~(u*)7*~D~u*iD~u*k=o, 1 < l < N ,

as one can easily verify. The geometrical data n, N, M, 2, /l, ~o, K for U and U* are the same, but d

has been replaced by 1. Thus, by (2.40),

lu*(y)- -u*(y ' ) l<cly-y ' l ~ for Y, Y' 6.r whence

l u ( x ) - u ( x ' ) l < c d - ~ l x - x ' l '~ for all x ,x ' e~J d,

and where the number c depends on n, N, M, 2,/1, o2, and ~c, but not on d. Thus, we have verified the estimate (2.7), and Theorem 4 is proved.

Proof of Theorem 1

If 2F is compact, the assertion of the theorem is an immediate consequence of the maximum principle since [u(x)[ 2 is subharmonic on Y" because of (2.22).

If Y" is simple, the statement of Theorem 1 follows directly from the estimate (2.7) by letting d tend to infinity.

Harmonic Mappings and Minimal Submanifotds 287

3. The Grassmannian Manifold G(n, p)

The purpose of this section is to provide the reader with a survey on results about the Grassmannian manifold which will be used in the next section. In contrast to some other authors, we shall treat the Grassmannian manifold G(n, p) of oriented n-spaces in E "+p instead of the Grassmannian G*(n, p) of unoriented n-spaces in E "+p since those will occur in our applications. Of course, results on G(n, p) are closely related to corresponding statements on G*(n,p), and vice versa, since G(n,p) is a two-fold regular covering of G*(n,p). in particular, we shall sketch a proof of a pinching result for the sectional curvature which has been stated by Wong without proof. Most of our results have been taken from Leichtweiss [28], Wong [42, 43], Wolf [41], Jensen [23], and Fischer-Colbrie [11].

3.1. In the following, denote by E "+p the (n+p)-dimensional euclidean space. Let b f[ stand for the euclidean length of the vector f~E, "+p, and denote by ( f , g) the euclidean inner product of J~ g~ E n+p.

By G(n,p) we denote the (real) Grassmann manifold consisting of all n- dimensional oriented subspaces (called n-planes) of IR "+p. Leichtweiss [28], p. 338, has proved the following: G(n, p) carries a Riemannian metric which is invariant under all transformations of the group of motions induced in G(n, p) by O(n+p). Except for n = p = 2 , this metric is, up to a constant positive factor, the only invariant metric of this kind on G(n, p).

For n+p+4 this result follows also from a general theorem due to E. Caftan. since we have the representation

G(n, p)= SO(n + p)/SO(n) x SO(p).

Therefore, G(n, p) is in a uniquely defined sense a Riemannian manifold (except for n = p = 2 ) . This manifold forms a symmetric, homogeneous, complete space. Moreover G(I, n) and G(n, 1) can be identified with S". Except for G(1, 1), G(n,p) is simply connected, and dim G(n, p)=np. (For details, cf. [28] and [41].)

3.2. Crittenden [8], Theorem 5, has shown that the minimum locus of a point Po of a Riemannian manifold . ~ coincides with the first conjugate locus of Po, provided that Jg is a simply connected, symmetric space. On account of 3.1, this assertion holds in particular for ./r p). Let K>0 be an upper bound for the sectional curvature K of .//g, and let Bv(Q) be a closed ball with center Q and radius M in ~ ' , such that M < rc / l~ . Then Rauch's comparison theorem implies that the first conjugate locus of Q does not meet BM(Q), cf. [4], pp. 29-35. Thus we obtain the following result:

Suppose that to>_0 is an upper bound for the sectional curvature K of G(n, p). Let BM(Q) be a bali-of radius M and center Q in G(n, p) such that M < ~/(21/~). Then we can introduce normal coordinates on B~t(Q) around each point P of BM(Q) as center.

3.3. We want to construct now this metric on G(n,p), which is most easily achieved by the approach of E. Cartan, as it was also done by Fischer-Colbrie [11].

288 s. Hildebrandt et al.

An oriented n-plane in E "+p can be described by an orthonormal n-frame, which can be complemented by an orthonormal p-frame to an oriented orthonormal base of E "+p. Fixing a special or thonormal base of E "+p gives rise to an identification of equioriented bases of E "+p and elements of SO(n +p), by writing each base in terms of the fixed one. The equivalence relation on such frames to describe the same oriented n-plane is induced by the operation of SO(n) x SO(p), and we have the representation

G(n, p)=SO(n + p)/SO(n) x SO(p). (3.1)

We can identify the tangential space of a point of SO(n +p) with the Lie algebra o(n+p) of skew symmetric matrices of order (n+p). An invariant metric on SO(n +p) can be chosen with the following normalization

(A, B) = 1 tr (AB*) (3.2)

where "tr" denotes the trace. The tangential space m of a point in G(n,p) can be identified with the

or thonormal complement of the tangent space o(n)x o(p) of SO(n)x SO(p) in o(n +p), i.e. with the space of skew symmetric matrices A, B . . . . of the form

(0 (0 A = - X * , B = _ y * . . . . (3.3)

where X and Y are arbitrary (n x p)-matrices. Consequently, for A, B ~ m, i.e. of the form (3.3), (3.2) becomes

(A, B) = t r (XY*), (3.4)

which is the wanted invariant metric on G(n, p). It corresponds to the quadratic form

n+p

2 (3.5) a = l i = n + l

on the space of (n x p)-matrices, which is just the formula of LeichtweiB in [28-1, p. 338.

Now we want to introduce local coordinates on G(n,p). We shall describe the metric in terms of these coordinates.

For this purpose, let Po be an oriented n-plane in E "+p. We represent it again by n vectors g,, c~=1 . . . . . n, which are complemented by p vectors dl, i = n + 1 . . . . . n +p , such that d o, d i form an or thonormal base of E "+p. Then we can span the n-planes P in a neighborhood ~A/" of Po by n vectors f , , ~ = 1, ..., n, given by

f~=d,+ z~d, (3.6)

(summation with respect to i from n + l to n+p). These vectors constitute a base of the solutions of the linear system

xi '~-ziotx~ , i=n+ 1 ... . . n+p (3.7)

(summation with respect to c~ from 1 to n).


Therefore, each point in Ar is uniquely described by a set of p equations of the form (3.7), and we have a coordinate patch (,W, Z) on G(n,p) with local coordinates Z = (zi~).

The vectors f~ are complemented by the vectors

(summation with respect to a from 1 to n). Since

( (L , f~)),,~= 1 ...... =I,+ZZ* and

(<f~, fj>)i,j=,+l ...... +p=Iv+ Z*Z,

where Z* denotes the transpose of Z, and I m is the identity on E ' , we get for the line element of our invariant metric in these local coordinates by the transfor- mation rules of linear algebra

dsZ=tr((I,+ZZ*) -a dZ.(lp+Z*Z) l dZ.). (3.8)

This is the formula of Wong [42, 43].

3.4. Employing the local representation (3.7) of G(n,p), Wong [42] has noted that a geodesic segment Z =Z(s) in the patch A/" is described as a solution of

Z=22Z*(1+ZZ*) 12 (3.9)

d where =d~ ' Let us assume that n<p . Let T be a unit tangent vector at the

point P0 described by Z =0 . Without loss of generality, we may assume that T is of the form

2.

(3.10)

where ~ 22=1. Otherwise, we would replace d 1 . . . . ,d, and d,+l . . . . ,d,+p by ~t=l

appropriately chosen different orthonormal bases for Span {Jl . . . . , d,} and for Span {g, + 1 . . . . . d, + p}, respectively.

The solution Z=Z(s) of (3.9) with the initial values Z(0)=0, Z ( 0 ) = T is given by

tan(21s ) 0 i

I Z (s) = tan (2 2 S) I �9 ,

0 tan (2. s) I I

for 0 <s < ~/2, where s is the parameter of arc length.

(3.11)


The vectors d 1 . . . . . d, describe the fixed n-plane P0, while the vectors f~ . . . . , f , with

f~=do+z'=(s)d,, 1 <o~<n, (3.12)

describe the n-plane P=P(s) in Y corresponding to the coordinates Z=Z(s). Let

where /l=o,,f,, / 2 = (.02 f 2 . . . . . L - ~ - gOn f n ( 3 . 1 3 )

c% = cos (2~ s). (3.14)

Since ILl = 1/coo, the vectors f , . . . . , f , are o r thonormal . Therefore, we can define the inner p roduc t (Po, P ) of the n-planes Po = el A

... A ~, and P = f l A. . . A J~, by

(Po, P ) = det ((do, fp)) .

It follows f rom (3.11-15) that

(i 1 (Po, P ) = d e t (2)2

(3.15)

. ~ (~0~, �9 . o = l

(-On

max {(Po, P>: Pe(?BM(Po)}=cos" ~nn) (3.19)

for 0_-<M<=/2, and n<p. I f p < n , we have to replace n by p in (3.19). Hence, for

m = m i n {n, p}, (3.20)

whence

that is,

(Po, P(s)) = f l cos(2~s), 0 ~ s < n / 2 . (3.16) e = l

This formula is due to Fischer-Colbr ie [11] who derived it with a m o r e abst ract a rgument invoking the max imal torus theorem. In 3.5 we shall p rove that the sectional curvature K of G(n,p) furnished with the canonical metric of 3.3, is pinched by 0_< K _< 2. Thus,

s=d i s t (Po , P(s)) for 0 < s < ~ / 2 , (3.17)

taking 3.2 into account. Let 0 < s < r t / 2 . By virtue of a straight forward com- pu ta t ion carried out by Fischer-Colbr ie [11], L e m m a 3.4, it follows that

max cos (2os)" 22 = 1 = cos" (3.18) o ~ l o = l

H a r m o n i c M a p p i n g s a n d M i n i m a l S u b m a n i f o l d s 291

it follows that

for 0 < M < re/2.

max { ( Po, P ) : P ~ ~? BM(Po)} =cos~ ( 5 ) (3.21)

3.5. From the theory of Lie groups (cf. [4], Chap. 3), we obtain for the sectional curvature of a tangent plane at Po~ G(n, p), spanned by two linearly independent vectors X = (xi~), Y = (yi)

I1 [A, B] [12 K(X, Y)=K(A,B)=HALI2. HB[12 [(A,B)I2 , (3.22)

employing the representation (3.3) and setting

1 liAH = (A, A) ~ = ~ {tr (AA*)} ~ (3.23)

I '

(cf. (3.2)). This is equivalent with

tr {(X Y* - YX*)(X Y* - YX*)* + (X* Y - Y* X)(X* Y - Y* X)*} K(X, Y ) -

2 {tr (XX*) tr ( Y Y * ) - [ t r (X Y*)I 2}

o r

2 tr {(X Y* - YX*) (X Y* - YX*)* + (X* Y - Y* X)(X* Y - Y* X)*} K(X, r ) -

(3.24)

4 t r (XX*) t r (YY*)- {tr(X Y* + YX*)} 2

which is formula 6 of Wong [43]. This in turn agrees with

. -

(xcty fl i j K ( X , y ) ~t, f l = l I i = n + l i,k=n+l

L n + p e i 2k k i x2 I,x~.3 t~ -- xt~Y~J

a, f l = l i.k=n+l

which is the formula of Leichtweiss [28], p. 350. Clearly, K >_ 0. Let

•=sup K(X, Y)

If max (n, p) > 2 and min (n, p) = 1, then

K(X, Y)--- re= 1

}2 i k k i (G Y~ - x~ y~)

ct=l

(3.25)

, (3.26)

since G(n,p) can be identified with S N, N=max(n ,p) . If min (n, p) > 2, there are two-planes X A Y on which K vanishes, and ~c = 2. Since the last result has been stated by Wong [43], cf. Theorem 3(a) and Theorem 5(i), without proof, a sketch of a proof that ~c = 2 might be welcome.

292 S.Hildebrandt et al.

Firstly, let n = p = 2, and consider the tangent vectors

~ r -;) at the point PoeG(2, 2) with the coordinates Z = 0 . Then, X X * = X , X Y * = Y*, Y Y * = X , X Y * - Y X * = - 2 Y , X * Y - Y * X = 2 Y which implies that t r (XX*) = t r (YY*)=2 , t r (XY*)=0, and the nominator in (3.24) becomes 8tr(YY*). Therefore, K(X, Y)=2, whence to>2 for G(2, 2). Since G(2, 2) can be canonically imbedded into G(n,p) if rain {n,p} >2, we get ~c>2 also in the general case.

Secondly, we will show that K(X, Y)<2 for all tangent vectors X =(x~,), Y =(y~,). Let us assume that n<p. For notational convenience, we set

~ i ~ n+i n+i x~ , G~=y, , l <-e<n, l <i<p.

By the reasoning pointed out in Sect. 3.4, we may assume that X=({=~) is of the form

X =

(4 I' ~22 ".. 00) Moreover, since K(X, Y) has the same value for all X, Y spanning the same 2- plane, we may assume that tr (X Y*)---0. Then, the sectional curvature takes the form

where

and

K(X, Y)=Z/N z= y ( ~ _ ~ ) 2 + ~ ({ii,hj-~j/7~,)L

~t, fl i, j ( a , p ) i * j

/~,j

Here, we have set {ii=O for i>n, and summation with respect to greek or latin indices is to be taken from 1 to n, or from 1 to p, respectively. By virtue of Schwarz' inequality, we see that

On the other hand,

p ~ j

~2 2

>2~{2={~r /~=+ ~" qgj}>�89 ~ J ( j* ~t)


Therefore,

K(X, Y)=<2,

whence K=2 for min {n, p} ~ 2, since the case p<=n can be handled in an analogous way.

4. Minimal Submanifolds

By virtue of the results of Sect. 3, Theorem 1 yields the following Liouville theorem for harmonic mappings into Grassmannian manifolds:

Theorem5. Let U: X--*G(n,p) be a harmonic map of a simple Riemannian manifold X into a Grassmannian manifold G(n,p) such that the range U (X) of U is contained in a closed ball BM(Po) in G(n, p) with center Po and radius m<1r / (2] f~) where ~=1 for r e = l , ~ = 2 for m>=2, and m=min{n ,p} . Then U is a constant map.

The link between minimal submanifolds in the euclidean space E "+p and harmonic mappings is formed by an important result due to Ruh and Vilms [37].

Let F: X---,E "+p denote an immersion of an n-dimensional manifold X into the (n+p)-dimensional euclidean space E n+p. As usual, X is considered to be a Riemannian manifold the metric of which is the induced metric of E n~-p. The Gauss map associated with the immersion F is the map fr X ~ G ( n , p ) which assigns to each point x of X the tangent plane to F(X) at F(x). The mean curvature field H(x), x ~ X, is the trace of the second fundamental form on F(X). It is zero, if Y" is immersed into E "+~ as a minimal submanifold. The mean curvature field H is said to be parallel if ( ~ H ) ~ = 0 for all tangent vectors v ~ TF~x)F(X) where Vdenotes the standard connection on E "+p, and _L indicates the orthogonal projection of V~,H~T~(=)E"+V~-E n+v onto the normal space Ne(x)F(X ). Then, Ruh and Vilms have proved the following:

Theorem 6. The Gauss map (#: X--*G(n,p) of a C3-immersion F: X--+E "+v of an n-dimensional manifold X into E "+p is harmonic if and only if X is immersed with parallel mean curvature field. In particular, ~ is harmonic if F(X) is a minimal n- dimensional submanifold in E "+v.

A simple proof of this theorem, using the method of moving frames, can be found in [6], pp. 138-140; cf. also [10], p. 30.

There is a canonical isometry J : G(n, p)--*G(p, n) assigning to each oriented n-plane the properly oriented orthogonal p-plane. We then define the adjoint Gauss map N*: X--*G(p,n) of the Gauss map f#: X ~ G ( n , p ) of an immersion F: X ~ E "+v by f 4 * = ~ o fr Since f#* is harmonic if and only if N is harmonic, we obtain the following

Corollary. The adjoint Gauss map N*: X--*G(p, n) is harmonic if and only if X is immersed into E n+p with parallel mean curvature field.

Now we can state our first Bernstein theorem.

294 S. H i l d e b r a n d t e t a l .

Theorem 7. Let 9J~=F(lR") be a minimal, n-dimensional subman!Jbld in E "+p given by a C3-immersion F: IR"~IR "+p. Suppose that there exists a fixed oriented n- plane Po, and a number C~o,

CX 0 ~ COS m

such that

m = min {n, p}, ~c = if 2 m > '

(P , Po) > c% (4.1)

holds for all appropriately oriented tangential n-planes P of ~Jl (cf. (3.15)). Assume also that W=(IR", dcr 2) with

do" 2 =(dF(x) , dF(x ) )= F~(x) . Fx~(x)dx'dx ~

(1 < ~, fl < n, summation over ~ and fl) is a simple Riemannian manifold. Then, the associated Gauss map ~: Yf--*G(n, p) is constant, and ?0l is an n-dimensional affine linear subspace of E n+p.

Proof. On account of Theorem 6, the associated Gauss map .(~: ~ G ( n , p) is harmonic. By (3.21), condit ion (4.1) implies that there exists a number M = M ( % ) , 0 < M < ~ / ( 2 I / ~ ) , such that PeBM(Po) holds for all (properly oriented) tangential n-planes P of 9J /=F(X). Thus we conclude from Theorem 5 that ff is constant, and the theorem is proved.

Now we turn to a special representat ion 9)l=F(2~'). We assume that Y'=IR", and that, for x = (x 1, x2, . . . , x")~ IR",

F(x)=(x, f ( x ) )=(x ~, fi(x)), 1 <e<n , n+ 1 < i < n + p (4.2)

and F e C 3 (IR"). This is called a non-parametric representation of 9)l. Clearly, F is an immersion.

Osserman [34] has proved:

Theorem 8. Let a manifold 9Jl be given in the non-parametric form. Set

7~t~ (x) = 6,~ + (D~ f (x) , Dp f ( x ) ) = 6~ +fi~(x) fj~ (x),

7 = d e t ( 7 ~ ) , and (7~ )= (7~) -~. (4.3)

Then the following statements are equivalent:

(i) 9J~ is a minimal submanifold of E"+P ;

(ii) the functions f i satisfy the equations

D~{]//77"OD, f~}=O on ~-,", i = n + l . . . . , n+p; (4.4)

(iii) the functions f i satisfy on IR" the Eq. (4.4) together with

D, {]/~ 7"t~} =0 , /~=1 . . . . . n; (4.5)

iv) the functions f i satisfy on IR"

7~D~Dr i = n + l . . . . . n+p. (4.6)

H a r m o n i c M a p p i n g s and M i n i m a l Submani fo lds 295

Morrey [31] has proved that weak solutions f e C * (N,", IR p) of (4.4) are real analytic. On the other hand, Lawson and Osserman [27] have shown that there exist Lipschitz continuous weak solutions of (4.4) which are not of class C ~.

Let f e C I ( I R " , I R p) be a (weak) solution of (4.4). Suppose that {d~,e~ 1 <ot<_n, n+ 1 <_iNn+p} are the standard basis vectors o f E "+p. Choose P0 as an n-plane which is represented by the simple normalized n-vector dl A d 2 A ... A d,. Furthermore,

whence

F(x) = (x, f (x)) = x ~ ~ + f i (x) di,

F=F,, =d~+j;idi, f i = f j , , l<_o~<_n.

The vectors F1, F 2 . . . . , F n are linearly independent and span the tangent plane P =P(x ) of 9J~=F(IR") at F(x)=(x, f (x)) . Thus, P is represented by the simple normalized n-vector e 1 A e z A ... A e, where

and e~=Afl/"F=, c~=l, . . . ,n,

A f = I F 1 A F 2 A . . . AF. I=(F 1 A . . . A F., F 1 A . . . A F n ) 1/2

= {det ((F~, Fe))} 1/2

or, equivalently,

+ix=( )'f~'~( ))} (4.7) A/(x)={det(a~e -i X i X 1/2

(summation with respect to i from n + 1 to n+p). Then

(P, Po )= det ((e~, 4 ) ) = A f ' (4.8)

Therefore, we can replace condition {4.1) of Theorem 7 in case of a non- parametric representation

z i= f i ( x l . . . . . x"), i = n + l . . . . . n+p, x~lR",

of a minimal n-dimensional submanifold OJl by the following equivalent condition:

There is a number/30, 0 </3 o < cos-"0z/(2 l/~cm)) such that

d/(x)<flo for all x e l R n. (4.9)

Together with Theorem 7, we thus obtain a proof of Theorem 2. Finally we shall derive an analogue to an interesting rigidity theorem for

minimal compact n-dimensional submanifolds 9.R in the (n +p)-dimensional unit sphere S "+p due to Fischer-Colbrie [11] which, in turn, generalizes previous work of DeGiorg i [9], Simons [39], and Reilly [35]. The approach of [11] is quite similar to ours but in her case it suffices to apply the maximum principle since ~1l is compact while a Liouville theorem is needed for the noncompact case. Our analogue deals with noncompact minimal immersions into S "+p.

296 s. Hildebrandt et al.

Le t 93/be a min ima l n -d imens iona l submani fo ld of S "+p which is given by a C3- immers ion F : IR"-~S "+p. To ~ , we assign the cone

C(?OI)={zEE"+P: z=ry , y~?0l, r > 0 } .

Moreover , we extend F : I R " ~ S "+p to a C3- immers ion if: IR" • IR+--*IR "+I+p by

if(x, r)=rF(x), xe~," , r > 0 .

Clearly, F(IR" • I R + ) = C(~R). It is we l l -known that ~Jl is min ima l in S "+p if and only if C ( ~ ) is min ima l in IR "+1 +P. Thus, by the coro l la ry to T he o re m 6, the ad jo in t Gauss m a p if*: {IR"x I ( +, d~2}~G(p, n+ 1) is harmonic . Moreover , we note tha t the no rma l space to ~Jl in S "+p at each po in t y~gJl coincides with the no rma l space of C(~l) in IR "+p+I for all poin ts z= ry , r > 0 . Hence ~*[~, : {lR, da2}-~G(p,n+ 1) is harmonic . Then, essent ial ly with the same reasoning as in the p roo f of Theo rem 7, we ob ta in the fol lowing result :

Theorem 9. Let gJ] = F (F,") be a minimal, n-dimensional submanifold in S "+p given by a C3-immersion F: IR"~S "+p. Suppose that there is a fixed oriented p-plane Po, and a number 7 0 > 0 ,

such that

k 1} k = m i n { n + l , p } , ~c= if k > 2 '

(P, P0> --> 70 (4.10)

holds for all normal p-planes P of C(gJl) in IR "+p. Assume also that 5~=(IR", d a 2) with

d a 2 = (dF(x), dF(x)) = Fx.(x ) �9 Fx, (x) dx ~ dx ~

is a simple Riemannian manifold. Then !Ol is contained in a totally geodesic subsphere orS "+p, that is, in a "great

subsphere" of S "+'.

References

1. Almgren, F.J.: Some interior regularity theorems for minimal surfaces and an extension of Bernstein's theorem. Ann. of Math. (2), 84, 277-292 (1966)

2. Bernstein, S.N.: Ober ein geometrisches Theorem und seine Anwendung auf die partiellen Differentialgleichungen vom elliptischen Typus. Math. Z. 26, 551-558 (1927)

3. Bombieri, E., DeGiorgi, E., Giusti, E.: Minimal cones and the Bernstein theorem. Invent. math. 7, 243-269 (1969)

4. Cheeger, J., Ebin, D.: Comparison theorems in Riemannian geometry. Amsterdam-Oxford: North-Holland, New York: American Elsevier Publ. 1975

5. Cheng, S.-Y.: Liouville theorem for harmonic maps, Preprint 6. Chern, S.-S., Goldberg, S.I.: On the volume decreasing property of a class of real harmonic

mappings. Amer. J. Math. 97, 133-147 (1975)


7. Chern, S.-S., Osserman, R.: Complete minimal surfaces in euclidean n-space. J. Analyse Math 19, 15-34 (1967)

8. Crittenden, R.: Minimum and conjugate points in symmetric spaces. Canad. J. Math. 14, 320- 328 (1962)

9. DeGiorgi, E.: Una estensione del teorema di Bernstein. Ann. Scuola Norm. Sup. Pisa 19, 79-85 (1965)

10. Eelts, J., Lemaire, L.: A report on harmonic maps. Bull. London Math. Soc. 10, 1-68 (1978) 11. Fischer-Colbrie, D.: Some rigidity theorems for minimal submanifolds of the sphere. Preprint.

To appear in Acta Math. 12. Frehse, J.: Essential selfadjointness of singular elliptic operators. Bol. Soc. Brasil. Mat. 8, 87-107

(1977) 13. Gather, W.-D., Ruijsenaars, S.N.M., Seiler, E., Burns, D.: On finite action solutions of the

nonlinear a-model. Ann. Physics 119, 305-325 (1979) 14. Hildebrandt, S., Kaul, H.: Two-dimensional variational problems with obstructions, and

Plateau's problem for H-surfaces in a Riemannian manifold. Comm. Pure Appl. Math. 25, 187- 223 (1972)

15. Hildebrandt, S., Widman, K.-O.: Some regularity results for quasilinear elliptic systems of second order. Math. Z. 142, 67-86 (1975)

16. Hildebrandt, S., Widman, K.-O.: On the H61der continuity of weak solutions of quasilinear elliptic systems of second order. Ann. Scuola Norm. Sup. Pisa (IV), 4, 145-178 (1977)

17. Hildebrandt, S., Kaul, H., Widman, K.-O.: An existence theorem for harmonic mappings of Riemannian manifolds. Acta Math. 138, 1-16 (1977)

18. Hildebrandt, S.: Entire solutions of nonlinear elliptic systems, and an approach to Bernstein theorems. Lecture at the Colt+ge de France, March 1979, to appear

19. Hildebrandt, S., Widman, K.-O.: S~itze vom Liouvilleschen Typ ftir quasilineare elliptische Gleichungen und Systeme. Nachr. Akad. Wiss. G6ttingen, I1. Math.-Phys. Klasse, Nr. 4, 41-59. Jahrgang 1979

20. Hoffman, D.A., Osserman, R.: The geometry of the generalized Gauss map. Preprint 21. Hoffman, D.A., Osserman, R.: The area of the generalized Gaussian image and the stability of

minimal surfaces in S" and ~". Preprint 22. Ivert, P.A.: On quasilinear elliptic systems of diagonal form. Math. Z. 170, 283-286 (1980) 23. Jensen, G.R.: lmbeddings of Stiefel manifolds into Grassmannians. Duke Math. J. 42, 394-407

(1975) 24. Jost, J.: Eineindeutigkeit harmonischer Abbildungen. Diplomarbeit 1979 25. Jost, J.: Eine geometrische Bemerkung zu Siitzen tiber harmonische Abbildungen, die ein

Dirichletproblem 16sen. Manuscripta Math., in press (1980) 26. Karp, L.: Differential inequalities on Riemannian manifolds: Applications to isometric immer-

sions and harmonic mappings. Preprint 27. Lawson, H.B., Osserman, R.: Non-existence, non-uniqueness and irregularity of solutions to the

minimal surface system. Acta Math. 139, 1-17 (1977) 28. Leichtweiss, K.: Zur Riemannschen Geometrie in Grassmannschen Mannigfaltigkeiten. Math. Z.

76, 334-366 (1961) 29. Meier, M.: Liouville theorems for nonlinear elliptic equations and systems. Manuscripta Math.

29, 207-228 (1979) 30. Meier, M.: Liouville theorems for nondiagonal elliptic systems in arbitrary dimensions. Preprint

(1980) 31. Morrey, C.B.: Second order elliptic systems of differential equations. In: Contributions to the

theory of partial differential equations, pp. 101-160. Ann. of Math~ Studies No. 33. Princeton: Princeton University Press 1954

32. Morrey, C.B.: Multiple integrals in the calculus of variations. Heidelberg-New York: Springer- Verlag 1966

33. Moser, J.: On Harnack's theorem for elliptic differential equations. Comm. Pure Appl. Math. 14, 577-591 (1961)

34. Osserman, R.: Minimal Varieties. Bull. Amer. Math. Soc. 75, 1092-1120 (1969) 35. Reilly, R.: Extrinsic rigidity theorems for compact submanifolds of the sphere. J. Differential

Geometry 4, 487-497 (1970)


36. Ruh, E.A.: Asymptotic behaviour of non-parametric minimal hypersurfaces. J. Differential Geometry 4, 509-513 (1970)

37. Ruh, E.A., Vilms, J.: The tension field of the Gauss map. Trans. Amer. Math. Soc. 149, 569-573 (1970)

38. Schoen, R., Yau, S.-T.: Harmonic maps and the topology of stable hypersurfaces and manifolds with nonnegative Ricci curvature. Comm. Math. Helv. 39, 333-341 (1976)

39. Simons, J.: Minimal varieties in riemannian manifolds. Ann. of Math. 88, 62-105 (1968) 40. Wiegner, M.: A-priori Schranken ftir L/3sungen gewisser elliptischer Systeme. Manuscripta

Math. 18, 279-297 (1976) 41. Wolf, J.A.: Spaces of constant curvature. New York: McGraw-Hill 1966 42. Wong, Y.-C.: Differential Geometry of Grassmann manifolds. Proc. Nat. Acad. Sci. USA 57,

589-594 (1967) 43. Wong, Y.-C.: Sectional curvatures of Grassmann manifolds. Proc. Nat. Acad. Sci. USA 60, 75-

79 (1968)

Received May 13, 1980

Harmonic mappings and minimal submanifolds

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