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1 Max-Planck-Institut fur Mathematik in den Naturwissenschaften Leipzig Bernstein type theorems for higher codimension by Jurgen Jost and Yuan-Long Xin Preprint-Nr.:

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3 BERNSTEIN TYPE THEOREMS FOR HIGHER CODIMENSION J. Jost Max Planck Institute for Mathematics in the Sciences, Leipzig and Y.L.Xin* Institute of Mathematics, Fudan University, Shanghai Abstract. We show a Bernstein theorem for minimal graphs of arbitrary dimension and codimension under a bound on the slope that improve previous results and is independent of the dimension and codimension. The proof depends on the regularity theory for the harmonic Gauss map and the geometry of Grassmann manifolds. 1. Introduction The celebrated theorem of Bernstein says that the only entire minimal graphs in Euclidean 3;space are planes. More precisely, let z = f(x y) be a smooth function dened on all of R, whose graph in R 3 is a minimal surface. Then f is a linear function, and the graph is a plane. The eorts to generalize Bernstein's theorem to higher dimensions led to profound developments in analysis and geometric measure theory. The nal result in a successive series of achievements by several mathematicians was the theorem of J. Simons [S] that an entire mimimal graph has to be planar for dimension 7 while Bombieri, de Giorgi and Giusti [B-G-G] shortly afterwards produced a counterexample to such an assertion in dimension 8 and higher. By way of contrast, J. Moser [M] had earlier proved a Bernstein type result in arbitrary dimension under the additional assumption that the slope of the graph is uniformly bounded. All these results hold for hypersurfaces, i.e. minimal graphs of codimension 1: For higher codimension, the situation becomes more complicated, and Lawson- Osserman [L-O] had given explicit counterexamples to Bernstein type results in higher codimension: for example they showed that the cone over a Hopf map is an entire Lipschitz solution to the minimal surface system. On the other hand, Hildebrandt-Jost-Widman [H-J-W] had obtained a Bernstein type result in arbitrary codimension under the assumption of a certain quantitative bound for the slope. Their method was to show that the Gauss map of *Thanks the Max Planck Institute for Mathematics in Sciences in Leipzig for providing good conditions during the preparation of this paper, and also NNSFC and SFECC for support Typeset by AMS-TEX 1

4 J. JOST AND Y.L. XIN such a minimal submanifold which was known to be a harmonic map into a Grassmannian has to be constant, if its range is contained in a suciently restricted set of the Grassmannian. This restriction required the bound on the slope. In fact, that result applies not only to minimal graphs but also to ones of parallel mean curvature as the Gauss map continues to stay harmonic under that condition. Actually, Chern [C] had shown that a hypersurface in Euclidean space which is an entire graph of constant mean curvature necessarily is a minimal hypersurface. Thus, by Simons' theorem, it is a hyperplane for dimension 7: Chern's result was generalized by Chen-Xin [C-X]. The proof of Hildebrandt-Jost-Widman derived Holder estimates for harmonic maps with values in Riemannian manifolds with an upper bound for sectional curvature and by a scaling argument then concluded a Liouville type theorem for harmonic maps under assumptions including the above mentioned harmonic Gauss maps. The Holder estimates needed a bound on the radius of the image, and examples show that [H-J-W] had achieved the optimal bound in the general framework for that paper. Nevertherless, when applied to the special case of interest here, namely harmonic maps with values in Grassmannians, it turns out that the bound for the slope reqired in [H-J-W] for their Bernstein type result depends on the dimension and codimension of the minimal graphs. In the present paper, we obtain such a Bernstein type result under a bound for the slope which is better than the one in [H-J-W] and independent of dimension and codimension. Theorem 1. Let z i = f i (x 1 x n ) i = 1 m be smooth functions dened everywhere inr n : Suppose their graph M =(x f(x)) is a submanifold with parallel mean curvature in R m+n : Suppose that there exists a number 0 with when m 0 < (1.1) 1 when m =1 such that where f 0 for all x R n (1.) f (x) =fdet( + X i Then f 1 f m are linear functions on R n R m+n : f i x (x)f i x (x))g 1 : (1.3) representing an ane n;plane in The case m =1of course is simply Moser's result. In the framework of geometric measure theory the Bernstein problem is transfered to the rigidity problem of m;dimensional minimal submanifolds M in the sphere S m+n : Namely, it follows from the general work of Allard [A] on varifolds that the tangent cone at innityofsuchanm;dimensional submanifold of R m+n+1 is the cone over a minimal submanifold of S m+n : J. Simons [S] proved that a compact minimal submanifold of the sphere whose normal planes lie in a suciently small neighborhood is a totally geodesic subsphere. Reilly [R] and Fischer-Colbrie [FC] improved the previous work successively. Like [H-J-W], however, these results require bounds that depend on dimension and codimension, and we are able here to get a similar generalizations as above.

5 BERNSTEIN TYPE THEOREMS FOR HIGHER CODIMENSION 3 Theorem. Let M be anm;dimensional compact or simple 1 Riemannian manifold which is a minimal submanifold in S m+n : Suppose that there is a xed oriented n;plane P 0 and a number 0 1 if min(m n) 0 > 0 if min(m n) =1 such that hp P 0 i 0 holds for all normal n;planes P of M in S m+n : Then M is contained inatotally geodesic subsphere of S m+n : As mensioned, the bounds in our results ( in Thm. 1 and 1 in Thm. ) are sharper than the ones previously known. As the examples of Lawson-Osserman show some such bound is necessary. However, these counterexamples are not sharp for our bounds, and so one may askwhether our bounds are optimal. This is probably not the case because the essential point in our argument is to construct a strictly convex function on a suciently large region in a Grassmannian. The question of nding the optimal bound - at least for the strategy pursued here - then amounts to nding the largest such convex supporting region in a Grassmannian. Geometric intuition suggests that such an optimal region may on one hand be larger than the one constructed in our paper, but on the other hand by nomeans as explicitely presentable as ours and possibly also nonunique in a general way. In other words, the optimal bound will probably turn out to be some number that may virtually be impossible to write down explicitely, and thus perhaps also be of comparatively little signicance. Our general strategy is the same as in [H-J-W]. By using the Ruh-Vilms theorem [R-V], weprove the harmonic Gauss map is constant under certain conditions. The main renement in this paper is a detailed study of geodesic convex sets in the Grassmannian manifold G n m : We dene a new specic geodesic convex set B G in G n m which is larger than the usual geodesic convex ball and interesting in its own right. We expect further applications.. Preliminaries In this section we describe some basic notions and results on harmonic maps and submanifols in the Euclidean space and the sphere which will be used in later sections. Let (M g) and (N h) be Riemannian manifolds with metric tensors g and h, respectively. Harmonic maps are described as critical points of the following energy functional Z E(f) = 1 e(f) 1 (.1) 1 as dened below M

6 4 J. JOST AND Y.L. XIN where e(f) stands for the energy density. The Euler-Lagrange equation of the energy functional is (f) =0 (.) where (f) is the tension eld. In local e(f) j h (.3) (f) =( M f + g @x (.4) where ; denotes the Christoel symbols of the target manifold N: Here and in the sequel we use the summation convention. For more details on harmonic maps consult [E-L] A Riemannian manifold M is said to be simple, if it can be described by coordinates x on R n with a metric ds = g ij dx i dx j (.5) for which there exist positive numbers and such that jj g ij i j jj (.6) for all x and in R n : In other words, M is topologically R n with a metric for which the associated Laplace operator is uniformly elliptic on R n : Hildebrandt-Jost-Widman proved a Liouville-type theorem for harmonic maps in [H-J-W]: Theorem.1. Let f be a harmonic map from a simple or compact Riemannian manifold M into a complete Riemannian manifold N the sectional curvature of which is bounded above by a constant 0: Denote by B R (Q) a geodesic ball in N with radius R< p which does not meet the cut locus of its center Q: Assume also that the range f(m) of the map f is contained inb R (Q): Then f is a constant map. By using the composition formula for the tension eld, one easily veries that the composition of a harmonic map f : M! N with a convex function : f(m)! R is a subharmonic function on M: The maximum principle then implies Proposition.. Let M be a compact manifold without boundary, f : M! N a harmonic map with f(m) V N: Assume that there exists a strictly convex function on V. Then f is a constant map. Let M! R m+n be an n-dimensional oriented submanifold in Euclidean space. For any point x M by parallel translation in the ambient Euclidean space, the tangent space T x M is moved to the origin of R m+n to obtain an n-subspace in R m+n namely, apoint of the Grassmannian manifold (x) G n m : Thus, we dene a generalized Gauss map : M! G n m : E. Ruh and J. Vilms discovered the relation between the property of the submanifold and the harmonicity of its Gauss map in [R-V].

7 BERNSTEIN TYPE THEOREMS FOR HIGHER CODIMENSION 5 Theorem.3. Let M be a submanifold in R m+n : Then the mean curvature vector of M is parallel if and only if its Gauss map is a harmonic map. Let M! S m+n,! R m+n+1 be an m-dimensional submanifold in the sphere. For any x M by parallel translation in R m+n+1 the normal space N x M of M in S m+n is moved to the origin of R m+n+1 : We then obtain an n-subspace in R m+n+1 : Thus, the so-called normal Gauss map : M! G n m+1 has been dened. There is a natural isometry between G n m+1 and G m+1 n which maps any n-subspace into its orthogonal complementary (m + 1)-subspace. The map = maps any point x M into an (m + 1)-subspace consisting of T x M and the position vector of x: On the other hand, to study properties of the submanifold M in the sphere we may investigate the cone CM generated by M: CM is the image under the map from M [0 1) into R m+n+1 dened by (x t)! tx where x M t [0 1): CM has a singularity att = 0: To avoid the singularity we consider the truncated cone CM " which is the image of M [ " 1 ) under the same map, where " is any positive number. J. Simons showed in [S] that M is a minimal m-submanifold if and only if CM " is an (m + 1)-dimensional minimal cone. In fact, we have (see [X], Prop 3.): Proposition.4. CM " has parallel mean curvature in R m+n+1 if and only if M is a minimal submanifold in S m+n : From Theorem.3 and Proposition.4 it follows that Proposition.5. M is a minimal m;dimensional submanifold in the sphere S m+n if and only if its normal Gauss map : M! G n m+1 is a harmonic map. 3. Geometry of Grassmannian Manifolds Let N be a Riemannian manifold with curvature tensor R( ): Let be a geodesic issueing from x 0 with (0) = x 0 and (t) = x where t is the arc length parameter. Dene a self-adjoint map R _ : w! R(_ w)_: (3.1) Let v be a unit eigenvector of R _(0) with eigenvalue and hv _(0)i =0. Let v(t) be the vector eld obtained by parallel translation of v along. In the case of N being a locally symmetric space with nonnegative sectional curvature, v(t) is an eigenvector of R _(t) with eigenvalue 0 namely R(_(t) v(t)) _ = v(t): Thus, J(t) = ( 1 p sin( p t)v(t) when >0 tv(t) when =0

8 6 J. JOST AND Y.L. XIN is a Jacobi eld along (t) with J(0) = 0. On the other hand, the Hessian of the distance function r from x 0 can be computed by those Jacobi elds. Now, we assume is a geodesic without a conjugate point up to distance r from x 0 : For orthonormal vectors X Y T (r) B r (x 0 ) there exist unique Jacobi elds J 1 and J such that J 1 (0) = J (0) = 0 J 1 (r) =X J (r) =Y since there is no conjugate point of x 0 along. We then have Hess(r)(X Y )=hr X _ Y i = hr J1 _ J ij (r) = = = = Z r 0 Z r 0 Z r 0 Z r 0 (0) d dt hr J 1 _ J i dt (hr _ r J1 _ J i + hr J1 _ r _ J i)dt (hr(j 1 _)_ J i + hr _ J 1 r _ J i)dt d dt hr _J 1 J i;hr _ r _ J 1 + R(_ J 1 )_ J i = hr _ J 1 J i Assume that i and v i (t) are eigenvalues and orthonormal eigenvectors of R _(t). Then J i (t) = 1 p i sin( p i t)v i (t) are n ; 1 orthogonal Jacobi elds, where i > 0: and Hess(r)(J i J j )=hr _ J i J j i = cos( p 1 i r)v i (r) p sin( p j r)v j (r) j = 1 p j sin( p j r) cos( p i r) ij Hess(r)(v i (r) v j (r)) = p i cot( p i r) ij (3.) (In the case i =0 Hess(r)(v i (r) v i (r)) = 1 r ). On the other hand dt Ric( _ _) = X i hr(_ v i )_ v i i = X i i : Let R m+n be an (m + n);dimensional Euclidean space. The set of all oriented n;subspaces (called n;planes) constitutes the Grassmannian manifold G n m,which is the irreducible symmetric space SO(m + n)=so(m) SO(n). Our discussion of the geometry of G n m will be based on the work of Wong [W]. References for other contributions on the geometry of Grassmannian are listed in [H-J-W].

9 BERNSTEIN TYPE THEOREMS FOR HIGHER CODIMENSION 7 Let P and Q be two points in G n m : Wong dened the angles between P and Q as the critical values of the angle between a nonzero vector x in P and its orthogonal projection x in Q as x runs through P: Assume that e 1 e n are orthonormal vectors which span P and f 1 f n for Q: For a nonzero vector x = X x e its orthogonal projection in Q is x = X x f : Thus, for any y in Q we have hx ; x yi =0: Assume that a = he f i : We then have x = X a x and cos = = = DP P E x e x f pp q P = x x P pp qp x qp A x x pp x a a x x a a x x P a x x pp q P x x (3.3) where A = P a a is symmetric in and : It follows that the angles between P and Q are = cos ;1 ( a ) 0 where are the eigenvalues of the symmetric matrix (A ): Let fe e n+i g be a local orthonormal frame eld in R m+n where i j = 1 m = 1 n a b = 1 m + n (say, n m). Let f!! n+i g be its dual frame eld so that the Euclidean metric is g = X! + X i! n+i:

10 8 J. JOST AND Y.L. XIN The Levi-Civita connection forms! ab of R m+n are uniquely determined by the equations d! a =! ab ^! b (3.4)! ab +! ba =0: Now, we consider any point P G n m and a suciently close point Q: The canonical Riemannian metric on G n m can be dened by the sum of the squares of the n angles between P and Q namely ds = X i! n+i (3.5) >From (3.4) and (3.5)itiseasily seen that the curvature tensor of G n m is R i j k l = ik jl + ij kl ; il kj ; ij kl in a local orthonormal frame eld fe i g which is dual to f! n+i g Let _ = x i e i and v = v i e i : Then hr( _ e j )_ vi = x i x k v l hr(e i e j )e k e l i = x i x k v l ( ik jl + ij kl ; il kj ; ij kl ) (3.6) By an action of SO(m) SO(n) = x i x i v j + x j x l v l ; x l x j v l x i = i where P =1: In fact, there exist an n n orthogonal matrix U and an m m matrix U 0, such that UXU 0 = n 1 A where X =(x ai ) is an n m matrix. From (3.1) and (3.6) we have R _ v =( i i v j + j l v l ; j l v l )e j =( v j + j v ; j v )e ( j ( v + v ; v )e when j = =1 n = v se s when j = s = n +1 m: For any n m matrix V,thereis an orthogonal decomposition V =(V 1 0) + (V 0) + (0 V 3 )

11 BERNSTEIN TYPE THEOREMS FOR HIGHER CODIMENSION 9 where V 1 is an n n symmetric matrix, V is an n n skew-symmetric matrix and V 3 is an n (m ; n) matrix. For v = v R _ v e =( ; ) v e : For v = ;v R _ v e =( + ) v e : In summary, R _ has eigenvalues: 1 with multiplicity m ; n. n. with multiplicity m ; n ( + ) with multiplicity 1 ( ; ) with multiplicity 1 0 with multiplicity n ; 1 for each pair and with 6= : From (3.) it follows that the eigenvalues of the Hessian of the distance function r from a xed point at the direction X =(x i )= ( i ) are the same as the ones at X 1 =(j j i ): They are as follows. 1 cot( 1 r) with multiplicity m ; n. n cot( n r). with multiplicity m ; n ( + ) cot( + )r with multiplicity 1 ( ; ) cot( ; )r with multiplicity 1 1 r with multiplicity n ; 1 (3.7) where > 0 without loss of generality. Let P 0 be an oriented n-space in R m+n. We represent itby n vectors e which are complemented by m vectors e n+i, such that fe e n+i g form an orthonormal base of R m+n : Then we can span the n-planes P in a neighborhood U of P 0 by n vectors f : f = e + z i e n+i where (z i ) are the local coordinates of P in U: Let (x i ) = ( i ) be a unit tangent vector at P 0 : The geodesic from P 0 at the direction (x i ) in U is (see [W]) (z i (t)) = 0 tan( 1 t) tan( n t) 1 C A (3.8)

12 10 J. JOST AND Y.L. XIN where t is the arc length parameter and 0 t< j with j j j = max(j 1 j j n j): Thus, a geodesic in G n m between two n;spaces is simply obtained by rotating one into the other in Euclidean space, by rotating corresponding basis vectors. We also see that in the case of codimension > 1 the above zero eigenvalues of the curvature tensor are obtained for a pair of tangent vectors in the Grassmannian that correspond to rotating two given orthogonal basis vectors of an n;plane separately into two dierent mutually orthogonal directions orthogonal to that n;plane. Take as an example the ;plane spanned by e 1 e in R 4 : One tangent direction in G 4 would be to move e 1 into e 3 and keep e xed, and the other tangent direction would move e into e 4 and keep e 1 xed. The largest possible eigenvalue, namely in case of codimension > 1 (e. g. for 1 = = p 1 3 = = n = 0), is realized if one takes two orthogonal directions, say e 1 e in a given n;plane and two other such directions, say f 1 f orthogonal to that n;plane, and the two tangent directions corresponding to rotating e 1 to f 1 e to f and e 1 to f e to ;f 1, respectively. This geometric picture is useful for visualizing our subsequent constructions. Now, let us dene an open set B G (P 0 ) in U G n m : In U we have the normal coordinates around P 0 and then the normal polar coordinates around P 0 : Dene B G (P 0 ) in normal polar coordinates around P 0 as follows: B G (P 0 )= (X t) X =( i ) 0 t<t X = (j 0j + j 0j) (3.9) where 0 and 0 are two eigenvalues with largest absolute values. >From (3.8) we see that B G (P 0 ) lies inside the cut locus of P 0 : We also know from (3.7) that the square of the distance function r from P 0 is a strictly convex smooth function in B G (P 0 ): Remark. The above denition of B G (P 0 ) is for the case of m n > 1: If n = 1 G 1 m is the usual sphere S m and the dened set is the open hemisphere as usual. Let P = (( i ) t) and Q = (( 0 i ) t 0 ) be two points in B G (P 0 ): Then the local expression of P in U is the n m matrix (tan( t) i ) similarly that of Q is (tan( 0 t 0 ) i ) : Consider a curve ; between P and Q dened by (tan( t(1 ; h)+ 0 t 0 h) i ) in U there 0 h 1 is the parameter for ;: We claim that ; is a geodesic. Let P 0 be the middle point dened by t + 0 tan t 0 i in U: In R m+n the n;plane P is spanned by orthogonal vectors f = e + tan( t)e n+

13 BERNSTEIN TYPE THEOREMS FOR HIGHER CODIMENSION 11 whose unit ones are ~f = cos( t)e +sin( t)e n+ : Similarly, P 0 is spanned by n vectors ~f 0 = cos t + 0 t 0 e +sin t + 0 t 0 e n+ : By using (3.3) we obtain the angles between P and P 0 is 1j t ; 0 t 0 j and their distance is d(p P 0 )= 1 s X ( t ; 0 t 0 ) : (3.10) Take any point P 00 between P and P 0 on ; P 00 = (tan( t(1 ; h)+ 0 t 0 h) i ) where 0 <h< 1 : By computation we know that d(p P 00 )=h s X ( t ; 0 t 0 ) (3.11) and d(p 00 P 0 )=( 1 ; h) s X ( t ; 0 t 0 ) : (3.1) (3.10), (3.11) and (3.1) show that the restriction of ; to the segment between P and P 0 is the minimal geodesic. The same holds for the other segment of ;: Hence, the entire curve ; is a geodesic between P and Q: We claim that geodesic ; has the following properties: (1) ; B G (P 0 ) () ; B G (P 0 ): For any h [0 1] ;(h) lies on a geodesic starting from P 0 in the direction ; 1A ( t(1 ; h)+ 0 t 0 h) i where A>0 is the normalizer. By our construction of B G (P 0 ) the radius in this direction is s = A (j 1 t(1 ; h)+ 0 1 t0 hj + j t(1 ; h)+ 0 t0 hj) where for notational simplicity we assume that the rst and the second components are the two with largest absolute values. As P and Q are in B G (P 0 ) and by the conditions for t and t 0 j 1 t(1 ; h)+ 0 1t 0 hj + j t(1 ; h)+ 0 t 0 hj (j 1 j + j j)t(1 ; h)+(j 0 1j + j 0 j)t 0 h < (j 1j + j j) (j 0 + j 0j) (1 ; h)+ (j0 1j + j 0 j) (j 0 + j 0 0 j) h 0

14 1 J. JOST AND Y.L. XIN so s>a which means ;(h) B G (P 0 ) and we conrmed the rst claim. Now, we move the origin of the coordinates from P 0 to P 0 : The n;plane P 0 is spanned by n orthonormal vectors Dene f 0 = cos t + 0 t 0 f 0 n+ = ; sin t + 0 t 0 e +sin t + 0 t 0 e n+ : e + cos t + 0 t 0 e n+ and the remaining m ; n vectors do not change. Thus, ff f 0 n+i 0 g forms an orthonormal base of R m+n. Each point ;(h) of the geodesic ; is an n;plane in R m+n. It is spanned by cos( t(1 ; h)+ 0 t 0 h)e +sin( t(1 ; h)+ 0 t 0 h)e n+ : By the above changed base, ;(h) is also spanned by cos( t( 1 ; h) ; 0 t 0 ( 1 ; h))f 0 + sin( t( 1 ; h) ; 0 t 0 ( 1 ; h))f 0 n+ which means that the geodesic ; in the coordinate neighborhood U 0 around P 0 can be described by tan( t( 1 ; h) ; 0 t 0 ( 1 ; h)) i : Its tangent direction at P 0 is (( 0 t 0 ; t) i ) : By a similar argument we obtain the second property, that is ; B G (P 0 ): These two properties of the geodesic ; mean that any minimal geodesic from P to Q lies in B G (P 0 ) and has length less than t X where X is the unit tangent vector of at P: On the other hand, we already computed all Jacobi elds along any geodesic in G n m which means that any geodesic from P of length < t X has no conjugate points and that the squared distance function from P remains strictly convex along this geodesic for length <t X : Thus, by a similar reasoning as Prop..4.1 in [J] we conclude that B G (P 0 ) shares all the properties of the usual convex geodesic ball. In fact, the situation here is even simpler because, in a symmetric space, by a result of Crittenden [Cr], the rst cut point along a geodesic always has to be a conjugate point. In summary, we have

15 BERNSTEIN TYPE THEOREMS FOR HIGHER CODIMENSION 13 Theorem 3.1. In B G (P 0 ) the square of the distance function from its center P 0 is a smooth strictly convex function. Furthermore, B G (P 0 ) is a convex set, namely any two points in B G (P 0 ) can be joined in B G (P 0 ) by a unique geodesic arc. This arc is the shortest connection between its end points and thus in particular does not contain a pair of conjugate points. Remark. In the Grassmannian manifold there is the usual convex geodesic ball B R (P 0 ) of radius R< ( p when min(m n) > 1 when min(m n) =1: >From (3.9) it is seen that B R (P 0 ) B G (P 0 ): Let P (t) be any n;plane in U of P 0 which is spanned by where z i is dened by (3.8). Let f = e + z i e n+i ~f 1 = cos( 1 t)f 1 ~ fn = cos( n t)f n : Since jf j = 1 cos( t) the vectors ~ f 1 ~ f n are orthonormal. Therefore, we can dene the inner product hp 0 Pi of n;planes P 0 = e 1^^e n and P = ~ f 1 ^^ ~f n by It follows that hp 0 P(t)i = det 0 E hp 0 Pi =det De f ~ : cos( 1 t) 0 cos( t)... 0 cos( n t) 1 C Y n A = cos( t): =1 Theorem 3.. maxfhp 0 Pi G g = 1 : (3.13) Proof. Suppose now for notational simplicity that 1 n > n =1: Let us maximize f( 1 n )= ny =1 cos : ( 1 + )

16 14 J. JOST AND Y.L. XIN Let F ( 1 n )= ny =1 cos where is a Lagrange multiplier. Then F 1 = ; + + ( 1 + ) ( 1 + ) sin 1 ( 1 + ) cos ( 1 + ) ( 1 + ) cos 1 ( 1 + ) sin ( 1 + ) nx t=1 + (1 ; ny ny s=3 s=3 cos cos nx =1 ) s ( 1 + ) s ( 1 + ) t ( 1 + ) cos 1 ( 1 + ) cos ( 1 + ) sin t ( 1 + ) ny s6=t s=3 cos s ( 1 + ) ; 1 = ; ( 1 + ) tan 1 ( 1 + ) f()+ ( 1 + ) tan ( 1 + ) f() + nx t=3 At a critical point of f t ( 1 + ) tan t ( 1 + ) f() ; 1: 1 f() = ; ( 1 + ) tan 1 ( 1 + ) + ( 1 + ) tan ( 1 + ) + nx t=3 t ( 1 + ) tan t ( 1 + ) (3.14) and similarly, f() = 1 ( 1 + ) tan 1 ( 1 + ) ; 1 ( 1 + ) tan ( 1 + ) nx (3.15) t + ( 1 + ) tan t ( 1 + ) : t=3 Adding both sides of (3.14) and (3.15) gives ( 1 + ) f() = ( 1 ; ) ( 1 + ) tan 1 ( 1 + ) ; ( 1 ; ) ( 1 + ) tan ( 1 + ) nx (3.16) t + ( 1 + ) tan t ( 1 + ) : t=3 If < 1, then (3.16) means >0:

17 BERNSTEIN TYPE THEOREMS FOR HIGHER CODIMENSION 15 On the other hand, subtracting (3.15) from (3.14) gives ( 1 ; ) f() = ; ( 1 + ) tan 1 ( 1 + ) + ( 1 + ) tan ( 1 + ) : (3.17) If < 1 the left side of (3.17) is positivebutitsright side is negativewhich gives a contradiction. Consequently, the critical points of f occur only when 1 = : Now, from (3.16) we have 4 1 X n f() = t tan t : (3.18) 4 t= We also have F t = ; ( 1 + ) ny 6=t =1 sin where t =3 n: At a critical point ( 1 n ) of f ; ( 1 + ) ny 6=t =1 sin t ( 1 + ) cos ( 1 + ) ; t t ( 1 + ) cos ( 1 + ) ; t =0: It follows that t f() = ; tan t : (3.19) If there exists t such that t > 0, then from (3.19) wehave <0 whichcontradicts with (3.18). Consequently, the only critical point of f() occurs when 1 = = p 3 = = n =0 and f( 1 n )= ny =1 cos ( 1 + ) 1 : Q.E.D.

18 16 J. JOST AND Y.L. XIN 4. Proofs of the Main Theorems Theorem 4.1. Let M be an n;dimensional simple Riemannian manifold which is immersed in Euclidean space R m+n with parallel mean curvature. Let : M! G n m be Gauss map. Suppose that there exists a xed oriented n;plane P 0, and a number 0 1 when min(m n) 0 > 0 when min(m n) =1 such that hp P 0 i 0 (4.1) holds for all point P (M) G n m : Then M has to be an n;dimensional ane linear subspace. Proof. : M! G n m is harmonic by Thm.3. The point now is to show that such a harmonic map is constant if its range is contained in a region B G (P 0 ): In the case where B G (P 0 ) is replaced by a geodesiccally convex ball, this was shown by an iteration argument in [H-J-W]. The properties of B G (P 0 ) as established in x3 are strong enough to make that iteration technique still applicable (for example, a general version of that iteration technique that directly applies here has been given in [G-J]). Alternatively, at least in the case of minimal graphs, one uses Allard's result [A] mentioned in the introduction that the tangent cone at innity of M is the cone over a compact minimal submanifold M 0 of the sphere. The Gauss map of M 0 again is a harmonic map with values in a region B G (P 0 ). Since B G (P 0 ) supports a strictly convex function by Thm 3.1, the composition of that Gauss map with that function is a subharmonic function on M 0 which then has to be constant as M 0 is compact, as in [FC]. Possibly, there is even a third method to reach the conclusion. Namely, it is quite likely that Kendall's result [K] can be generalized to show that for any Q B G (P 0 ) there exists a strictly convex function on all of B G (P 0 ) with its minimum at Q: If that can be shown, the Holder estimates and Liouville theorems for harmonic maps with values in B G (P 0 ) can be shown by a considerably simplied version of the method of [H-J-W], namely in a single step without the need to iterate. Although this would yield the simplest proof, we refrain here from studying the technical details as we have already described two other ways to reach the conclusion that the Gauss map is constant. Obviously, if the Gauss map is constant, the submanifold has to be ane linear. Q.E.D. We are now in a position to prove Theorem 1 and Theorem stated in the introduction. Proof of Theorem 1 Since M = (x f(x)) is a graph in R m+n dened by m fuctions, the induced metric on M is ds = g dx dx

19 BERNSTEIN TYPE THEOREMS FOR HIGHER CODIMENSION 17 where g = + X @x : It is obvious that the eigenvalues of the matrix (g ) at each point are 1: The condition (1.) implies that the eigenvalues of the matrix (g ) are 0: The condition (.6) is satised and M is a simple Riemannian manifold. Let fe e n+i g be the standard orthonormal base of R m+n : Choose P 0 as an n;plane spanned by e 1 ^^e n : At each point inm its image n;plane P under the Gauss map is spanned by f = e n+i: It follows that and jf 1 ^^f n j = det( + X i f = jf 1 ^^f n i ) The n; plane P is also spanned by p = ; 1 n f f furthermore, We then have jp 1 ^^p n j =1: hp P 0 i = det(he p i) = 0 = ;1 f ; 1 n f ; 1 n f 1 C A which is 1 0 > 1 Proof of Theorem by (1.1) and (1.). Thus Theorem 1 follows from Thorem 4.1. >From Proposition.5 the normal gauss map : M! G n m+1 is harmonic. The condition (1.5) means that the image (M) under the normal Gauss map is contained inside of B G (P 0 ) G n m+1 : Theorem.1 and Theorem 3.1 imply that is constant. We thus complete the proof of Theorem.

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