CONVEX FUNCTIONS ON GRASSMANNIAN MANIFOLDS AND LAWSON-OSSERMAN PROBLEM. 1. Introduction

Transcription

1 CONVEX FUNCTIONS ON GRASSANNIAN ANIFOLDS AND LAWSON-OSSERAN PROBLE Y. L. XIN AND LING YANG arxiv: v1 [math.dg] 6 Jun 008 Abstract. We derive estimates of the Hessian of two smooth functions defined on Grassmannian manifold. Based on it, we can derive curvature estimates for minimal submanifolds in Euclidean space via Gauss map as [4]. In this way, the result for Bernstein type theorem done by Jost and the first author could be improved. 1. Introduction The celebrated theorem of Bernstein [] says that the only entire minimal graphs in Euclidean 3-space are planes. Its higher dimensional generalization was finally proved by J. Simons [19], which says that an entire minimal graph has to be planar for dimension 7, while Bombieri- De Giorgi-Giusti [3] shortly afterwards produced a counterexample to such an assertion in dimension 8 and higher. Schoen-Simon-Yau [18] gave us a direct proof for Bernstein type theorems for n 5 dimensional minimal graphs with the aid of curvature estimates for stable minimal hypersurfaces. There is a weak version of Bernstein type theorem in arbitrary dimension. It was J. oser [15] who proved that the entire solution f to the minimal surface equation is affine linear, provided f is uniformly bounded. Afterward Ecker-Huisken [9] obtained curvature estimates by a geometric approach, as a corollary oser s result had been improved for the controlled growth of f. oser s theorem had been generalized to certain higher codimensional cases by Chern-Osserman [6] for dimension and Babosa, Fischer-Colbrie for dimension 3 [1], [10]. But the counterexample constructed by Lawson-Osserman [13] prevents us going further. They also raised in the same paper a question for finding the best constant possible in the theorem. In contrast, the first author with J. Jost [1] proved the following Bernstein type theorem without the restriction of dimension and codimension, which is an improvement of the work done by Hildebrandt-Jost-Widman [11] athematics Subject Classification. 49Q05, 53A07, 53A10. The research was partially supported by NSFC and SFECC. 1

2 Y.L. XIN AND LING YANG Theorem 1.1. Let z = f x 1,,x n ), = 1,,m, be smooth functions defined everywhere in R n. Suppose their graph = x,fx)) is a submanifold with parallel mean curvature in R n+m. Suppose that there exists a number β 0 with { when m, 1.1) β 0 < when m = 1; such that 1.) f = [ det δ ij + )]1 f f x i x j β 0. Then f 1,,f m has to be affine linear representing an affine n-plane. The key point of the proof is to find a geodesic convex set B JX P 0 ) = { P G n,m : sum of any two Jordan angles between P and P 0 < π in a geodesic polar coordinate of the Grassmannian manifold, where P 0 denotes a fixed n plane. It is larger than the largest geodesic convex ball of radius π in 4 G n,m. The geometric meaning of the condition of the above result is that the image under the Gauss map of lies in a closed subset S B JX P 0 ). Recently, the authors [4] studied complete minimal submanifolds whose Gauss image lies in an open geodesic ball of radius π. They carried out the Schoen- 4 Simon-Yau type curvature estimates and the Ecker-Huisken type curvature estimates, and on the basis, the corresponding Bernstein type theorems with dimension limitation or growth assumption could be derived. It is natural to study the situation when β 0 in the condition 1.1) and 1.) of the Theorem 1.1 approach to. The present paper will devote to this problem. We shall follow the mainidea of our previous paper [4]. But, we view now the Grassmannian manifolds as submanifolds in Euclidean space via Plücker imbedding. The auxiliary functions are constructed from this viewpoint. As shown before, B JX P 0 ) is defined in a coordinate neighborhood U of the Grassmannian G n,m. We introduce two functions v and u in U. Via the Gauss map we can obtain useful functions on our minimal n submanifold in R m+n with m. Then, we can carry out the Schoen-Simon-Yau type curvature estimates and the Ecker-Huisken type curvature estimates, which enable us to get the corresponding Bernstein type theorems and other geometrical conclusions. In Section, we give some facts of a Grassmannian manifold G n,m, which can be isometric imbedding into a Euclidean space. There is the height function for a submanifold in Euclidean space. Such a height function is called w function on G n,m. Then we have an open domain of U G n,m, where the w function is positive. Every point in U has a one-to-one correspondence to an n m matrix. We describe canonical metric and the corresponding connection on U with respect }

3 CONVEX FUNCTIONS AND LAWSON-OSSERAN PROBLE 3 to the coordinate. On the basis, the Hessian of an arbitrary smooth function could be calculated. In Section 3 we define v = 1 on U. We also define another function u on U. In w the section, we shall show v and u are convex on B JX P 0 ) and give estimates of the Hessian of them. The estimates are quite delicate. We use the radial compensation technique to accurate the estimates. In Section 4, we define four auxiliary functions, h 1, h, h 3 and h 4. They are defined on the minimal submanifolds of R n+m whose Gauss image is confined, and they are expressed in term of v and u. We also estimate the Laplacian of them, which is useful for the next sections. Later in Section 5, not only we give the Schoen-Simon-Yau type curvature estimateswiththeaidof h 1 and h 3, butalsoweobtaintheecker-huisken typecurvature estimates with the aid of h and h 4. Our method is completely similar to the previous paper [4], so we only describe the outline of process. From the estimates several geometrical conclusions follow, including the following Bernstein type theorems. Theorem 1.. Let = x,fx)) be an n-dimensional entire minimal graph given by m functions f x 1,,x n ) with m,n 4. If [ f = det δ ij + )]1 f f <, x i x j then f has to be affine linear functions representing an affine n-plane. Theorem 1.3. Let = x,fx)) be an n-dimensional entire minimal graph given by m functions f x 1,,x n ) with m. If [ f = det δ ij + )]1 f f <, x i x j and 1.3) f ) 1 = or 4 3 ), where R = x + f. Then f has to be affine linear functions and hence has to be an affine linear subspace. Those are what shall be done in Section 6. It is worthy to note that Theorems still hold true when is a submanifold with parallel mean curvature. Dong generalized Chern s result [5] [4] to higher codimension, which states that a graphic submanifold = x,fx)) with parallel mean curvature has to be minimal if the slope of f is uniformly bounded. Hence our results improve Theorem 1.1. It is natural to ask what is the relations between the results here ) and that )) of the previous paper [4]. Since the v function varies in sec,sec p π 4 π p on the open geodesic ball of radius 4 in G n,m, where p = minn,m), the results

4 4 Y.L. XIN AND LING YANG of the present article do not generalize those in the previous one. Both results are complementary.. Preliminaries on the Grassmannian manifold G n,m Let R n+m be an n + m-dimensional Euclidean space. All oriented n-subspaces constitute the Grassmannian manifolds G n,m, which is an irreducible symmetric space of compact type. Fix P 0 G n,m in the sequel, which is spanned by a unit n vector ε 1 ε n. For any P G n,m, spanned by a n vector e 1 e n, we define an important function on G n,m w def. = P,P 0 = e 1 e n,ε 1 ε n = detw, where W = e i,ε j ). It is well known that W T W = O T ΛO, where O is an orthogonal matrix and µ 1 0 Λ =..., p = minm,n), 0 µ p where each 0 µ i 1. The Jordan angles between P and P 0 are defined by θ i = arccosµ i ). Denote U = {P G n,m : wp) > 0}, let {ε n+ } be m-vectors such that {ε i,ε n+ } form an orthornormal basis of R m+n. Then we can span arbitrary P U by n vectors f i : f i = ε i +z i ε n+, where Z = z i ) are the local coordinate of P in U. Here and in the sequel we use the summation convention and agree the range of indices: 1 i,j,k,l n; 1,β,γ,δ m. The canonical metric on G n,m in the local coordinate can be described as see [] Ch. VII).1) g = tr I n +ZZ T ) 1 dzi m +Z T Z) 1 dz T). Let P U determined by an n m matrix Z 0 = λ δ i ), where λ = tanθ and θ 1,,θ m be the Jordan angles between P and P 0. Here and in the sequel we

5 CONVEX FUNCTIONS AND LAWSON-OSSERAN PROBLE 5 assume n m without loss of generality; for it is similar for n < m.) Let X,Y,W denote arbitrary n m matrices. Then.1) tells us X,Y P = tr I n +Z 0 Z0 T ) 1 XI m +Z0 T Z 0) 1 Y T).) = 1+λ i ) 1 1+λ ) 1 X i Y i. i, Note that if m+1 i n, λ i = 0.) Furthermore, from In +Z 0 +tw)z 0 +tw) T) 1 X I m +Z 0 +tw) T Z 0 +tw) ) 1 Y T = I n +Z 0 Z T 0 +twz T 0 +Z 0 W T )+Ot ) ) 1 X Im +Z T 0 Z 0 +tw T Z 0 +Z T 0 W)+Ot ) ) 1 Y T = I n +ti n +Z 0 Z T 0 ) 1 WZ T 0 +Z 0W T )+Ot ) ) 1 I n +Z 0 Z T 0 ) 1 X Im +ti m +Z T 0 Z 0 ) 1 W T Z 0 +Z T 0 W)+Ot ) ) 1 Im +Z T 0 Z 0 ) 1 Y T = I n ti n +Z 0 Z T 0 ) 1 WZ T 0 +Z 0W T )+Ot ) ) I n +Z 0 Z T 0 ) 1 X Im ti m +Z T 0 Z 0) 1 W T Z 0 +Z T 0 W)+Ot ) ) I m +Z T 0 Z 0) 1 Y T = I n +Z 0 Z T 0 ) 1 XI m +Z T 0 Z 0) 1 Y T t [ I n +Z 0 Z T 0 ) 1 WZ T 0 +Z 0W T )I n +Z 0 Z T 0 ) 1 XI m +Z T 0 Z 0) 1 Y T we have.3) +I n +Z 0 Z T 0 ) 1 XI m +Z T 0 Z 0) 1 W T Z 0 +Z T 0 W)I m +Z T 0 Z 0) 1 Y T] +Ot ), W X,Y P = tr [ I n +Z 0 Z T 0 ) 1 WZ T 0 +Z 0W T ) I n +Z 0 Z T 0 ) 1 XI m +Z T 0 Z 0) 1 Y T +I n +Z 0 Z T 0 ) 1 XI m +Z T 0 Z 0) 1 W T Z 0 +Z T 0 W)I m +Z T 0 Z 0 ) 1 Y T]. We let E i be the matrix with 1 in the intersection of row i and column and 0 otherwise. Denote g i,jβ = E i,e jβ and let g i,jβ) be the inverse matrix of gi,jβ ). Denote by the Levi-Civita connection with respect to the canonical matric on G n,m, and by Then from.),.4) and obviously Ei E jβ = Γ kγ i,jβ E kγ. g i,jβ P) = 1+λ i ) 1 1+λ ) 1 δ β δ ij.5) g i,jβ P) = 1+λ i )1+λ )δ βδ ij. oreover, a direct calculation from.3) and.5) shows.6) Γ kγ i,jβ = 1 gkγ,lδ E lδ E i,e jβ +E i E jβ,e lδ +E jβ E lδ,e i ) = λ 1+λ ) 1 δ j δ βγ δ ik λ β 1+λ β) 1 δ βi δ γ δ jk.

6 6 Y.L. XIN AND LING YANG From.5), we see that.7) 1+λ i )1 1+λ ) 1 Ei 1 i n,1 m) form an orthonormal basis of T P G n,m. Denote its dual basis in TP G n,m by.8) ω i 1 i n,1 m), then.9) g = i, ω i at P. 3. Hessian estimates of two smooths functions on G n,m On U, w > 0, then we can define 3.1) v = w 1 on U. For arbitrary Q U determined by an n m matrix Z, it is easily seen that 3.) vq) = [ m deti n +ZZ T ) ]1 = secθ. =1 where θ 1,,θ m denotes the Jordan angles between Q and P 0. Now we calculate the Hessian of v at P whose corresponding matrix is Z 0. At first, by noting that for any n n orthogonal matrix U and m m orthogonal matrix V, Z UZV induces an isometry of U which keeps v invariant, we can assume Z 0 = λ δ i ) without loss of generality, where λ = tanθ and θ 1,,θ m denotes the Jordan angles between P and P 0. We also need a Lemma as follows. Lemma 3.1. Let be a manifold, A be a smooth nonsingular n n matrix-valued function on, X,Y be local tangent fields, then 3.3) X logdeta = tr X A A 1 ) and 3.4) Y X logdeta = tr Y X A A 1 ) tr X A A 1 Y A A 1 ). Proof. Assume that e 1,,e n is a standard basis in R n, then deta e 1 e n = Ae 1 Ae n.

7 CONVEX FUNCTIONS AND LAWSON-OSSERAN PROBLE 7 Hence X deta e 1 e n = i Ae 1 Ae i 1 X Ae i Ae i+1 Ae n = i Ae 1 Ae i 1 X A A 1 )Ae i Ae i+1 Ae n =tr X A A 1 )Ae 1 Ae n =tr X A A 1 )deta e 1 e n. Thereby 3.3) immediately follows. 3.4) follows from 3.3) and A Y A 1 + Y A A 1 = Y AA 1 ) = 0. Now we let = U, AZ) = I n + ZZ T, then logv = 1 logdeta. A direct calculation shows X A = XZ T +ZX T, Y X A = XY T +YX T. Hence we compute from Lemma 3.1 that at P X logv = 1 tr XZ0 T +Z 0X T )I n +Z 0 Z0 T ) 1) = λ 1+λ ) 1 X, X Y logv = 1 tr XY T +YX T )I n +Z 0 Z0 T ) 1) 1 tr XZ0 T +Z 0X T )I n +Z 0 Z0 T ) 1 YZ0 T +Z 0Y T )I n +Z 0 Z0 T ) 1) = 1+λ i) 1 X i Y i i, = = 1 m+1 i n, XZ0 T +Z 0 X T ) ij 1+λ j) 1 YZ0 T +Z 0 Y T ) ji 1+λ i) 1 i,j X i Y i λ ) 1 X β Y β λ β ) 1 X β Y β,β 1 λ β X β +λ X β )1+λ β ) 1 λ Y β +λ β Y β )1+λ ) 1,β λ 1+λ ) 1 X i Y i m+1 i n, m+1 i n,1+λ ) 1 X i Y i +,β,β 1+λ ) 1 1+λ β ) 1 X β Y β

8 8 Y.L. XIN AND LING YANG,β λ λ β 1+λ ) 1 1+λ β ) 1 X β Y β. Furthermore, X v =v X logv = λ 1+λ ) 1 X )v, X Y v =v X Y logv+ X logv Y logv) = m+1 i n,1+λ ) 1 X i Y i + 1+λ ) 1 1+λ β) 1 X β Y β,β + ) λ λ β 1+λ ) 1 1+λ β ) 1 X Y ββ X β Y β ) v,β = 1+λ i ) 1 1+λ β ) 1 X iβ Y iβ i,β +,β ) λ λ β 1+λ ) 1 1+λ β) 1 X Y ββ X β Y β ) v. In particular, 3.5) Ei vp) = λ 1+λ ) 1 vδ i and 1+λ i ) 1 1+λ ) 1 v i = j, = β; λ 3.6) Ei Ejβ vp) = λ β 1+λ ) 1 1+λ β ) 1 v i = β,j =, β; λ λ β 1+λ ) 1 1+λ β ) 1 v i =,j = β, β; 0 otherwise. Then, from.6), 3.5) and 3.6) we obtain Hessv)E i,e jβ )P) = Ei Ejβ v Ei E jβ )v 3.7) = Ei Ejβ v Γ kγ i,jβ E kγ v 1+λ i) 1 1+λ ) 1 v i = j, = β,i ; 1+λ )1+λ ) v i = j = = β; = λ λ β 1+λ ) 1 1+λ β ) 1 v i = β,j =, β; λ λ β 1+λ ) 1 1+λ β ) 1 v i =,j = β, β; 0 otherwise.

9 In other words CONVEX FUNCTIONS AND LAWSON-OSSERAN PROBLE 9 Hessv) P = v ωi + 1+λ )v ω + λ β vω ω ββ +ω β ω β ) i βλ = ωi m+1 i n,v + 1+λ )v ω + λ β v ω ω ββ βλ 3.8) + [ ) 1+λ λ β )v ω β +ω β ) <β ) ] +1 λ λ β )v ω β ω β ). 3.8) could be simplified further. Please note 3.5), which also tells us 3.9) dv = λ v ω ; then 3.10) dv dv = λ v ω + βλ λ β v ω ω ββ. Substituting 3.10) into 3.8) yields Hessv) P = ωi m+1 i n,v + 1+λ )v ω +v 1 dv dv + [ ) 3.11) 1+λ λ β )v ω β +ω β ) <β ) ] +1 λ λ β )v ω β ω β ). Note that λ 0 and 1 λ λ β = 1 tanθ tanθ β = cosθ+θ β) cosθ cosθ β ; which implies that Hessv) P is positive definite if and only if θ +θ β < π for arbitrary β, i.e., P B JX P 0 ). By 3.), v = 1+λ )1, then λ λ β [ 1+λ )1+λ β) ]1 1 v 1, the equality holds if and only if λ = λ β and λ γ = 0 for each γ,β. Hence, we have 1 λ λ β v. Finally we arrive at an estimate 3.1) Hessv) v v)g +v 1 dv dv. Now we introduce another smooth function on U. For any Q U, 3.13) u def. = tanθ.

10 10 Y.L. XIN AND LING YANG where θ 1,,θ m denotes the Jordan angles between Q and P 0. Denote by Z the coordinate of Q, then it is easily seen that 3.14) uq) = trzz T ). We can calculate the Hessian of u at P U whose corresponding matrix is Z 0 in the same way. Similar to above, we can assume Z 0 = λ a δ i ), where λ = tanθ and θ 1,,θ m are the Jordan angles between P and P 0. Obviously 3.15) Then, at P X u =trxz T )+trzx T ), X Y u =trxy T )+tryx T ). Hessu)E i,e jβ ) = Ei Ejβ u Ei E jβ )u = Ei Ejβ u Γ kγ i,jβ E kγ u =δ ij δ β + λ 1+λ ) 1 δ j δ βγ δ ik +λ β 1+λ β) 1 δ βi δ γ δ jk ) 3.16) In other words λ γ δ kγ [ =δ ij δ β +λ λ β 1+λ ) 1 +1+λ β ) 1] δ j δ βi i = j, = β,i ; = +4λ [ 1+λ ) 1 i = j = = β; λ λ β 1+λ ) 1 +1+λ β ) 1] i = β,j =, β. Hessu) P = i 1+λ i)1+λ )ω i + +6λ )1+λ )ω + βλ λ β +λ +λ β )ω β ω β 3.17) = m+1 i n,1+λ i)1+λ )ω i + +6λ )1+λ )ω + [ 1+λ )1+λ β )+λ λ β +λ +λ β )][ ω β +ω β ) + [ ] 1+λ )1+λ β ) λ λ β +λ +λ β )][ ω β ω β ) ] By computing, 3.18) [ 1+λ )1+λ β ) λ λ β +λ +λ β )] = 1 λ λ β )λ +λ β λ λ β +1). It is positive if and only if 1 λ λ β = 1 tanθ tanθ β = cosθ+θ β) cosθ cosθ β 0, i.e., θ +θ β < π. Hence Hessu) P is positive definite if and only P B JX P 0 ).

11 CONVEX FUNCTIONS AND LAWSON-OSSERAN PROBLE 11 oreover, the right side of 3.18) can be estimated by 1 λ λ β )λ +λ β λ λ β +1) 1 λ +λ β)λ +λ β = 1 λ +λ β ) ) u ) = 1 u. +1 ) Here we used the fact u = tan θ = λ.) By combining it with 3.17) and.9), we arrive that 3.19) Hessu) 1 ) u g. For later applications the estimates 3.1) and 3.19) are not accurate enough. Using the radial compensation technique we could refine those estimates which are based on the following lemmas. Lemma 3.. Let V be a real linear space, h be a nonnegative definite quadratic form on V and ω V. V = V 1 V, h is positive definite on V 1, hv 1,V ) = 0 and ωv ) = 0. Denote by ω the unique vector in V 1 such that for any z V 1, Then we have ωz) = hω,z). 3.0) h ωω ) 1 ω ω. Proof. For arbitrary y V, there exist λ R, z 1 V 1 and z V, such that y = λω +z 1 +z and hω,z 1 ) = 0. Then and hy,y) = λ hω,ω )+hz 1,z 1 )+hz,z ) λ hω,ω ) = λ ωω ) Hence 3.0) holds. ωω ) 1 ω ωy,y) = ωω ) 1 ωy) = λ ωω ). Lemma 3.3. Let Ω be a compact and convex subset of R k, such that for every σ Sk) and x = x 1,,x k ) Ω, 3.1) T σ x) = x σ1),,x σk) ) Ω; where Sk) denotes the permutation group of {1,,k}. If f : Ω R is a symmetric C function, and D f) is nonpositive definite everywhere in Ω, then there exists x 0 = x 1 0,,x k 0) Ω, such that x 1 0 = x 0 = = x k 0 and 3.) fx 0 ) = supf. Ω

12 1 Y.L. XIN AND LING YANG Proof. By the compactness of Ω, there exists x = x 1,,x k ) Ω, such that fx) = sup Ω f. Furthermore we have f T σ x) ) = fx) = supf Ω σ Sk) from the fact that f is symmetric. Denote by C σ x) the convex closure of {T σ x) : σ Sk)}, then C σ x) Ω and fy) sup σ Sk) f T σ x) ) = sup Ω f for arbitrary y C σ x), since D f) 0; which implies Denote x 1 0 = = xk 0 = 1 k k i=1 xi, then f sup Cσx) f. Ω x 0 = x 1 0,,x k 0) = 1 k From which 3.) follows. k x s,x s+1,,x k,x 1,x,,x s 1 ) C σ x); s=1 By 3.1), 3.3) h def. = Hessv) v v)g v 1 dv dv is nonnegative definite on T P G n,m. Denote 3.4) V 1 = E, V = i E i ; then T P G n,m = V 1 V, and 3.11),.9), 3.9) tell us hv 1,V ) = 0, dvv ) = 0 and 3.5) h V1 = v 1+λ )v ω. ispositivedefinite. Denoteby v theuniqueelement inv 1 such thatforanyx V 1, h v,x) = dvx). From 3.5) and 3.9), it is not difficult to obtain and v = λ 1+λ ) E v 1+λ 3.6) dv v) = λ v. v 1+λ

13 CONVEX FUNCTIONS AND LAWSON-OSSERAN PROBLE 13 Then Lemma 3. and 3.3) tell us 3.7) Hessv) v v)g + [1+ λ ) ] 1 v 1 dv dv. v 1+λ It is necessary to estimate the upper bound of 3.8) ν = log1+λ ), then λ = ; since v = 1+eν 1+λ, )1 logv = 1 log1+λ ) = 1 and Now we define λ v 1+λ = 3.9) Ω = { ν 1,,ν m ) R m : ν 0, λ. Denote v 1+λ ν 1+e ν v +e ν. ν = logv }, and f : Ω R by ν 1,,ν m ) 1+e ν v +e ν. Then obviously Ω is compact and convex, T σ Ω) = Ω for every σ Sm) cf. Lemma 3.3), f is a symmetric function and a direct calculation shows i.e., Then from Lemma 3.3, sup Ω f ν ν β = v 1)eν v e ν ) v +e ν ) 3 δ β ; D f) 0 when v 1,]. f = f logv m,, logv m λ v 1+λ which is an upper bound of Hessv) v v)g + In summary, we have the following Proposition. ) m 1+v m) = ; v +v m. Substituting it into 3.7) gives ) dv dv. v 1 mvv m 1) + m+1 mv Proposition 3.1. v is a convex function on B JX P 0 ) U G n,m, and v ) Hessv) v v)g + + p+1 ) dv dv pvv p 1) pv on {P U : vp) }, where p = minn,m).

14 14 Y.L. XIN AND LING YANG Similarly, we consider 3.31) h def. = Hessu) 1 u )g; which is nonnegative definite on T P G n,m. The definition of V 1 and V is similar to above. It is easily seen from 3.17) and.9) that hv 1,V ) = 0 and 3.3) h V1 = 8λ +6λ4 + 1 u )ω is positive definite. By 3.15), 3.33) du = λ 1+λ )ω, then duv ) = 0. Hence Lemma 3. can be applied for us to obtain 3.34) Hessu) 1 u )g + du u) ) 1 du du. where u denotes the unique element in V 1 such that for arbitrary X V 1, h u,x) = dux). From 3.3) and 3.33), we can derive and hence u = 3.35) du u) = λ 1+λ ) 8λ +6λ4 + 1 ue, λ 1+λ ) 3λ 4 +4λ u. 3.34) tells us it is necessary for us to estimate the upper bound of the right side of 3.35). Define Ω = {ν 1,,ν m ) R m : ν = u} and f : Ω R ν 1,,ν m ) ν 1+ν ) 3ν +4ν +C where C = 1 4 u. Thenitiseasytoseethatsupf isanupperboundofdu u),sinceu = tan θ = λ. Obviously Ω is compact and convex, T σ Ω) = Ω for every σ Sm), f is a symmetric function and a direct calculation shows f = 4 [ 3C 1)ν 3 +6Cν +9C 3C )ν +4C C ] δ ν ν β 3ν +4ν +C) 3 β.

15 CONVEX FUNCTIONS AND LAWSON-OSSERAN PROBLE 15 To show D f) 0 when u 0,], it is sufficient to prove F : [0,u] R t 3C 1)t 3 +6Ct +9C 3C )t+4c C is a nonnegative function, where C = u 4 0,1]. If F attains its minimum at t 0 0,u), then 3.36) 3.37) 0 = F t 0 ) = 33C 1)t 0 +1Ct 0 +9C 3C, 0 F t 0 ) = 63C 1)t 0 +1C. On the other hand, when 3C 1 0, we have F t 0 ) 9C 3C > 0, which causes a contradiction; when 3C 1 < 0, from 3.37), t 0 C, then 1 3C F t 0 ) F 0) = 9C 3C > 0, which also causes a contradiction. Therefore In conjunction with F0) = 4C C > 0 min F = min{ F0),Fu) }. [0,u] Fu) = 3C 1)u 3 +6Cu +9C 3C )u+4c C = 9 16 u u u3 +u > 0, F is a nonnegative function. Thereby applying Lemma 3.3 we have 3.38) du u) supf = f u m,, u m ) = u+m) m )u+4m. Substituting 3.38) into 3.34) gives Hessu) 1 ) u g m )u+4m du du. u+m) We rewrite the conclusion as the following Proposition. Proposition 3.. u is a convex function on B JX P 0 ) U G n,m and 3.39) Hessu) 1 ) u g p )u+4p du du u+p) on {P U : up) }, where p = minn,m). 4. The construction of auxiliary functions Let 4.1) h 1 = v k v) k,

16 16 Y.L. XIN AND LING YANG where k > 0 to be chosen, then h 1 = kv k 1 v) k kv k v) k 1 = kv k 1 v) k 1, h 1 =kk +1)v k v) k 1 +kk 1)v k 1 v) k =4kv k v) k k +1 v). Here denotes derivative with respect to v. Hence, from 3.30) Hessh 1 ) = kv k 1 v) k 1 Hessv)+4kv k v) k k +1 v)dv dv 4.) kv k v) k g kv k v) k [ v 1) v) pv p 1) + p+1 p v) k +1 v) ] dv dv. Please note that v 1 v p 1 even, since v 1 = 1+v v p 1 otherwise, when p is odd, v 1 = v 1 1 p 1 v p 1 it follows from Hence and moreover Now we take is an increasing function on [1,]: it is easily seen when p is v p 1 + v v v 1 1 p v 1 p p p +v 4 p + +v 1 p ; v p 1 = 1+v p +v 4 p + +v 1 3 p + v 1 1 p ) = 1 p +1 1 p )v 1 p v 1 p +1) v 1 v p 1 p, 0. v 1) v) pv p 1) + p+1 p v) k+1 v) 1 + p+1 p = 1 1 p p k. ) v) k+1 v) ) v + 3+ p 4.3) k = p, ) k +1) v 1 p +1

17 then v 1) v) pv p 1) CONVEX FUNCTIONS AND LAWSON-OSSERAN PROBLE 17 + p+1 v) k+1 v) 0 and then 4.) becomes p ) h 1 g. p 4.4) Hessh 1 ) kh 1 g = Denote 4.5) h = h 6p 3p+ 1 = v 3 v) 3, then Hessh ) = 6p 6p 3p+ h 3p+ 1 1 Hessh 1 )+ 6p 6p 3p+ 3p+ +1) h 6p 3p+ 1 dh 1 dh 1 3h 6p 3p+ 1 g + 3 ) 6p h 3p+ 4.6) 1 dh dh Let + 1 3p =3h g ) h 1 dh dh. 3p 4.7) h 3 = u+) 1 u), where > 0 to be chosen. A direct calculation shows h 3 = u+) u) u+) 1 = +)u+), h 3 =+)u+) 3. Here denotes derivative with respect to u. Combining with 3.39), we have 4.8) Choose Hessh 3 ) = +)u+) Hessu)++)u+) 3 du du +)u+) h 3 g u+) [ u+) )u+) 3 4 p )u+4p ) u+p) 4.9) = p, then and u+) p )u+4p ) Thereby 4.8) becomes u+p) = +)u+) u+) ] du du p) u+4p 4 0, u+p) +p p = 1+ p. 4.10) Hessh 3 ) 1+ p) h3 g.

18 18 Y.L. XIN AND LING YANG Denote 4.11) h 4 = h 3p p+ 3 = u+p) 3p p+ u) 3p p+, then Hessh 4 ) = 3p 3p p+ h p+ 1 3 Hessh 3 )+ 3p 3p p+ p+ +1) h 3p p+ 3 dh 3 dh 3 3h 3p p+ 3 g + 4 ) 3p h p+ 4.1) 3 dh 4 dh p =3h 4 g ) h 1 4 dh 4 dh 4. 3p Let be an n-dimensional submanifold in R n+m with m. The Gauss map γ : G n,m is defined by γx) = T x G n,m via the parallel translation in R m+n for arbitrary x. The energy density of the Gauss map see [1] Chap.3, 3.1) is eγ) = 1 γ e i,γ e i = 1 B. Ruh-Vilms proved that the mean curvature vector of is parallel if and only if its Gauss map is a harmonic map [17]. If the Gauss image of is contained in {P U G n,m : vp) < }, then the composition function h 1 = h 1 γ of h 1 with the Gauss map γ defines a function on. Using composition formula, we have 4.13) h 1 = Hessh 1 )γ e i,γ e i )+dh 1 τγ)) ) B h1, p where τγ) is the tension field of the Gauss map, which is zero, provided has parallel mean curvature by the Ruh-Vilms theorem mentioned above. Similarly, for h = h γ defined on, we have 4.14) h = Hessh )γ e i,γ e i )+dh τγ)) 3 h B ) h 1 3p h. If the Gauss image of is contained in {P U G n,m : up) < }, we can defined composition function h 3 = h 3 γ and h 4 = h 4 γ on. Again using composition formula, we obtain 4.15) h 3 1+ ) B h3 p

19 and CONVEX FUNCTIONS AND LAWSON-OSSERAN PROBLE ) h 4 3 h 4 B ) h 1 4 h 4. 3p With the aid of h 1 and h 3, we immediately have the following lemma. Lemma 4.1. Let be an n-dimensional minimal submanifold of R n+m needs not be complete), if the Gauss image of is contained in {P U G n,m : vp) < } or respectively, {P U G n,m : up) < }), then we have φ 1 3 B φ ) or respectively, p) + 1 φ 1 1+ ) p for any function φ with compact support D. ) B φ 1 Remark 4.1. For a stable minimal hypersurface there is the stability inequality, which is one of main ingredient for Schoen-Simon-Yau s curvature esimates for stable minimal hypersurfaces. For minimal submanifolds with the Gauss image restriction we have stronger inequality as shown in 4.17). Our proof is similar to [3] and [4], so we omit the detail of it. 5. Curvature estimates We are now in a position to carry out the curvature estimates of Schoen-Simon- Yau type. Let be an n-dimensional minimal submanifold in R n+m. Assume that the estimate 5.1) φ 1 λ B φ 1 holds for arbitrary function φ with compact support D, where λ is a positive constant. Replacing φ by B 1+q φ in 5.1) gives B 4+q φ 1 λ 1 B 1+q φ) 1 5.) = λ 1 1+q) B q B φ 1+λ 1 +λ 1 1+q) B 1+q B φ φ 1. B +q φ 1

20 0 Y.L. XIN AND LING YANG Using Bochner technique, the estimate done in [14][7], and the Kato-type inequality derived in [4], we obtain 5.3) B 1+ ) B 3 B 4. mn For the detail, see [4] Section.) It is equivalent to 5.4) B B B + 3 mn B 4. ultiplying B q φ with both sides of 5.4) and integrating by parts, we have B q B φ 1 mn 5.5) 1+q) B q B φ 1 B 1+q B φ φ 1+ 3 B 4+q φ 1. By multiplying 3 with both sides of 5.) and then adding up both sides of it and 5.5), we have 5.6) mn +1+q 3 λ 1 1+q) ) B q B φ 1 3 λ 1 B +q φ 1+ 3λ 1 1+q) ) B 1+q B φ φ 1. By using Young s inequality, 5.6) becomes mn +1+q 3 λ 1 1+q) ε ) B q B φ 1 5.7) C 1 ε,λ,q) B +q φ 1. If 5.8) λ > 3 then whenever ) 1, mn mn +1+q 3 λ 1 1+q) > 0 [ 5.9) q 0, 1+ 3 λ+ 1 4λ ) ) λ. mn Thus we can choose ε sufficiently small, such that 5.10) B q B φ 1 C B +q φ 1 where C only depends on n, m, λ and q.

21 CONVEX FUNCTIONS AND LAWSON-OSSERAN PROBLE 1 Combining with 5.) and 5.10), we can derive 5.11) B 4+q φ 1 C 3 n,m,λ,q) by again using Young s inequality. B +q φ 1 By replacing φ by φ +q in 5.11) and then using Hölder inequality, we have 5.1) B 4+q φ 4+q 1 C φ 4+q 1. where C is a constant only depending on n, m, λ and q. Similarly, replacing φ by φ 1+q in 5.11) and then again using Hölder inequality yields 5.13) B 4+q φ +q 1 C B φ +q 1. where C is a constant only depending on n, m, λ and q. Let r be a function on with r 1. For any R [0,R 0 ], where R 0 = sup r, suppose R = {x, r R} is compact. 5.1) and Lemma 4.1 enable us to prove the following results by taking φ C c R ) to be the standard cut-off function such that φ 1 in θr and φ C1 θ) 1 R 1. Theorem 5.1. Let be an n-dimensional minimal submanifolds of R n+m. If the Gauss image of R is contained in {P U G n,m : vp) < }, then we have the estimate 5.14) B L s θr ) Cn,m,s)1 θ) 1 R 1 Vol R ) 1 s for arbitrary θ 0,1) and [ s 4,4+ 4 3p + 3+ ) 6 3 p mn + ) ). p If p 4, and the Gauss image of R is contained in {P U G n,m : up) < }, then 5.14) still holds for arbitrary θ 0,1) and [ s 4, p + 1+ ) 1 3 p mn + 8 ) p ). We can also fulfil the curvature estimates of Ecker-Huisken type. Assume that h is a positive function on satisfying the following estimate 5.15) h 3h g +c 0 h 1 dh dh,

22 Y.L. XIN AND LING YANG where is a positive constant. We compute from 5.15) and 5.3): B s h q) c 0 > 3 1 mn 3q s) B s+ h q +ss 1+ mn ) B s B h q +qq +c 0 1) B s h q h +4sq B s 1 B h q 1 h. By Young s inequality, when ss 1+ ) qq+c mn 0 1) sq), i.e., 5.16) q s 1 1 mn c ) q, mn the inequality 5.17) B s h q) 3q s) B s+ h q holds. Especially, 5.18) B s 1 h s whenever 5.19) s ) 3 B s+1 h s mn 1 1 c mn ). Let η be a smooth function with compact support. Integrating by parts in conjunction with Young s inequality lead to B s h s η s 1 B s 1 h s η s B s 1 h) s 1 3 = B s 1 h) s η s ) B s 1 h s sη s 1 η B s 1 h) s s B s h s η s η 1. By Hölder inequality, 5.1) B s h s η s η 1 )s 1 )1 B s h s η s s 1 h s η s s 1. Substituting 5.1) into 5.0), we finally arrive at )1 5.) B s h s η s s 1 )1 3 s h s η s s 1.

23 CONVEX FUNCTIONS AND LAWSON-OSSERAN PROBLE 3 Take η C c R ) to be the standard cut-off function such that η 1 in θr and η C1 θ) 1 R 1 ; then from 5.) we have the following estimate. Theorem 5.. Let be an n-dimensional minimal submanifolds of R n+m. If there exists a positive function h on satisfying 5.15), then there exists C 1 = C 1 n,m,c 0 ), such that 5.3) B h C L s θr ) s)1 θ) R h L s R ) whenever s C 1 and θ 0,1). By 4.14) and 4.16), if the Gauss image of is contained in {P U G n,m : vp) < }, or p 4 and the Gauss image of is contained in {P U G n,m : up) < }, there exists a positive function on, which is h or respectively h 4, satisfying 5.15). Hence the estimate 5.3) holds for both cases. Furthermore, the mean value inequality for any subharmonic function on minimal submanifolds in R m+n ref. [8], [16]) can be applied to yield an estimate of the upper bound of B. We write the results as the following theorem without detail of proof, for it is similar to [4]. Please note that B R x) R m+n denotes a ball of radius R centered at x and its restriction on is denoted by D R x) = B R x). Theorem 5.3. Let x, R > 0 such that the image of D R x) under the Gauss map lies in {P U G n,m : vp) < }. Then, there exists C 1 = C 1 n,m), such that 5.4) B s x) Cn,s)R n+s) sup h ) s VolD R x)), D R x) for arbitrary s C 1. If p 4, the image of D R x) under the Gauss map lies in {P U G n,m : up) < }, then there exists C = C n,m) such that 5.5) B s x) Cn,s)R n+s) sup h 4 ) s VolD R x)), D R x) holds for any s C. 6. Bernstein type theorems and related results If is a submanifold in R n+m, then the function w defined on G n,m see Section )andthegaussmapγ couldbecomposed, yielding asmoothfunctionon, which is also denoted by w. By studying the properties of w-function, we can obtain:

24 4 Y.L. XIN AND LING YANG Proposition 6.1. [4] Let be a complete submanifoldin R n+m. If the w function is boundedbelow by a positive constantw 0. Then is an entire graph with Euclidean volume growth. Precisely, 6.1) VolD R x)) 1 w 0 Cn)R n. Now we let be a complete minimal submanifold in R n+m whose Gauss image lies in {P U G n,m : vp) < }. Then w = v 1 > 1 on and Proposition 6.1 tells us is an entire graph. Precisely, the immersion F : R m+n is realized by a graph x,fx)) with f : R n R m. At each point in its image n-plane P under the Gauss map is spanned by Hence the local coordinate of P in U is By 3.), Hence 6.) vp) = [ det [ f i = ε i + f x i ε. det Z = δ ij + f x i ). δ ij + )]1 f f. x i x j )]1 f f x i x j at each x R n. Conversely, if = x,fx)) is a minimal graph given by f : R n R m which satisfy 6.), then the Gauss image of lies in {P U G n,m : vp) < }. Let P U such that up) = tan θ <, then cos θ = 1+tan θ ) 1 > 1 3 and wp) = cosθ > 3 p. Hence Proposition 6.1 could be applied when is a complete minimal submanifold in R n+m whose Gauss image lies in {P U G n,m : vp) < }; which is hence a minimal graph given by f : R n R m. Thereby 3.14) shows ) f 6.3) <. x i And vice versa. i, Theorem 5.1 and Proposition 6.1 give us the following Bernstein-type theorem. <

25 CONVEX FUNCTIONS AND LAWSON-OSSERAN PROBLE 5 Theorem 6.1. Let = x,fx)) be an n-dimensional entire minimal graph given by m functions f x 1,,x n ) with m,n 4. If [ f = det δ ij + )]1 f f < x i x j or Λ f = ) f <, x i i, then f has to be affine linear functions representing an affine n-plane. Proof. If f <, then the Gauss image of is contained in {P U G n,m : vp) < }. We choose s = p > 4. Fix x and let r be the Euclidean distance function from x and R = D R x). Hence, letting R + in 5.14) yields B L = 0. s ) i.e., B = 0. has to be an affine linear subspace. For the case Λ f <, the proof is similar. Theorem 5.3 and Proposition 6.1 yield Bernstein type results as follows. Theorem 6.. Let = x,fx)) be an n-dimensional entire minimal graph given by m functions f x 1,,x n ) with m. If [ f = det δ ij + )]1 f f <, x i x j and 6.4) f ) 1 = or 4 3 ), where R = x + f. Then f has to be affine linear functions and hence has to be an affine linear subspace. Theorem 6.3. Let = x,fx)) be an n-dimensional entire minimal graph given by m functions f x 1,,x n ) with p = min{n,m} 4. If Λ f = ) f <, x i i, and 6.5) Λ f ) 1 = or p+) 3p ) where R = x + f. Then f has to be affine linear functions and hence has to be an affine linear subspace.

26 6 Y.L. XIN AND LING YANG Proof. Here we only give the proof of Theorem 6., for the proof of Theorem 6.3 is similar. From 4.5), it is easily seen that h C v) 3, where C is a positive constant. Thus, for any point q, by Theorem 5.3 and Proposition 6.1, we have B s q) Cn,s)R s v γ) 3 s Letting R + in the above inequality forces Bq) = 0. Remark 6.1. If n = or 3, the conclusion of Theorem could be inferred from the work done by Chern-Osserman [6], Babosa [1] and Fischer-Colbrie [10]. From 5.13) it is easy to obtain the following result. Theorem 6.4. Let = x,fx)) be an n-dimensional entire minimal graph given by m functions f x 1,,x n ). Assume has finite total curvature. If f <, or p 4 and Λ f <, then has to be an affine linear subspace. There are other applications of the strong stability inequalities 4.17), besides its key role in S-S-Y s estimates. We state following results, whose detail proof can be found in the previous paper of the first author [3]. Theorem 6.5. Let = x,fx)) be an n-dimensional entire minimal graph given by m functions f x 1,,x n ). If f < or Λ f <, then any L -harmonic 1-form vanishes. Theorem 6.6. Let be one as in Theorem 6.5, N be a manifold with non-positive sectional curvature. Then any harmonic map f : N with finite energy has to be constant. References 1. J.L..Babosa: An extrinsic rigidity theorem for minimal immersion from S into S n. J. differntial Geometry 143) 1980), S. Bernstein: Sur un théorème de géométrie et ses application aux équations aux dérivés partielles du type elliptique. Comm. de la Soc ath. de Kharkov é sér.) ), E. Bombieri, E. De Giorgi and E. Guusti: inimal cones and Bernstein problem. Invent. ath ), Yuxin Dong: On graphic submanifolds with parallel mean curvature in Euclidean space. Preprint. 5. S. S. Chern: On the curvature of a piece of hypersurfaces in Euclidean space, Abh. ath. Sem. Univ. Hamberg ), S. S. Chern and R. Osserman: Complete minimal surfaces in Euclidean n space. J. d Anal. ath ),

27 CONVEX FUNCTIONS AND LAWSON-OSSERAN PROBLE 7 7. Qing Chen and Senlin Xu: Rigidity of compact minimal submanifolds in a unit sphere. Geom. Dedicata 45 1)1993), S. Y. Cheng, P. Li and S. T. Yau: Heat equations on minimal submanifols and their applications. Amer. J. ath ), K. Ecker and G. Huisken: A Bernstein result for minimal graphs of controlled growth. J. Diff. Geom ), D. Fischer-Colbrie: Some rigidity theorems for minimal submanifolds of the sphere. Acta math ), S. Hildebrandt, J. Jost, J and K. O. Widman: Harmonic mappings and minimal submanifolds. Invent. math ), J. Jost and Y. L. Xin: Bernstein type theorems for higher codimension. Calculus. Var. PDE ), H. B. Lawson and R. Osserman: Non-existence, non-uniqueness and irregularity of solutions to the minimal surface system. Acta math ), An-in Li and Jimin Li: An intrinsic rigidity theorem for minimal submanifolds in a sphere. Arch. ath ), J. oser: On Harnack s theorem for elliptic differential equations. Comm. Pure Appl. ath ), Lei Ni: Gap theorems for minimal submanifolds in R n+1. Comm. Analy. Geom. 9 3)001), E. A. Ruh and J. Vilms: The tension field of Gauss map. Trans. Amer. ath ), R. Schoen, L. Simon and S. T. Yau: Curvature estimates for minimal hypersurfaces. Acta ath ), J. Simons: inimal varieties in Riemannian manifolds. Ann. ath ), K. Smoczyk, Guofang Wang and Y. L. Xin: Bernstein type theorems with flat normal bundle. Calc. Var. and PDE. 61)006), Y. L. Xin: Geometry of harmonic maps. Birkhäuser PNLDE 3, 1996).. Yuanlong Xin: inimal submanifolds and related topics. World Scientific Publ., 003). 3. Y. L. Xin: Bernstein type theorems without graphic condition. Asia J. ath. 91)005), Y. L. Xin and Ling Yang: Curvature estimates for minimal submanifolds of higher codimension. arxiv: Institute of athematics, Fudan University, Shanghai 00433, China and Key Laboratory of athematics for Nonlinear Sciences Fudan University), inistry of Education address: ylxin@fudan.edu.cn