Calculus of Variations

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1 Calc. Var :7 737 DOI 0.007/s Calculus of Variations The Gauss image of entire graphs of higher codimension and Bernstein type theorems J. Jost Y. L. Xin Ling Yang Received: 9 September 00 / Accepted: 0 July 0 / Published online: 6 June 0 The Authors 0. This article is published with open access at Springerlink.com Abstract Under suitable conditions on the range of the Gauss map of a complete submanifold of Euclidean space with parallel mean curvature, we construct a strongly subharmonic function and derive a-priori estimates for the harmonic Gauss map. The required conditions here are more general than in previous work and they therefore enable us to improve substantially previous results for the Lawson Osseman problem concerning the regularity of minimal submanifolds in higher codimension and to derive Bernstein type results. athematics Subject Classification 99 58E0 53A0 Introduction We consider an oriented n-dimensional submanifold in R n+m with n 3, m. The Gauss map γ : G n,m maps into a Grassmann manifold. In fact, for codimension m =, this Grassmann manifold G n, is the unit sphere S n. In this paper, however, we are interested in the case m where the geometry of this Grassmann manifold is more complicated. By the theorem of Ruh and Vilms [6], γ is harmonic if and only if has parallel mean curvature. This result applies thus in particular to the case where is a minimal submanifold of Euclidean space. Communicated by L. Ambrosio. J. Jost B L. Yang ax Planck Institute for athematics in the Sciences, Inselstr., 0403 Leipzig, Germany jost@mis.mpg.de Y. L. Xin Institute of athematics, Fudan University, Shanghai 00433, China ylxin@fudan.edu.cn Present Address: L. Yang Institute of athematics, Fudan University, Shanghai 00433, China yanglingfd@fudan.edu.cn

2 7 J. Jost et al. Now, the Bernstein problem for entire minimal graphs is one of the central problems in geometric analysis. Let us summarize the status of this problem, first for the case of codimension. The central result is that an entire minimal graph of dimension n 7and codimension has to be planar, but there are counterexamples to such a Bernstein type theorem in dimension 8 or higher. However, when the additional condition is imposed that the slope of the graph be uniformly bounded, then a theorem of oser [5], called a weak Bernstein theorem asserts that such an in arbitrary dimension has to be planar. Thus, the counterexamples arise from a non-uniform behavior at infinity. In fact, by a general scaling argument, the Bernstein theorems are intimately related to the regularity question for the minimal hypersurface equation. A natural and important question then is to what extent such Bernstein type theorems generalize to entire minimal graphs of codimension m. oser s result has been extended to higher codimension by Chern and Osserman [3] for dimension n = and Barbosa [] and Fisher and Colbrie [5] for dimension n = 3. For dimension n = 4 and codimension m = 3, however, there is a counterexample given by Lawson and Osserman [4]. In fact, their paper emphasizes the stark contrast between the cases of codimension and greater than for the minimal submanifold system, concerning regularity, uniqueness, and existence. The Lawson Osserman problem then is concerned with a systematic understanding of the analytic aspects of the minimal submanifold system in higher codimension. As in the case of codimension, the Bernstein problem provides a key towards this aim. While the work of Lawson Osserman produced a counterexample for a general Bernstein theorem, there are also some positive results in this direction which we shall now summarize. Hildebrandt et al. [0] started a systematic approach on the basis of the aforementioned Ruh Vilms theorem. That is, they developed and employed the theory of harmonic maps and the convex geometry of Grassmannian manifolds, and obtained Bernstein type results in general dimension and codimension. Their main result says that a Bernstein result holds if the image of the Gauss map is contained in a strictly convex distance ball. Since the Riemannian sectional curvature of G n,m is nonnegative, the maximal radius of such a convex ball is bounded. In codimension, this in particular reproduces oser s theorem, and in this sense, their result is optimal. For higher codimension, their result can be improved, for the following reason. Since the sectional curvature of G n,m for n, m is not constant, there exist larger convex sets than geodesic distance balls, and it turns out that harmonic e.g. Gauss maps with values in such convex sets can still be well enough controlled. In this sense, the results of [0] could be improved by Jost and Xin [], Wang [8] and Xin and Yang []. In [], the largest such geodesically convex set in a Grassmannian manifold was found. Formulating it somewhat differently, the harmonic map approach is based on the fact that the composition of a harmonic map with a convex function is a subharmonic function, and by using quantitative estimates for such subharmonic functions, regularity and Liouville type results for harmonic maps can be obtained. Thus, convex functions were systematically utilized in []. In this paper, also the fundamental connection between estimates for the second fundamental form of minimal submanifolds and estimates for their Gauss maps was systematically explored. On this basis, the fundamental curvature estimate technique, as developed by Schoen et al. [7] and Ecker and Huisken[4], could be used in []. Still, there remains a large quantitative gap between those positive results and the counterexample of Lawson Osserman. In this situation, it could either be that Bernstein theorems can be found under more general conditions, or that there exist other counterexamples in the so far unexplored range.

3 The Gauss image and Bernstein type theorems 73 In the present paper, we make a step towards closing this gap in the positive direction. We identify a geometrically natural function v on a Grassmann manifold and a natural quantitative condition under which the precomposition of this function with a harmonic Gauss map is strongly subharmonic Theorem 3.. When the precomposition of v with the Gauss map of a complete minimal submanifold is bounded, then that submanifold is an entire graph of bounded slope. On one hand, this is the first systematic example in harmonic map regularity theory where this auxiliary function is not necessarily convex. On the other hand, the Lawson Osserman s counterexample can also be readily characterized in terms of this function. Still, the range of values for v where we can apply our scheme is strictly separated from the value of v in that example. Therefore, still some gap remains which should be explored in future work. Our work also finds its natural position in the general regularity theory for harmonic maps. Also, once we have a strongly subharmonic function, we could derive Bernstein type results within the frame work of geometric measure theory, by the standard blow-down procedure and appeal to Allard s regularity theorem []. By building upon the work of many people on harmonic map regularity, we can obtain more insight, however. In particular, we shall use the iteration method of [0], we can explore the relation with curvature estimates, and we shall utilize a version of the telescoping trick Theorem 4. to finally obtain a quantitatively controlled Gauss image shrinking process Theorems 5. and 6.. In this way, we can understand why the submanifold is flat as the Bernstein result asserts. ore precisely, we obtain the following Bernstein type result, which substantially improves our previous results. Theorem. Let z = f x,...,x n, =,...,m, be smooth functions defined everywhere in R n n 3, m. Suppose their graph = x, f x is a submanifold with parallel mean curvature in R n+m. Suppose that there exists a number β 0 < 3 such that [ f = det δ ij + ] f f x i x j β 0.. Then f,..., f m has to be affine linear, i.e., it represents an affine n-plane. The essential point is to show that v := f is subharmonic when <3. In fact, when v β 0 < 3, then v K 0 B where K 0 is a positive constant and B is the second fundamental form of in R n+m. This principle is not new. Wang [8] has given conditions under which log v is subharmonic and has derived Bernstein results from this, as indicated above. He only needs that v be uniformly bounded by some constant, not necessarily <3, but in addition that there exist some δ>0such that for any two eigenvalues λ i,λ j with i = j, the inequality λ i λ j δholds the latter condition means in geometric terms that df is strictly area decreasing on any two-dimensional subspace. Since subharmonicity of log v is a weaker property than subharmonicity of v itself, his computation is substantially easier than ours, and our results cannot be deduced from his. In fact, v = + λi, and while the condition of [] which can be reformulated as v being bounded away from 4 implies the condition of [8] so that the latter result generalizes the former, the condition needed in the present paper is only the weaker one that v be bounded away from 9. In fact, somewhat more refined results can be obtained, as will be pointed out in the final remarks of this paper.

4 74 J. Jost et al. Geometry of Grassmann manifolds Let R n+m be an n + m-dimensional Euclidean space. Its oriented n-subspaces constitute the Grassmann manifold G n,m, which is the Riemannian symmetric space of compact type SOn + m/son SOm. G n,m can be viewed as a submanifold of some Euclidean space via the Plücker embedding. The restriction of the Euclidean inner product on is denoted by w : G n,m G n,m R wp, Q = e e n, f f n =det W where P is spanned by a unit n-vector e e n, Q is spanned by another unit n-vector f f n, and W = e i, f j. It is well-known that W T W = O T O with O an orthogonal matrix and μ =.... Here each 0 μi. Putting p := min{m, n}, then at most p elements in {μ,...,μ n } are not equal to 0. Without loss of generality, we can assume μi = whenever i > p. We also note that the μi can be expressed as μi = + λi.. The Jordan angles between P and Q are defined by The distance between P and Q is defined by Thus,. becomes μ n θ i = arccosμ i i p. dp, Q = θ i.. λ i = tan θ i..3 In the sequel, we shall assume n m without loss of generality. We use the summation convention and agree on the ranges of indices: i, j, k, l n,, β, γ m, a, b,...=,...,n + m. Now we fix P 0 G n,m. We represent it by n vectors ɛ i, which are complemented by m vectors ɛ n+, such that {ɛ i,ɛ n+ } form an orthonormal base of R m+n. Denote U := {P G n,m,wp, P 0 >0}. We can span an arbitrary P U by n-vectors f i : f i = ɛ i + Z i ɛ n+.

5 The Gauss image and Bernstein type theorems 75 The canonical metric in U can be described as ds = tri n + ZZ T dzi m + Z T Z dz T,.4 where Z = Z i is an n m-matrix and I n res. I m denotes the n n-identity res. m m matrix. It is shown that.4 can be derived from. in [0]. For any P U, the Jordan angles between P and P 0 are defined by {θ i }. Let E i be the matrix with in the intersection of row i and column and 0 otherwise. Then, sec θ i sec θ E i form an orthonormal basis of T P G n,m with respect to.4. Denote its dual frame by ω i. Our fundamental quantity will be v, P 0 := w, P 0 on U..5 For arbitrary P U determined by an n m matrix Z, it is easily seen that vp, P 0 = [ deti n + ZZ T ] = m sec θ = = m = μ..6 where θ,...,θ m denote the Jordan angles between P and P 0. In this terminology, Hessv, P 0 has been estimated in []. By 3.8 in [], we have Hessv, P 0 = vωi + + λ vω + λ λ β vω ω ββ +ω β ω β i = =β = + λ vω + λ λ β vω ω ββ It follows that vωi + m+ i n, + + λ λ β v <β + λ λ β v ω β + ω β =β ω β ω β..7 v, P 0 Hessv, P 0 = g + λ ω + =β λ λ β ω ω ββ + ω β ω β..8 The canonical Riemannian metric on G n,m derived from. can also be described by the moving frame method. This will be useful for understanding some of the sequel. Let {e i, e n+ } be a local orthonormal frame field in R n+m. Let {ω i,ω n+ } be its dual frame field so that the Euclidean metric is g = i ω i + ω n+. The Levi-Civita connection forms ω ab of R n+m are uniquely determined by the equations dω a = ω ab ω b, ω ab + ω ba = 0.

6 76 J. Jost et al. It is shown in [0] that the canonical Riemannian metric on G n,m can be written as ds = i,ω in Subharmonic functions Let m R n+m be an isometric immersion with second fundamental form B. Around any point p, we choose an orthonormal frame field e i,...,e n+m in R n+m, such that {e i } are tangent to and {e n+ } normal to. The metric on is g = i ω i. We have the structure equations ω in+ = h ij ω j, 3. where h ij are the coefficients of second fundamental form B of in R n+m. Let 0 be the origin of R n+m, SOm + n be the Lie group consisting of all orthonormal frames 0; e i, e n+, TF = { p; e,...,e n : p, e i T p, e i, e j =δ ij } be the principle bundle of orthonormal tangent frames over, and NF = { p; e n+,...,e n+m : p, e n+ N p } be the principle bundle of orthonormal normal frames over. Then π : TF NF is the projection with fiber SOn SOm. The Gauss map γ : G n,m is defined by γp = T p G n,m via the parallel translation in R n+m for every p. Then the following diagram commutes i TF NF SOn + m π π γ G n,m where i denotes the inclusion map and π : SOn + m G n,m is defined by It follows that 0; e i, e n+ e e n. dγ =,i, j h ij = B was computed for the metric.4 whose corresponding coframe field is ω i. Since.4 and.9 are equivalent to each other, at any fixed point P G n,m there exists an isotropic group action, i.e., an SOn SOm action, such that ω i is transformed to ω in+, namely, there are a local tangent frame field and a local normal frame field such that at the point under consideration, In conjunction with 3.and3.3 we obtain ω in+ = γ ω i. 3.3 γ ω i = h ij ω j. 3.4 By the Ruh Vilms theorem [6], the mean curvature of is parallel if and only if its Gaussmapisaharmonicmap.Now,weassumethat has parallel mean curvature.

7 The Gauss image and Bernstein type theorems 77 We define v := v, P 0 γ, 3.5 This function v on will be the source of the basic inequality for this paper. Its geometric significance is seen from the following observation. If the v-function has an upper bound or the w-function has a positive lower bound, can be described as an entire graph on R n by f : R n R m, provided is complete. In this situation, λ i is the singular values of df and [ v = det δ ij + ] f f x i x j 3.6 Using the composition formula, in conjunction with.8, 3. and3.4, and the fact that τγ = 0 the tension field of the Gauss map vanishes [6], we deduce the important formula of Lemma. in [5] and Prop.. in [8]. Proposition 3. Let be an n-submanifold in R n+m with parallel mean curvature. Then v = v B + v λ h,j + v λ λ β h,j h β,βj + h,β j h β,j, 3.7, j =β,j where h,ij are the coefficients of the second fundamental form of in R n+m see 3.. A crucial step in this paper is to find a condition which guarantees the strong subharmonicity of the v-function on. ore precisely, under a condition on v, we shall bound its Laplacian from below by a positive constant times squared norm of the second fundamental form. Looking at the expression 3.7, we group its terms according to the different types of the indices of the coefficients of the second fundamental form as follows. v v = h,ij + I j + II jβ + III βγ + IV i, j>m j>m j>m,<β <β<γ 3.8 where I j = + λ h,j + λ λ β h,j h β,βj, 3.9 =β II jβ = h,β j + h β,j + λ λ β h,β j h β,j, 3.0 and III βγ = h,βγ + h β,γ + h γ,β + λ λ β h,βγ h β,γ + λ β λ γ h β,γ h γ,β + λ γ λ h γ,β h,βγ 3. IV = + λ h, + β = h,ββ + + λ β h β,β + λ β λ γ h β,β h γ,γ + λ λ β h,ββ h β,β. 3. β =γ β = It is easily seen that I j = λ h,j + + λ h,j h,j. 3.3

8 78 J. Jost et al. Obviously II jβ = λ λ β h,β j + h β,j + λ λ β h,β j + h β,j. 3.4 v = + λ implies + λ + λ β v. Assume + λ + λ β C v, then differentiating both sides implies Therefore λ dλ + λ + λ βdλ β + λ β = 0. dλ λ β = λ β dλ + λ dλ β [ = λ β + λ λ + λ β = λ β λ ] dλ λ β + λ dλ λ β + λ. 3.5 It follows that λ,λ β λ λ β attains its maximum at the point satisfying λ = λ β, which is hence C,C. Thus λ λ β C v and moreover II jβ 3 v h,β j + h β,j. 3.6 Lemma 3. III βγ 3 v h,βγ + h β,γ + h γ,β. Proof It is easily seen that III βγ 3 v h,βγ + h β,γ + h γ,β = λ h,βγ + λ β h β,γ + λ γ h γ,β + v λ h,βγ + v λ β h β,γ + v λ γ h γ,β. If λ,λ β,λ γ v, then III βγ 3 vh,βγ + h β,γ + h γ,β is obviously nonnegative definite. Otherwise, we can assume λ γ >v without loss of generality, then + λ + λ β + λ γ v implies λ <v,λ β <v. Denote s = λ h,βγ + λ β h β,γ, then by the Cauchy Schwarz inequality, s = λ h,βγ + λ β h β,γ = λ v λ v λ h λ β,βγ + v λ v λ β h β,γ β λ λ v β v λ + λ h v λ,βγ + v λ β h β,γ β i.e. v λ h,βγ + v λ β h β,γ λ λ β v λ + v λ s. β 3.7

9 The Gauss image and Bernstein type theorems 79 Hence III βγ 3 v h,βγ + h β,γ + h γ,β λ λ s + λ γ h γ,β + β v λ + λ = λ β + v λ + v λ β v λ β s + v λ γ h γ,β s + v h γ,β + λ γ sh γ,β. 3.8 It is well known that ax + bxy + cy is nonnegative definite if and only if a, c 0and ac b 0. Hence the right hand side of 3.8 is nonnegative definite if and only if λ v λ β + v λ + v λ λ γ β i.e. v λ + v λ β + v λ γ v. 3.0 Denote x = + λ, y = + λ β, z = + λ γ. Let C be a constant v, denote = { x, y, z R 3 : x, y <v, z >v, xyz = C } and f : R We claim f v x, y, z v x + v y + v z. on. Then 3.0 follows and hence III βγ 3 v h,βγ + h β,γ + h γ,β is nonnegative definite. We now verify the claim. For arbitrary ε>0, denote f ε = v + ε x + v + ε y + v + ε z, then f ε is obviously a smooth function on ε = { x, y, z R 3 : x, y v, z v + ε, xyz = C }. The compactness of ε implies the existence of x 0, y 0, z 0 ε satisfying f ε x 0, y 0, z 0 = sup ε f ε. 3. Fix x 0, then 3. implies that for arbitrary y, z R satisfying y v, z v +ε and yz = x C 0, we have f ε,x0 y, z = v + ε y + v + ε z +. v + ε y 0 v + ε z 0

10 70 J. Jost et al. Differentiating both sides of yz = x C 0 yields dy y d v + ε y + = v + ε z [ = y v + ε y z v + ε z + dz z = 0. Hence dy v + ε y + dz v + ε z ] dy y = v + ε yzy z dy v + ε y v + ε z y. 3. It implies that f ε,x0 y, yx C 0 is decreasing in y and y 0 =. Similarly, one can derive x 0 =. Therefore sup f ε = f ε,, C = ε v + ε + v + ε C < v + ε. Note that f ε f and lim ε 0 + ε. Hence by letting ε 0 one can obtain f v. Lemma 3. There exists a positive constant ε 0, such that if v 3, then IV ε 0 h, + β,β h,ββ + h. β = Proof For arbitrary ε 0 [0,, denote C = ε 0, then IV ε 0 h, + β,β h,ββ + h β = = β λ β h β,β + C + λ h, + β = ] [Ch,ββ + C + λ β h β,β + λ λ β h,ββ h β,β. 3.3 Obviously Ch,ββ + C + λ β Ch,ββ + C λ λ β h β,β + λ λ β h,ββ h β,β C h,ββ + C λ λ β h β,β 0, hence, the third term of the right hand side of 3.3 satisfies h β,β + λ λ β h,ββ h β,β and If there exist distinct indices β,γ = satisfying C + λ β C λ λ β 0 C + λ γ C λ λ γ 0, C + λ β C λ λ β h β,β 3.4

11 The Gauss image and Bernstein type theorems 7 then λ > C and It implies λ β C λ C, λ γ C λ C. + λ λ + λβ + λ γ + λ + C C λ C. Define f : x, + x+x+c C, then a direct calculation shows x C log f = x + + x + C C x C It follows that f x f CC + 3 = C+3 C+, i.e. v + λ + λ β = x CC + 3x + C x + x + C Cx C. + λ γ C + 3 C If C =, then C+3 C+ = 7 > 9; hence there is ε > 0, once ε 0 ε, then C = ε 0 satisfies C+3 C+ > 9, which causes a contradiction to v 9. Hence, one can find an index γ =, such that C + λ β C λ λ β > 0 for arbitrary β =, γ. 3.6 Denote s = β =γ λ βh β,β, then by using the Cauchy Schwarz inequality, C + λ h, + C + λ β C λ λ β h β,β β =,γ λ C + λ + λ β C + λ β =,γ β C λ s. 3.7 λ β Substituting 3.7and3.4into3.3 yields IV ε 0 h, + β,β h,ββ + h β = s + λ γ h γ,γ + λ C + λ + λ β C + λ β =,γ β C λ s λ β + C + λ γ C λ λ γ h γ,γ + λ C + λ + λ β s C + λ β =,γ β C λ λ β + C + λ γ C λ λ γ h γ,γ + λ γ sh γ,γ. 3.8 Note that when m =, s = λ h, and β =,γ λ β C+λ β C λ λ β = 0.

12 7 J. Jost et al. The right hand side of 3.8 is nonnegative definite if and only if and + λ C +λ + β =,γ C + λ γ C λ λ γ λ β C + λ β C λ C +λ λ γ C λ λ γ λ γ 0. β 3.30 Assume C + λ γ C λ λ γ < 0, then λ > C and λ γ > C, which implies λ C + λ + λ γ λ +λ +CC λ C. Define f : x C, + x+x+cc x C, then log f = x + + x + CC x C = x 4Cx C C x + x + CC x C. and hence min f = f C + 4C + = C + C + + C 4C +. In particular, when C =, min f = > 9. There exists ε > 0, such that once ε 0 ε, one can derive min f > 9 and moreover v + λ + λ γ >9, which contradicts v 3. Therefore 3.9 holds. If C + λ γ C λ λ γ 0, 3.30 trivially holds. At last, we consider the situation when there exists γ, γ =, such that In this case, 3.30 is equivalent to C + λ γ C λ λ γ < 0. λ C + λ λ β + C + λ β = β C λ λ β. 3.3 Noting that λ β C + λ β C λ λ β = C C 3 C λ C λ + λ β + λ +CC C λ and let x β = + λ β, then 3.3 is equivalent to x x + C + C C 3 C + x β = C + x. x β x +C C x C 3.3 Denote ψx = x x + C, ϕx = x + C C, x C C C 3 ζx =, ξx = C + x C + x.

13 The Gauss image and Bernstein type theorems 73 Let = x,...,x m R m : x > C +, x β <ϕx for all β =, γ, x γ >ϕx, β x β = v 3.33 and define f : R x,...,x m ψx + β = We point out that in 3.33, and γ are fixed indices. Now we claim [ ζx ξx ]. x β ϕx where sup f = sup Ɣ f 3.34 Ɣ = x,...,x m R m : x C +, x β = forallβ =, γ, x γ ϕx, β x β = v When m =, obviously Ɣ = and 3.34 is trivial. We put ϕ ε x = ϕx + ε, ζ ε x = ζx + ε, ξ ε x = ξx + ε for arbitrary ε>0. If m 3, as in the proof of Lemma 3.,wedefine f ε = ψx + [ ζ ε x ξ ] εx, x β = β ϕ ε x then f ε is well-defined on ε = x,...,x m R m : x C +, x β ϕ ε x for all β =, γ, x γ ϕ ε x, β x β = v. The compactness of ε enables us to find y,...,y m ε, such that f ε y,...,y m = sup ε f ε Denote b = ϕ ε y, then 3.36 implies for arbitrary β =, γ that x β b + x γ b y β b + y γ b

14 74 J. Jost et al. holds whenever x β x γ = y β y γ, x β ϕ ε y and x γ ϕ ε y. Differentiating both sides yields dx β x β + dx γ x γ = 0, thus d x β b + x γ b = dx β x β b dx γ x γ b = b x β x γ x γ x β x β b x γ b dx β x β Similarly to 3.5, one can prove y b = y y +ε+c C > 9whenε y +ε C 0 ε note that C = ε 0 andε is sufficiently small. In conjunction with y x β x γ = y y β y γ v < 9, we have b x β x γ > 0. Hence 3.37 implies y β = forallβ =, γ. In other words, if we put Ɣ ε = x,...,x m R m : x C +, x β = forallβ =, γ, x γ ϕ ε x, β x β = v, then max ε f ε = max Ɣε f ε. Therefore, 3.34 follows from ε>0 ε,ɣ ε>0 Ɣ ε and lim ε 0 f ε = f. To prove 3.30, i.e. f, it is sufficient to show on Ɣ, ψx + ζx ξx 3.38 v x ϕx whenever x > C + and x v inequality is equivalent to >ϕx. After a straightforward calculation, the above x 3 + C C x + C3 3C + C + x v x C + x C C It is easily seen that if t 3 + C C t + C 3 3C + C + t inf t C+t C C >0 t C + t C > C then 3.39 naturally holds and furthermore one can deduce that IV ε 0 h, + β = h,ββ + h β,β is nonnegative definite. When C =, 3.40 becomes t t inf t> 3+ 5 t > t + If this is true, one can find a positive constant ε 3 to ensure 3.40 holds true whenever ε 0 ε 3. Finally, by taking ε 0 = min{ε,ε,ε 3 } we obtain the final conclusion. 3.4 is equivalent to the property that ht = t t 9t 3t+ = t 3 0t +7t 9 has no zeros on 3+ 5, +. h t = 3t 0t +7 implies h t <0on 3+ 5, and h t >0on , +, hence

15 The Gauss image and Bernstein type theorems 75 inf h = h t> 3+ 5 and 3.4 follows = > In conjunction with 3.3, 3.6, Lemma 3. and 3., we can arrive at Theorem 3. Let n be a submanifold in R n+m with parallel mean curvature, then for arbitrary p and P 0 G n,m, once vγp, P 0 3, then v, P 0 γ 0 at p. oreover, if vγp, P 0 β 0 < 3, then there exists a positive constant K 0, depending only on β 0, such that at p. v, P 0 γ K 0 B 3.4 We also express this result by saying that the function v satisfying 3.4 is strongly subharmonic under the condition vγp, P 0 β 0 < 3. Remark 3. If log v is a strongly subharmonic function, then v is certainly strongly subharmonic, but the converse is not necessarily true. Therefore, the above result does not seem to follow from Theorem. in [8]. 4 Curvature estimates Let z = f x,...,x n, =,...,m be smooth functions defined on D R0 R n. Their graph = x, f x is a submanifold with parallel mean curvature in R n+m. Suppose there is β 0 [, 3, such that [ f = det δ ij + ] f f x i x j β Denote by ɛ,...,ɛ n+m the canonical basis of R n+m and put P 0 = ɛ ɛ n. Then by 4. v, P 0 γ β 0 holds everywhere on. Putting v = v, P 0 γ, Theorem 3. tells us v K 0 β 0 B. 4. Let η be a nonnegative smooth function on with compact support. ultiplying both sides of 4.byη and integrating on gives K 0 B η η v. 4.3 F : D R0 defined by x = x,...,x n x, f x is obviously a diffeomorphism. F = ɛ x i i + f ɛ x i n+ implies F x i, F x j = δ ij + f x i f x j.

16 76 J. Jost et al. Hence F g = δ ij + f f x i x j dx i dx j 4.4 where g is the metric tensor on. In other words, is isometric to the Euclidean ball D R0 equipped with the metric g ij dx i dx j g ij = δ ij + f f. It is easily seen that for x i x arbitrary ξ j R n, ξ i g ij ξ j = ξ + f x i ξ i ξ. 4.5 On the other hand, f β 0 implies n i= μ i β0, with μ,...,μ n the eigenvalues of g ij, thus i ξ i g ij ξ j β 0 ξ 9 ξ. 4.6 In local coordinates, the Laplace Beltrami operator is = Gg ij G x i x j. Here g ij is the inverse matrix of g ij, and G = detg ij = f. From 4., 4.5and4.6 it is easily seen that 3 ξ β0 ξ ξ i Gg ij ξ j β 0 ξ 3 ξ. 4.7 Following [,3] we shall make use of the following abbreviations: For arbitrary R 0, R 0, let And for arbitrary h L denote h +,R def. = sup h, h p,r def. = = {x, f x : x D R }. 4.8 h p h,r def. ==inf h, p def. B h R = h = R h Vol p, shows that is a uniform elliptic operator. oser s Harnack inequality [5] for supersolutions of elliptic PDEs in divergence form gives Lemma 4. For a positive superharmonic function h on with R 0, R 0 ], p 0, and θ [,, we have the following estimate h p,θ R γ h,θ R. Here γ is a positive constant only depending on n, p and θ, but not on h and R. n n 4. shows the subharmonicity of v, and therefore v +,R v + ε is a positive superharmonic function on for arbitrary ε>0. With the aid of Lemma 4., one can follow [] to get

17 The Gauss image and Bernstein type theorems 77 Corollary 4. There is a constant δ 0 0,, depending only on n, such that v +, R δ 0 v +,R + δ 0 v R. Denote by G ρ the mollified Green function for the Laplace Beltrami operator on. Then for arbitrary p = y, f y, once B ρ p = {x, f x : x D R y} we have G ρ, p φ = B ρ p φ for every φ H, 0. The a priori estimates for mollified Green functions of [8] tellus Lemma 4. With o := 0, f 0, we have For arbitrary p 4, 0 G R, o c nr n on G R, o c nr n on. 4.0 G ρ, p CnR n on \. 4. oreover if ρ R 8, G ρ, p CnR n. 4. \ Basedon4.3, Corollary 4. and Lemma 4., we can use the method of [] toderivea telescoping lemma a la Giaquinta and Giusti [6] and Giaquita and Hildebrandt [7]. Theorem 4. There exists a positive constant C, only depending on n and β 0, such that for arbitrary R R 0, R n B C v +,R v +, R 4.3 oreover, there exists a positive constant C, only depending on n and β 0, such that for arbitrary ε>0, we can find R [exp C ε R 0, R 0 ], such that R n B ε. 4.4 Proof With ω R = R Vol G R, o where o = 0, f 0,

18 78 J. Jost et al. then ω R φ = R φ. Choosing ω R H, 0 as a test function in 4.3, we obtain K 0 B ω R ω R v = ω R ω R v v +,R = ω R ω R v v +,R v v +,R ω R ω R ω R v v +,R = R ω R v v +,R. By 4.0, there exist positive constants c 3, c 4, depending only on n, such that 0 ω R c 4 on, ω R c 3 on. Hence B K0 c 4 c 3 R v v +,R c 5 n,β 0 R n v +,R v R. 4.5 By Corollary 4., v +,R v R δ 0 v +,R v +, R ; substituting it into 4.5 yields 4.3. For arbitrary k Z +, 4.3gives k i R 0 n i=0 For arbitrary ε>0, we take B i R0 B C k =[C ε ], k v+, i R 0 v +, i R 0 i=0 = C v+,r0 v +, k R 0 C β 0 C 4.6 then we can find j k, such that j R 0 n B j R0 B k + C ε.

19 The Gauss image and Bernstein type theorems 79 Since j k C ε = exp [ log C ε ], it is sufficient to choose C = log C. 5 Gauss image shrinking property Lemma 5. For arbitrary a > and β 0 [, a, there exists a positive constant ε = ε a,β 0 with the following property. If P, Q G n,m satisfies vq, P b β 0, then we can find P G n,m, such that vp, P a for every P G n,m satisfying vp, P b, and { vq, P if b < + a 5. b ε otherwise. Proof Obviously wp, P = foreveryp G n,m, which shows G n,m is a submanifold in a Euclidean sphere via the Plücker embedding. Denote byr, the restriction of the spherical distance on G n,m, then by spherical geometry, w = cos r and hence v = sec r. Denote = arccosa and β = arccosb. Now we put γ = β and c = sec γ = Once vp, P c, the triangle inequality implies a b + a b. 5. rp, P rp, P + rp, P arccosb + arccosc = for every P satisfying vp, P b, and thus vp, P a follows. If b < + a, then a direct calculation shows β <, hence γ > β and moreover vq, P b < c. Thereby P = Q is the required point. Otherwise b + a and hence c b. Obviously one of the following two cases has to occur: Case I. vq, P <c. One can take P = Q to ensure v, P a whenever v, P b. In this case b vq, P = b + a. 5.3 Case II. vq, P c. Denote by θ,...,θ m the Jordan angles between Q and P, and put L = m θ, then as shown in [9], if we denote the shortest normal geodesic from Q to P by γ, then the Jordan angles between Q and γt are θ L t,..., θ m L t, while the Jordan angles between γt and P are θ L L t,..., θ m L L t. Hence vq,γt = θ sec L t, vγt, P = θ sec L t. L Since t sec θl L t is a strictly decreasing function, there exists a unique t 0 [0, L, such that sec θl L t 0 = c. Now we choose P = γt 0, then vp, P = c and b vq, P = b θ sec L t

20 730 J. Jost et al. It remains to show b sec θl t 0 is bounded from below by a universal positive constant ε. Once this holds true, in conjunction with 5.3and5.4, ε = min{ + a,ε } 5.5 is the required constant. t 0 can be regarded as a smooth function on { = b,θ,...,θ m R m+, + a b β0, 0 θ π, c secθ b } which is the unique one satisfying θ sec L L t 0 = c. By 5., c can be viewed as a function of b. The smoothness of t 0 follows from the implicit function theorem. Therefore F : R θ,...,θ m b θ sec L t 0 is a smooth function on. t 0 < L implies F > 0; then the compactness of gives inf F > 0, and ε = inf F is the required constant. Remark 5. ε is only depending on a and β 0, non-decreasingly during the iteration process in Theorem 6.. Theorem 5. Let = { x, f x : x D R0 R n} be a graph with parallel mean curvature, and f β 0 with β 0 [, 3. Assume there exists P 0 G n,m, such that v, P 0 γ b on with b β 0. If b < 6, then for arbitrary ε>0, one can find a constant δ 0, depending only on n,β 0 and ε such that v, P γ + ε on B δr0 5.6 for a point P G n,m. If b 6, then there are two constants δ 0 0, and ε > 0, only depending on n and β 0, such that v, P γ b ε on B δ0 R for a point P G n,m. Proof Let H be a smooth function on G n,m, then h = H γ gives a smooth function on. Let η be a nonnegative smooth function on with compact support and ϕ be a H, -function on, then by Stokes Theorem, 0 = divϕη h = ϕ η h + η ϕ h + = ϕ η h + ϕ ηh ϕη h h ϕ η + ϕη h.

21 The Gauss image and Bernstein type theorems 73 Hence ϕ ηh = ϕ η h + h ϕ η ϕη h. 5.8 G ρ, p η h + G ρ, pη h. For arbitrary R R 0, we take a cut-off function η supported in the interior of, 0 η, η onbr and η c 0 R. For every ρ R 8, denote by Gρ the mollified Green function on. For arbitrary p, inserting ϕ = G ρ, p into 5.8gives 4 G ρ, p ηh = h G ρ, p η We write 5.9as I ρ = II ρ + III ρ + IV ρ. By the definition of mollified Green functions, I ρ = ηh = Hence B ρ p B ρ p 5.9 h. 5.0 lim I ρ 0 + ρ = hp. 5. Noting that dγ = B, we have h G H dγ = G H B. Here and in the sequel, G denotes the Levi-Civita connection on G n,m. In conjunction with 4., we have II ρ G ρ, p η h Here T R def. = \ and T R sup T R G ρ, p sup T R CnR n sup V c n sup V which is a compact subset of U. η sup G H V B G H B Vol G H R n B. 5. V ={P G n,m : vp, P 0 3},

22 73 J. Jost et al. As shown in Sect., there is a one-to-one correspondence between the points in U and the n m-matrices. And each n m-matrix can be viewed as a corresponding vector in R nm. Define T : U R nm Z deti + ZZ T Z trzz T Note that trzz T = i, Z i equals Z when Z is treated as a vector in R nm. Since t [0, + [ det I + tztz T ] is a strictly increasing function and maps [0, + onto [, +, T is a diffeomorphism. By.6, T Z =vp, P 0. Via T, we can define the mean value of γ on by [ ] γ R = T T γ. 5.3 Vol Note that T maps sublevel sets of v, P 0 onto Euclidean balls centered at the origin. Hence the convexity of Euclidean balls gives v γ R, P 0 sup v, P 0 γ b. 5.4 The compactness of V ensures the existence of positive constants K and K, such that for arbitrary X T V, K X T X K X. The classical Neumann Poincaré inequality says φ φ CnR Dφ. D R D R As shown above, can be regarded as D R equipped with the metric g = g ij dx i dx j, and the eigenvalues of g ij are bounded. Hence it is easy to get φ φ CnR φ. Here φ can be a vector-valued function. Denote by d G the distance function on G n,m. Then, by using the above Neumann Poincaré inequality we have dg γ, γ R K T γ T γ R CnK R dt γ CnK K R = CnK K R dγ B. 5.5

23 The Gauss image and Bernstein type theorems 733 Now we write h = H γ = H γ R + H γ H γ R, then III ρ = H γ R G ρ, p η + H γ H γ R G ρ, p η. T R 5.6 Similar to 5.0, lim H γ R ρ 0 + G ρ, p η = lim H γ R ρ 0 + B ρ p η = H γ R. 5.7 The second term can be controlled by T R H γ H γ R G ρ, p η sup V G H sup η T R T R d G γ, γ R G ρ, p c 0 R sup G H dg γ, γ R G ρ, p V T R 5.8 Substituting 4.and5.5into5.8 yields H γ H γ R G ρ, p η c n sup G H R n B V T R From 5.9, 5., 5., 5.6, 5.7and5.9, letting ρ 0 we arrive at. 5.9 hp H γ R + c 3 n sup G H R n B V lim sup G ρ, pη h. 5.0 ρ 0 + for every p. 4 The compactness of G n,m implies the existence of a positive constant K 3, such that G v, P K 3 whenever v, P 3 5.

24 734 J. Jost et al. for arbitrary P G n,m. Hence by inserting H = v, P into 5.0 one can obtain vγp, P v γ R, P + c 3 K 3 R n B lim sup G ρ, pη v, P γ. 5. ρ 0 + By Lemma 5., if we put P = γ R, then v, P 3 whenever v, P 0 b provided that b < 6, which implies v, γ R γ is a subharmonic function on. Letting P = γ R in 5. yields vγp, γ R + c 3 K 3 R n B 5.3 for all p. By Theorem 4., foreveryε>0, there is δ 0,, depending only on 4 n,β 0 and ε, such that R n B c3 K 3 ε 5.4 for some R [4δR 0, R 0 ]. Substituting 5.4 into5.3gives5.6. If b 6, we put ε = ε 3,β 0 as given in Lemma 5.. Then Theorem 4. enables us to find R [4δ 0 R 0, R 0 ] such that R n B 4 c 3 K 3 ε, 5.5 where δ 0 only depends on n and β 0. Applying Lemma 5., one can find P G n,m, such that v γ R, P b ε 5.6 and v, P 3 whenever v, P 0 b. Theorem 3. ensures that v, P γ is a subharmonic function on. Taking P = P in 5. yields v γp, P v γ R, P + c 3 K 3 4 c 3 K 3 ε ε b 5.7 for all p. Here we have used 5.5 and5.6. From the above inequality immediately follows. 6 Bernstein type results Now we can start an iteration as in [0]and[9] to get the following estimates:

25 The Gauss image and Bernstein type theorems 735 Theorem 6. Let = { x, f x : x D R0 R n} be a graph with parallel mean curvature, and f β 0 with β 0 [, 3, then for arbitrary ε>0, there exists δ 0,, only depending on n,β 0 and ε, not depending on f and R 0, such that v,γo γ + ε on B δr0, where o = 0, f 0. In particular, if Df 0 = 0, then f + ε on D δr0. Proof Let {ɛ,...,ɛ n+m } be canonical orthonormal basis of R n+m and put P 0 = ɛ ɛ n. Then f β 0 implies v, P 0 β 0 on 0. If β 0 < 6, we put Q 0 = P 0. Otherwise by Theorem 5., one can find P G n,m, such that v, P γ β 0 ε on B δ0 R 0 6. with constants δ 0 and ε depending only on n and β 0. Similarly for each j, if β 0 jε < 6, then we put Q 0 = P j ; otherwise Theorem 5. enables us to find P j+ G n,m satisfying v, P j+ γ β 0 j + ε on B j+ δ R 0 Denoting [ ] 6 k = 6 3 then obviously β 0 kε <. Hence there exists Q 0 G n,m, such that 6 v, Q 0 γ b < on B δ k 0 R Again using Theorem 5., for arbitrary ε>0, there exists δ 0,, depending only on n,β 0 and ε, such that ε +, v, Q γ + + ε on B δ δ k 0 R for a point Q G n,m. With r, as in the proof of Lemma 5.,then r, Q γ = arccos v, Q γ arccos + ε. Using the triangle inequality we get r,γ0 γ r, Q γ + rγ 0, Q γ arccos + ε. Thus v,γ0 γ + ε on B δ δ k 0 R 0. It is sufficient to put δ = δ δ k 0. Letting R 0 + we can arrive at a Bernstein-type theorem: Theorem 6. Let z = f x,...,x n, =,...,m, be smooth functions defined everywhere in R n n 3, m. Suppose their graph = x, f x is a submanifold with parallel mean curvature in R n+m. Suppose that there exists a number β 0 < 3 with [ f = det δ ij + ] f f x i x j β Then f,..., f m has to be affine linear representing an affine n-plane.

26 736 J. Jost et al. Final remarks For any P 0 G n,m, denote by r the distance function from P 0 in G n,m. The eigenvalues of Hessr were computed in []. Then define { B JX P 0 = P G n,m : sum of any two Jordan angles between P and P 0 < π } in the geodesic polar coordinate neighborhood around P 0 on the Grassmann manifold. From 3., 3.7 and 3.9 in [] it turns out that Hessr > 0onB JX P 0. oreover, let B JX P 0 be a closed subset, then θ + θ β β 0 < π and Hessr cot β 0 g, where g is the metric tensor on G n,m. Hence, the composition of the distance function with the Gauss map is a strongly subharmonic function on, provided the Gauss image of the submanifold with parallel mean curvature in R n+m is contained in. The largest sub-level set of v, P 0 in B JX P 0 were studied in []. The Theorem 3. in [] shows that Therefore, and max{wp, P 0 ; P B JX P 0 }=. {P G n,m,v, P 0 <} B JX P 0, {P G n,m ; v, P 0 } B JX P 0 =. On the other hand, we can compute directly. From.7 wealsohave Hessv, P 0 = vωi + + λ vω + λ λ β vω ω ββ m+ i n, =β + + λ λ β v ω β + ω β <β + λ λ β v ω β ω β. It follows that v, P 0 is strictly convex on B JX P 0. oreover, if θ + θ β β 0 < π, then Hessv, P 0 tan θ tan θ β v g = cosθ + θ β v g cos β 0 v g cos θ cos θ β where g is the metric tensor of G n,m and vγ, P 0 cos β 0 v B cos β 0 B. Now, we define P 0 = B JX P 0 {P G n,m ; v, P 0 <3} G n,m. The function v, P 0 is not convex on whole P 0. But, its composition with Gauss map could be a strongly subharmonic function on. Therefore, we could obtain a more general result: Let be a complete submanifold in R n+m with parallel mean curvature. If its image under the Gauss map is contained in a closed subset of P 0 for some P 0 G n,m, then has to be an affine linear subspace.

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