The Gauss image of entire graphs of higher codimension and Bernstein type theorems

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.


Introduction
We consider an oriented n-dimensional submanifold M in R n+m with n ≥ 3, m ≥ 2. The Gauss map γ : M → G n,m maps M into a Grassmann manifold. In fact, for codimension m = 1, this Grassmann manifold G n,1 is the unit sphere S n . In this paper, however, we are interested in the case m ≥ 2 where the geometry of this Grassmann manifold is more complicated. By the theorem of Ruh-Vilms [16], γ is harmonic if and only if M has parallel mean curvature. This result applies thus in particular to the case where M is a minimal submanifold of Euclidean space. 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 1. The central result is that an entire minimal graph M of dimension n ≤ 7 and codimension 1 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 Moser [15], called a weak Bernstein theorem, asserts that such an M 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 ≥ 2. Moser's result has been extended to higher codimension by Chern-Osserman for dimension n = 2 [3] and Barbosa [2] and Fisher-Colbrie [5] for dimension n = 3. For dimension n = 4 1991 Mathematics Subject Classification. 58E20,53A10. The second named author is grateful to the Max Planck Institute for Mathematics in the Sciences in Leipzig for its hospitality and continuous support. He is also partially supported by NSFC and SFMEC . and codimension m = 3, however, there is a counterexample given by Lawson-Osserman [14]. In fact, their paper emphasizes the stark contrast between the cases of codimension 1 and greater than 1 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 1, 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-Jost-Widman [10] 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 Grassmann 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 1, this in particular reproduces Moser'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 ≥ 2 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 [10] could be improved by Jost-Xin [12], Wang [18] and Xin-Yang [21]. In [12], the largest such geodesically convex set in a Grassmann 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. The most natural such convex function is the squared distance from some point, when its domain is restricted to a suitably small ball. As mentioned, the largest such ball on which a squared distance function is convex was utilized in [10]. As also mentioned, however, this result is not yet optimal, and other convex functions were systematically utilized in [21]. In that 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-Simon-Yau [17] and Ecker-Huisken [4], could be used in [21].
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.
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 3.1). 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 [1]. 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 [10], we can explore the relation with curvature estimates, and we shall utilize a version of the telescoping trick (Theorem 4.1) to finally obtain a quantitatively controlled Gauss image shrinking process (Theorem 5.1 and Theorem 6.1). In this way, we can understand why the submanifold is flat as the Bernstein result asserts. More precisely, we obtain the following Bernstein type result, which substantially improves our previous results. Theorem 1.1. Let z α = f α (x 1 , · · · , x n ), α = 1, · · · , m, be smooth functions defined everywhere in R n (n ≥ 3, m ≥ 2). Suppose their graph M = (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 Then f 1 , · · · , 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| 2 where K 0 is a positive constant and B is the second fundamental form of M in R n+m . This principle is not new. Wang [18] 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 > 0 such that for any two eigenvalues λ i , λ j with i = j, the inequality |λ i λ j | ≤ 1 − δ holds (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 2 = (1 + λ 2 i ), and while the condition of [12] which can be reformulated as v 2 being bounded away from 4 implies the condition of [18] so that the latter result generalizes the former, the condition needed in the present paper is only the weaker one that v 2 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.

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 SO(n + m)/SO(n) × SO(m).
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 M is denoted by w : G n,m × G n,m → R w(P, Q) = e 1 ∧ · · · ∧ e n , f 1 ∧ · · · ∧ f n = det W where P is spanned by a unit n-vector e 1 ∧ · · · ∧ e n , Q is spanned by another unit n-vector f 1 ∧ · · · ∧ f n , and W = e i , f j . It is well-known that  Here each 0 ≤ µ 2 i ≤ 1. Putting p := min{m, n}, then at most p elements in {µ 2 1 , · · · , µ 2 n } are not equal to 1. Without loss of generality, we can assume µ 2 i = 1 whenever i > p. We also note that the µ 2 i can be expressed as The Jordan angles between P and Q are defined by The distance between P and Q is defined by In the sequel, we shall assume n ≥ m without loss of generality. We use the summation convention and agree on the ranges of indices: 1 ≤ i, j, k, l ≤ n, 1 ≤ α, β, γ ≤ m, a, b, · · · = 1, · · · , n + m. Now we fix P 0 ∈ G n,m . We represent it by n vectors ǫ i ,, which are complemented by m vectors ǫ n+α , such that {ǫ i , ǫ n+α } form an orthonormal base of R m+n .

Denote
U := {P ∈ G n,m , w(P, P 0 ) > 0}. We can span an arbitrary P ∈ U by n-vectors f i : The canonical metric in U can be described as where Z = (Z iα ) is an (n × m)-matrix and I n (res. I m ) denotes the (n × n)-identity (res. m × m) matrix. It is shown that (2.4) can be derived from (2.2) in [20].
For any P ∈ U, the Jordan angles between P and P 0 are defined by {θ i }. Let E iα be the matrix with 1 in the intersection of row i and column α and 0 otherwise. Then, sec θ i sec θ α E iα form an orthonormal basis of T P G n,m with respect to (2.4). Denote its dual frame by ω iα .
For arbitrary P ∈ U determined by an n × m matrix Z, it is easily seen that where θ 1 , · · · , θ m denote the Jordan angles between P and P 0 .
In this terminology, Hess(v(·, P 0 ) has been estimated in [21]. By (3.8) in [21], we have (2.7) It follows that The canonical Riemannian metric on G n,m derived from (2.2) can also be described by the moving frame method. This will be useful for understanding some of the sequel. Let {e i , e n+α } be a local orthonormal frame field in R n+m . Let {ω i , ω n+α } be its dual frame field so that the Euclidean metric is The Levi-Civita connection forms ω ab of R n+m are uniquely determined by the equations It is shown in [20] that the canonical Riemannian metric on G n,m can be written as

Subharmonic functions
Let M m → R n+m be an isometric immersion with second fundamental form B. Around any point p ∈ M, we choose an orthonormal frame field e i , · · · , e n+m in R n+m , such that {e i } are tangent to M and {e n+α } normal to M. The metric on M is g = i ω 2 i . We have the structure equations (3.1) where h αij are the coefficients of second fundamental form B of M in R n+m .
Let 0 be the origin of R n+m , SO(m + n) be the Lie group consisting of all orthonormal frames (0; e i , e n+α ), T F = (p; e 1 , · · · , e n ) : p ∈ M, e i ∈ T p M, e i , e j = δ ij be the principle bundle of orthonormal tangent frames over M, and NF = (p; e n+1 , · · · , e n+m ) : p ∈ M, e n+α ∈ N p M be the principle bundle of orthonormal normal frames over M. Thenπ : T F ⊕ NF → M is the projection with fiber SO(n) × SO(m).
The Gauss map γ : M → G n,m is defined by via the parallel translation in R n+m for every p ∈ M. Then the following diagram commutes It follows that (2.8) was computed for the metric (2.4) whose corresponding coframe field is ω iα . Since (2.4) and (2.9) are equivalent to each other, at any fixed point P ∈ G n,m there exists an isotropic group action, i.e., an SO(n) × SO(m) action, such that ω iα is transformed to ω i n+α , namely, there are a local tangent frame field and a local normal frame field such that at the point under consideration, In conjunction with (3.1) and (3.3) we obtain By the Ruh-Vilms theorem [16], the mean curvature of M is parallel if and only if its Gauss map is a harmonic map. Now, we assume that M has parallel mean curvature.
We define This function v on M 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), M can be described as an entire graph on R n by f : R n → R m , provided M is complete. In this situation, λ i is the singular values of df and Using the composition formula, in conjunction with (2.8), (3.2) and (3.4), and the fact that τ (γ) = 0 (the tension field of the Gauss map vanishes [16]), we deduce the important formula of Lemma 1.1 in [5] and Prop. 2.1 in [18].
Proposition 3.1. Let M be an n−submanifold in R n+m with parallel mean curvature. Then where h α,ij are the coefficients of the second fundamental form of M in R n+m (see (3.1).
A crucial step in this paper is to find a condition which guarantees the strong subharmonicity of the v− function on M. More 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.
and (3.12) It is easily seen that Obviously (3.14) .
It follows that (λ α , λ β ) → λ α λ β attains its maximum at the point satisfying Proof. It is easily seen that γα , then by the Cauchy-Schwarz inequality, It is well known that ax 2 + 2bxy + cy 2 is nonnegative definite if and only if a, c ≥ 0 and ac − b 2 ≥ 0. Hence the right hand side of (3.18) is nonnegative definite if and only if We claim f ≤ 2 v−1 on Ω. Then (3.20) follows and hence We now verify the claim. For arbitrary ε > 0, denote then f ε is obviously a smooth function on The compactness of Ω ε implies the existence of (x 0 , y 0 , z 0 ) ∈ Ω ε satisfying Differentiating both sides of yz = C x 0 yields dy y + dz z = 0. Hence It implies that f ε,x 0 y, C yx 0 is decreasing in y and y 0 = 1. Similarly, one can derive x 0 = 1. Therefore Note that f ε → f and Ω ⊂ lim ε→0 + Ω ε . Hence by letting ε → 0 one can obtain f ≤ 2 v−1 .
Lemma 3.2. There exists a positive constant ε 0 , such that if v ≤ 3, then Obviously hence, the third term of the right hand side of (3.23) satisfies If there exist 2 distinct indices β, γ = α satisfying , then a direct calculation shows .
Remark 3.1. 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 1.2 in [18].
Here γ 1 is a positive constant only depending on n, p and θ, but not on h and R.
(4.2) shows the subharmonicity of v, and therefore v +,R − v + ε is a positive superharmonic function on B R for arbitrary ε > 0. With the aid of Lemma 4.1, one can follow [11] to get Corollary 4.1. There is a constant δ 0 ∈ (0, 1), depending only on n, such that Denote by G ρ the mollified Green function for the Laplace-Beltrami operator on B R . Then for arbitrary p = (y, f (y)) ∈ B R , once Based on (4.3), Corollary 4.1 and Lemma 4.2, we can use the method of [11] to derive a telescoping lemma a la Giaquinta-Giusti [6] and Giaquita-Hildebrandt [7]. Theorem 4.1. There exists a positive constant C 1 , only depending on n and β 0 , such that for arbitrary R ≤ R 0 , Moreover, there exists a positive constnat C 2 , only depending on n and β 0 , such that for arbitrary ε > 0, we can find R ∈ [exp(−C 2 ε −1 )R 0 , R 0 ], such that (4.14) Proof. With By (4.10), there exist positive constants c 3 , c 4 , depending only on n, such that Hence (4.15) .
For arbitrary k ∈ Z + , (4.13) gives then we can find 1 ≤ j ≤ k, such that

A Gauss image shrinking property
Lemma 5.1. For arbitrary a > 1 and β 0 ∈ [1, a), there exists a positive constant ε 1 = ε 1 (a, β 0 ) with the following property. If P 1 , Q ∈ G n,m satisfies v(Q, P 1 ) ≤ b ≤ β 0 , then we can find P 2 ∈ G n,m , such that v(P, P 2 ) ≤ a for every P ∈ G n,m satisfying v(P, P 1 ) ≤ b, and Proof. Obviously w(P, P ) = 1 for every P ∈ G n,m , which shows G n,m is a submanifold in a Euclidean sphere via the Plücker embedding. Denote by r(·, ·) the restriction of the spherical distance on G n,m , then by spherical geometry, w = cos r and hence v = sec r.
Otherwise b ≥ √ 2(1 + a −1 ) − 1 2 and hence c ≤ b. Obviously one of the following two cases has to occur: Case I. v(Q, P 1 ) < c. One can take P 2 = Q to ensure v(·, P 2 ) ≤ a whenever v(·, P 1 ) ≤ b. In this case Case II. v(Q, P 1 ) ≥ c. Denote by θ 1 , · · · , θ m the Jordan angles between Q and P 1 , and put L 2 = 1≤α≤m θ 2 α , then as shown in [19], if we denote the shortest normal geodesic from Q to P 1 by γ, then the Jordan angles between Q and γ(t) are θ 1 L t, · · · , θm L t, while the Jordan angles between γ(t) and P 1 are Since t → α sec θα L (L − t) is a strictly decreasing function, there exists a unique t 0 ∈ [0, L), such that α sec θα L (L − t 0 ) = c. Now we choose P 2 = γ(t 0 ), then v(P 2 , P 1 ) = c and It remains to show b− α sec θα L t 0 is bounded from below by a universal positive constant ε 2 . Once this holds true, in conjunction with (5.3) and (5.4), t 0 can be regarded as a smooth function on which is the unique one satisfying (By (5.2), c can be viewed as a function of b.) The smoothness of t 0 follows from the implicit function theorem. Therefore F : Ω → R is a smooth function on Ω. t 0 < L implies F > 0; then the compactness of Ω gives inf Ω F > 0, and ε 2 = inf Ω F is the required constant.
Remark 5.1. ε 1 is only depending on a and β 0 , non-decreasingly during the iteration process in Theorem 6.1.
As shown in Section 2, 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 Note that tr(ZZ T ) 1 2 = ( i,α Z 2 iα ) 1 2 equals |Z| when Z is treated as a vector in R nm . Since t ∈ [0, +∞) → det I + (tZ)(tZ) T 1 2 is a strictly increasing function and maps [0, +∞) onto [1, +∞), T is a diffeomorphism. By (2.6), |T (Z)| = v(P, P 0 ) − 1. Via T , we can define the mean value of γ on B R by Note that T maps sublevel sets of v(·, P 0 ) onto Euclidean balls centered at the origin. Hence the convexity of Euclidean balls gives The compactness of V ensures the existence of positive constants K 1 and K 2 , such that for arbitrary X ∈ T V, The classical Neumann-Poincaré inequality says As shown above, B R 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 Here φ can be a vector-valued function.

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 Hess(r) were computed in [12]. Then define B JX (P 0 ) = P ∈ G n,m : sum of any two Jordan angles between P and P 0 < π 2 in the geodesic polar coordinate neighborhood around P 0 on the Grassmann manifold. From (3.2), (3.7) and (3.9) in [12] it turns out that Hess(r) > 0 on B JX (P 0 ). Moreover, let Σ ⊂ B JX (P 0 ) be a closed subset, then θ α + θ β ≤ β 0 < π 2 and Hess(r) ≥ 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 M, provided the Gauss image of the submanifold M with parallel mean curvature in R n+m is contained in Σ. The largest sub-level set of v(·, P 0 ) in B JX (P 0 ) were studied in [12]. The Theorem 3.2 in [12] shows that max{w(P, P 0 ); P ∈ ∂B JX (P 0 )} = 1 2 .
On the other hand, we can compute directly. From (2.7) we also have Hess(v(·, P 0 )) =