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 and 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 and Osserman [3] for dimension n = 2 and Barbosa [2] 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 [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 et al. [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 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 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 and Xin [12], Wang [18] and Xin and Yang [21]. In [12], 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 [21]. 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. [17] and Ecker and 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 (Theorems 5.1 and 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.
) is a submanifold with parallel mean curvature in R n+m . Suppose that there exists a number β 0 < 3 such that (1.1) . . , 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 d f 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 0. 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 (2.1) The Jordan angles between P and Q are defined by The distance between P and Q is defined by Thus, (2.1) becomes In the sequel, we shall assume n ≥ m without loss of generality. We use the summation convention and agree on the ranges of indices: Now we fix P 0 ∈ G n,m . We represent it by n vectors i , which are complemented by m vectors n+α , such that { i , 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 (2.4) 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 .
Our fundamental quantity will be v(·, P 0 ) := w −1 (·, P 0 ) on U. (2.5) For arbitrary P ∈ U determined by an n × m matrix Z , it is easily seen that v(P, P 0 ) = det(I n + Z Z T ) 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 It follows that v(·, P 0 ) −1 Hess(v(·, P 0 )) = g + α 2λ 2 α ω 2 αα + α =β λ α λ β (ω αα ⊗ ω ββ + ω αβ ⊗ ω βα ). (2.8) 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 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 where i denotes the inclusion map and π : SO(n + m) → G n,m is defined by 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.
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 : 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,i j 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.
where (3.11) and It is easily seen that Obviously ( 3.15) It follows that (λ α , λ β ) → λ α λ β attains its maximum at the point satisfying Proof It is easily seen that then by the Cauchy-Schwarz inequality, (3.17) Hence It is well known that ax 2 + 2bx y + 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 . Then (3.20) follows and hence is nonnegative definite. We now verify the claim. For arbitrary ε > 0, denote then f ε is obviously a smooth function on The compactness of ε implies the existence of ( Fix x 0 , then (3.21) implies that for arbitrary (y, z) ∈ R 2 satisfying 1 ≤ y ≤ v, z ≥ v +2ε and yz = C x 0 , we have 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 ⊂ lim ε→0 + ε . Hence by letting ε → 0 one can obtain f ≤ 2 v−1 .
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].

Curvature estimates
(4.1) Denote by 1 , . . . , n+m the canonical basis of R n+m and put P 0 = 1 ∧ · · · ∧ n . Then by (4 Let η be a nonnegative smooth function on M with compact support. Multiplying both sides of (4.2) by η and integrating on M gives is obviously a diffeomorphism. F * where g is the metric tensor on M. In other words, M is isometric to the Euclidean ball . It is easily seen that for arbitrary ξ ∈ R n , On the other hand, f ≤ β 0 implies n i=1 μ i ≤ β 2 0 , with μ 1 , . . . , μ n the eigenvalues of (g i j ), thus In local coordinates, the Laplace-Beltrami operator is Here (g i j ) is the inverse matrix of (g i j ), and G = det(g i j ) = 2 f . From (4.1), (4.5) and (4.6) it is easily seen that Following [11,13] we shall make use of the following abbreviations: For arbitrary R ∈ (0, R 0 ), let (4.9) (4.7) shows that is a uniform elliptic operator. Moser's Harnack inequality [15] for supersolutions of elliptic PDEs in divergence form gives Lemma 4.1 For a positive superharmonic function h on B R with R ∈ (0, R 0 ], p ∈ (0, n n−2 ) and θ ∈ [ 1 2 , 1), we have the following estimate 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 . The a priori estimates for mollified Green functions of [8] tell us (4.10) 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 and Giusti [6] and Giaquita and Hildebrandt [7].
For arbitrary k ∈ Z + , (4.13) gives For arbitrary ε > 0, we take then we can find 1 ≤ j ≤ k, such that 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

Gauss image shrinking property
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. Denote α = arccos(a −1 ) and β = arccos(b −1 ). Now we put γ = α − β and Once v(P 2 , P 1 ) ≤ c, the triangle inequality implies , then a direct calculation shows β < α 2 , hence γ > β and moreover v(Q, P 1 ) ≤ b < c. Thereby P 2 = Q is the required point.
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, . . . , θ m L t, while the Jordan angles between γ (t) and P 1 are Since t → α sec θ α L (L − t) is a strictly decreasing function, there exists a unique t 0 ∈ [0, L), such that α sec θ α 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 θ α 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), is the required constant. 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 : is a smooth function on . t 0 < L implies F > 0; then the compactness of gives inf F > 0, and ε 2 = inf 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. M = (x, f (x)) : x ∈ D R 0 ⊂ R n be a graph with parallel mean curvature, and f ≤ β 0 with β 0 ∈ [1, 3). Assume there exists P 0 ∈ G n,m , such that v(·,
which is a compact subset of U.
As shown in Sect. 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 .
Note that tr(Z Z T ) 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 bȳ Note that T maps sublevel sets of v(·, P 0 ) onto Euclidean balls centered at the origin. Hence the convexity of Euclidean balls gives 14) 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 i j dx i dx j , and the eigenvalues of (g i j ) are bounded. Hence it is easy to get 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 then Similar to (5.10), The second term can be controlled by  . By Theorem 4.1, for every ε > 0, there is δ ∈ (0, 1), depending only on n, β 0 and ε, such that for some R ∈ [4δ R 0 , R 0 ]. Substituting (5.24) into (5.23) gives (5.6). If b ≥ √ 6 2 , we put ε 1 = ε 1 (3, β 0 ) as given in Lemma 5.1. Then Theorem 4.1 enables us to find R ∈ [4δ 0 R 0 , R 0 ] such that where δ 0 only depends on n and β 0 . Applying Lemma 5.1, one can find P 1 ∈ G n,m , such that v(γ R , P 1 ) ≤ b − ε 1 (5.26) and 1 ≤ v(·, P 1 ) ≤ 3 whenever 1 ≤ v(·, P 0 ) ≤ b. Theorem 3.1 ensures that v(·, P 1 ) • γ is a subharmonic function on B R . Taking P = P 1 in (5.22) yields v (γ (p), Here we have used (5.25) and (5.26). From the above inequality (5.7) immediately follows.
Letting R 0 → +∞ we can arrive at a Bernstein-type theorem: