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Curvature estimates for graphs with prescribed mean curvature and flat normal bundle Steffen Fr¨ohlich, Sven Winklmann November 8, 2018

Abstract n

We consider graphs Σ ⊂ Rm with prescribed mean curvature and flat normal bundle. Using techniques of Schoen, Simon and Yau [14] and Ecker-Huisken [4], we derive the interior curvature estimate sup |A|2 ≤

Σ∩BR

C R2

up to dimension n ≤ 5, where C is a constant depending on natural geometric data of Σ only. This generalizes previous results of Smoczyk, Wang and Xin [16] and Wang [20] for minimal graphs with flat normal bundle. Mathematics Subject Classification (2000): 35J60, 53A10, 49Q05

1

Introduction

Let ψ : Ω → Rk be a smooth function defined on a domain Ω ⊂ Rn , and denote by Σ = {(x, ψ(x)) : x ∈ Ω} the corresponding graph in Rm=n+k . In this paper we assume the normal bundle of Σ to be flat and prove the interior curvature estimate sup |A|2 ≤

Σ∩BR

C R2

(1)

up to dimension n ≤ 5, where |A| denotes the length of the second fundamental form, BR ⊂ Rm is a closed ball of radius R centered at some point p ∈ Σ, and C is a constant depending on natural geometric data of Σ only, see Theorem 3.3. Recently, curvature estimates for minimal graphs with flat normal bundle have been established independently by Smoczyk, Wang and Xin [16] and Wang [20]. In particular, they have obtained higher dimensional analogues of the famous Schoen-Simon-Yau estimates [14] and Ecker-Huisken’s Bernstein result [3] for entire minimal graphs of controlled growth. 1

Without any geometric restrictions on the normal bundle the situation turns out to be more complicated as can be seen from the counter example of Lawson-Osserman [10]. In [8] Hildebrandt, Jost and Widman have studied entire solutions of the minimal surface system ∂ √ ij ∂ψ α gg = 0, α = 1, . . . , k. ∂xi ∂xj P α ∂ψ α , (g ij ) = (gij )−1 and g = det(gij ). Using a Here, gij = δij + α ∂ψ ∂xi ∂xj regularity estimate for harmonic maps they could prove a Bernstein result under a suitable lower bound on the function !#−1/2 " X α α . w = det δij + Di ψ Dj ψ α

Later, their result has been improved by Jost-Xin [9] and Wang [19]. In fact, Wang’s Bernstein result holds for the entire class of area decreasing maps with bounded gradient. For a detailed survey on minimal graphs in higher co-dimension and further comments on the literature we refer to the recent monograph of Giaquinta-Martinazzi [6, Chapter 11]. We also remark, that more explicit estimates for two-surfaces in Rm can been obtained by using strictly two-dimensional techniques, cf. Osserman [13] and Bergner-Fr¨ ohlich [1]. The paper is organized as follows: In section 2 we first collect some basic facts on graphs with flat normal bundle. Using ideas of Ecker-Huisken [4] we then prove a rather general Simons inequality (Lemma 2.4) for the Laplacian of the length of the second fundamental form. In section 3 we use this estimate to derive the Lp curvature bound Z |A|p dHn ≤ CRn−p Σ∩BR

for some p > n with a constant C depending only on the geometric data of the problem, see Theorem 3.1. Here, we can proceed similarly as Winklmann [21] who established a corresponding estimate for hypersurfaces of prescribed anisotropic mean curvature. In view of a general mean value inequality (Lemma 3.2), which is of independent interest on its own, this leads to the desired curvature estimate (1). As an application of our results we recover the Bernstein result of Smoczyk, Wang and Xin [16] and Wang [20] for minimal graphs with flat normal bundle. Acknowledgement. The second author was financially supported by the Alexander von Humboldt foundation and the Centro di Ricerca Matematica Ennio De Giorgi via a Feodor Lynen research scholarship.

2

2

Notation and preliminary results

Let f : Σn → Rm=n+k be a smooth immersion of an n-dimensional, oriented manifold without boundary into euclidean m-space of arbitrary co-dimension k ≥ 1. We denote by g(X, Y ) = hdf (X), df (Y )i the induced metric with corresponding Levi-Civita connection ∇X Y = (DX Y )⊤ and curvature tensor R(X, Y )Z = ∇X ∇Y Z − ∇Y ∇X Z − ∇[X,Y ] Z. Here, X, Y, Z are smooth vectorfields on Σ, D denotes the covariant derivative on Rm and (·)⊤ is the projection onto T Σ, the tangent bundle of Σ, which we will always identify with df (T Σ). The second fundamental form is given by A(X, Y ) = (DX Y )⊥ = DX Y − ∇X Y, where (·)⊥ is the projection onto the normal bundle N Σ. Taking its trace defines the mean curvature vector H = trace(A). We also have an induced connection on the normal bundle N Σ defined by the relation ⊥ ∇⊥ X η = (DX η) for any normal section η. The corresponding curvature tensor is given by ⊥ ⊥ ⊥ ⊥ R⊥ (X, Y )ζ = ∇⊥ X ∇Y ζ − ∇Y ∇X ζ − ∇[X,Y ] ζ.

We remark that these connections extend naturally to higher order tensor bundles formed from T Σ and N Σ. For example, for an (0, r)-tensor T with values in N Σ the covariant derivative ∇T is given by (∇X T )(Y1 , . . . , Yr ) = ∇⊥ X T (Y1 , . . . , Yr ) − T (∇X Y1 , . . . , Yr ) − . . . − T (Y1 , . . . , ∇X Yr ).

Let us now choose local orthonormal frames {ei }i=1,...,n and {eα }α=n+1,...,m for T Σ and N Σ, respectively. In these frames the coefficients of the second fundamental form are given by hαij = hA(ei , ej ), eα i = −hDei eα , ej i 3

and the mean curvature vector by H = Hα eα

with Hα = hαii .

Here and in the following we are using Einstein’s summation convention: Repeated Latin and Greek indices are automatically summed from 1 to n and from n+1 to m, respectively, unless not otherwise stated. We also write ∇k hαij = h(∇ek A)(ei , ej ), eα i, Rijkl = g(R(ei , ej )ek , el ) and ⊥ = hR(ei , ej )eα , eβ i Rijαβ

for the coefficients of ∇A, R and R⊥ . The fundamental equations of Gauß, Codazzi and Ricci then take the form Rijkl = hαil hαjk − hαik hαjl ,

(2)

∇k hαij = ∇i hαjk

(3)

⊥ = hβik hαjk − hαik hβjk . Rijαβ

(4)

and We also write ∇i ∇j ϕ for the coefficients of ∇∇ϕ, the second covariant derivative of a smooth function ϕ. The Laplace-Beltrami operator is then given by ∆ϕ = ∇i ∇i ϕ. More generally, for any (0, r)-tensor with values in N Σ we write P ∇i ∇j Tαk1 ...kr for the coefficients of ∇∇T . Finally, we denote by |T |2 = i1 ,...,ir |T (ei1 , . . . , eir )|2 the square of the length of T . The following identity was first proved by Simons [15] and is a direct consequence of (2), (3) and (4). For further details see also Wang [17, Section 7]. Lemma 2.1 For an arbitrary immersion f : Σn → Rm the second fundamental form satisfies 1 ∆|A|2 = |∇A|2 + hαij ∇i ∇j Hα + Hα hαij hβjk hβki 2 X − (hαij hαkl )2 − |R⊥ |2 .

(5)

i,j,k,l

put

Next, we consider the parallel n-form Ω = dx1 ∧ . . . ∧ dxn on Rm and w = ∗Ω = Ω(e1 , . . . , en ),

where ∗ is the Hodge operator. The following equation is due to FischerColbrie [5] and Wang [18], [19]. For an alternative exposition we also refer to Giaquinta-Martinazzi [6, Chapter 11]: 4

Lemma 2.2 For an arbitrary immersion f : Σn → Rm the function w = ∗Ω satisfies X ⊥ ∆w + |A|2 w = Ωαi ∇i Hα − 2 Ωαβij Rijαβ , (6) α<β,i

where Ωαi = Ω(e1 , . . . , eα , . . . , en ) with eα occupying the i-th position, and Ωαβij = Ω(e1 , . . . , eα , . . . , eβ , . . . , en ) with eα , eβ occupying the i-th and j-th position, respectively. In this paper we are particularly interested in immersions with flat normal bundle, that is the case R⊥ = 0. The above equations then simplify as follows: 1 ∆|A|2 = |∇A|2 + hαij ∇i ∇j Hα + Hα hαij hβjk hβki 2 X − (hαij hαkl )2

(7)

i,j,k,l

and ∆w + |A|2 w = Ωαi ∇i Hα .

(8)

Suppose now that Σ = {(x, ψ(x)) : x ∈ Ω} is the graph of a smooth function ψ : Ω → Rk over some domain Ω ⊂ Rn . In this case one easily checks the identity w = [det (δij + Di ψ α Dj ψ α )]−1/2 . In particular we have w > 0. Define the quantity K1 by + K1 = w−1 Ωαi ∇i Hα ,

(9)

where g + denotes the positive part of the function g. Moreover, denote by Hn the n-dimensional Hausdorff measure. Then we can state an energy-type estimate as follows: Lemma 2.3 Suppose Σn ⊂ Rm is a graph with flat normal bundle. Then we have Z Z 2 2 n |∇ϕ|2 + K1 ϕ2 dHn (10) |A| ϕ dH ≤ Σ

Σ

for all testfunctions ϕ ∈

Cc∞ (Σ).

Proof: We test (8) with w−1 ϕ2 and perform a partial integration. This leads to Z Z Z −1 n 2 2 n w−2 |∇w|2 ϕ2 dHn w ϕ∇ϕ∇w dH − |A| ϕ dH = 2 Σ Σ ZΣ w−1 Ωαi ∇i Hα ϕ2 dHn . + Σ

5

The desired estimate now follows from the Cauchy-Schwarz inequality.

The next inequality generalizes the Simons inequality of Schoen, Simon and Yau [14] and Ecker-Huisken [4] for hypersurfaces in Rm to immersions with arbitrary co-dimension. Note that for H = 0 we can let ε ց 0 in (11) to obtain a corresponding estimate of Smoczyk, Wang and Xin [16] and Wang [20] for minimal immersions with flat normal bundle. Lemma 2.4 Let f : Σn → Rm be an immersion with flat normal bundle. Then we have the estimate 1 2 2 ∆|A| ≥ 1+ |∇|A||2 + hαij ∇i ∇j Hα 2 n+ε +Hα hαij hβjk hβki − |A|4 − C(n, ε)|∇H|2

(11)

for all ε > 0. Proof: From (7) we infer the estimate 1 ∆|A|2 ≥ |∇A|2 + hαij ∇i ∇j Hα + Hα hαij hβjk hβki − |A|4 . 2

(12)

In any point p0 ∈ Σ where |A| does not vanish, we have X ∇k hαij hαij . ∇k |A| = |A|−1 α,i,j

⊥ = 0 we infer from the Ricci equation that we may choose our Since Rijαβ frames such that in p0 all hαij , α = n+1, . . . , m, are simultaneously diagonal. Hence, we obtain 2 X X |∇|A||2 = |A|−2 ∇k hαii hαii α,i

k

≤

=

X

(∇k hαii )2

α,i,k

X

(∇k hαii )2 +

X (∇k hαkk )2 .

(13)

α,k

α,i,k i6=k

Moreover, we have |∇A|2 − |∇|A||2 ≥ =

X

α,i,j,k

X

(∇k hαij )2 − 2

X

(∇k hαii )2

α,i,k

(∇k hαij )

α,i,j,k i6=j

≥ 2

X

α,i,k i6=k

6

(∇k hαii )2 ,

(14)

where the last line P follows from the Codazzi equation. From ∇k Hα = i ∇k hαii we infer for fixed α and k

2 X X ∇k hαii . ∇k hαii + (∇k hαkk )2 = (∇k Hα )2 − 2∇k Hα i i6=k

i i6=k

Applying Young’s inequality and summing over α and k leads to X X n−1 2 2 (∇k hαkk ) ≤ (n − 1 + ε) (∇k hαii ) + 1 + |∇H|2 . ε α,i,k α,k

(15)

i6=k

Combining (12), (13), (14) and (15) now gives the desired estimate 1,∞ (11) in all points where |A|(p0 ) 6= 0. However, since |A| ∈ Wloc with ∇|A|(p0 ) = 0 whenever |A|(p0 ) = 0, we see that (11) must be globally true in the weak sense.

3

Curvature estimates

Following Ecker-Huisken [4] we define a quantity K2 by ( h ∇ i ∇ j Hα − , if |A| > 0 − αij |A| , K2 := 0 , if |A| = 0

(16)

where g− denotes the negative part of the function g. Clearly, we have the estimate K2 ≤ |∇∇H|. We will now prove the following integral curvature estimate: Theorem 3.1 If Σn ⊂ Rm is a graph with flat normal bundle, then we have Z |A|p ϕp dHn Σ Z p/2 p/3 |∇ϕ|p + |H|p + |∇H|p/2 + K1 + K2 ϕp dHn (17) ≤ C Σ

p for all p ∈ [4, 4 + 8/n) and for all non-negative testfunctions ϕ ∈ Cc∞ (Σ), the constant C depending on n and p only.

7

Proof: We test (10) with |A|q+1 ϕ, where ϕ ∈ Cc∞ (Σ) is a non-negative testfunction and q ≥ 0 is yet to be chosen, and obtain Z Z 2q+4 2 n 2 |A| ϕ dH ≤ (q + 1) |A|2q |∇|A||2 ϕ2 dHn Σ Σ Z |A|2q+1 ϕ∇|A|∇ϕ dHn +2(q + 1) Σ Z 2q+2 |A| (|∇ϕ|2 + K1 ϕ2 ) dHn . (18) + Σ

On the other hand, multiplying the Simons inequality (11) by |A|2q ϕ2 , integrating by parts and applying Young’s inequality in the form |Hα hαij hβjk hβki | ≤ C(n)|H||A|3 ≤ ε|A|4 + leads to

C(n) |H|2 |A|2 ε

Z 2 |A|2q |∇|A||2 ϕ2 dHn + 2q 1+ n+ε Σ Z Z 2q+4 2 n hαij ∇i ∇j Hα |A|2q ϕ2 dHn |A| ϕ dH − ≤ (1 + ε) Σ Σ Z Z 2q+2 2 2 n |A|2q |∇H|2 ϕ2 dHn |A| |H| ϕ dH + C +C Σ ZΣ 2q+1 n ϕ∇|A|∇ϕ dH −2 |A|

(19)

Σ

with C = C(n, ε). Combining (18) and (19) and recalling the definition of K2 we arrive at Z 2 2 1+ + 2q − (1 + ε)(q + 1) |A|2q |∇|A||2 ϕ2 dHn n+ε Σ Z |A|2q+2 (|∇ϕ|2 + |H|2 ϕ2 + K1 ϕ2 ) dHn ≤ C ΣZ Z |A|2q+1 K2 ϕ2 dHn |A|2q |∇H|2 ϕ2 dHn + C +C Σ ZΣ 2q+1 n |A| ϕ|∇|A|||∇ϕ| dH (20) +C Σ

with Cp = C(n, q, ε). We now choose q such that p = 4 + 2q. Then we have q ∈ [0, 2/n) and thus we can find ε > 0 small enough depending on n and q only such that 1+

2 + 2q − (1 + ε)(q + 1)2 > 0. n+ε

8

Hence, with this choice of ε we obtain Z |A|2q |∇|A||2 ϕ2 dHn Σ Z |A|2q+2 (|∇ϕ|2 + |H|2 ϕ2 + K1 ϕ2 ) dHn ≤ C ΣZ Z |A|2q+1 K2 ϕ2 dHn |A|2q |∇H|2 ϕ2 dHn + C +C Σ ZΣ 2q+1 n |A| ϕ|∇|A|||∇ϕ| dH +C Σ

with C = C(n, q). In view of Young’s inequality and (18) this leads to Z |A|2q+4 ϕ2 dHn Σ Z |A|2q+2 (|∇ϕ|2 + |H|2 ϕ2 + K1 ϕ2 ) dHn ≤ C ΣZ Z 2q 2 2 n |A|2q+1 K2 ϕ2 dHn (21) |A| |∇H| ϕ dH + C +C Σ

Σ

with C = C(n, q). To complete the proof we replace ϕ by ϕq+2 in (21) and obtain Z |A|2q+4 ϕ2q+4 dHn Σ Z |A|2q+2 ϕ2q+2 (|∇ϕ|2 + |H|2 ϕ2 + K1 ϕ2 ) dHn ≤ C ΣZ Z 2q 2q 2 4 n |A|2q+1 ϕ2q+1 K2 ϕ3 dHn |A| ϕ |∇H| ϕ dH + C +C Σ

Σ

with C = C(n, q). The desired inequality Z |A|2q+4 ϕ2q+4 dHn Σ Z |∇ϕ|2q+4 dHn ≤ C Σ Z 2q+4 2q+4 2q+4 2q+4 2 3 2 +C |H| + |∇H| + K1 + K2 ϕ2q+4 dHn Σ

t

now follows easily in view of the interpolation inequality ab ≤ γas + γ − s bt for all a, b ≥ 0, γ > 0 and s, t > 1 with 1s + 1t = 1. Denote by BR = BR (p) ⊂ Rm the closed ball of radius R > 0 with center p ∈ Σ. In order to obtain a sup curvature estimate we need the following mean value inequality. The proof is similar to [7, Theorem 8.17], however we assume less regularity on the coefficients of (22). For the convenience of the reader we sketch the argument below. 9

Lemma 3.2 Let Σn ⊂ Rm be an arbitrary graph, and suppose that u is a non-negative solution of ∆u + Qu ≥ g

on Σ,

(22)

where Q ∈ Lq/2 (Σ) and g ∈ Lp/2 (Σ) with q, p > n. If Σ ∩ B2R ⊂⊂ Σ then we have the estimate sup u ≤ C R−n/2 kukL2 (Σ∩B2R ) + k(R) , (23) Σ∩BR

where

k(R) = R2(1−n/p) kgkLp/2 (Σ∩B2R ) ,

(24)

the constant C depending on n, q, p, R2(1−n/q) kQkLq/2 (Σ∩B2R ) , R supΣ∩B2R |H| and R−n Hn (Σ ∩ B2R ). Proof: First, note that by scaling Rm → Rm , p 7→ Rp it suffices to consider the case R = 1. We now put v = u + k, where k = kgkLp/2 (Σ∩B2 ) , and let η ∈ Cc∞ (Σ) be a non-negative function supported in Σ ∩ B2 . For β ≥ 1 we multiply (22) with v β η 2 and perform a partial integration. This leads to Z Z v β−1 |∇u|2 η 2 dHn ≤ −2 v β η∇η∇u dHn β Σ Z Σ (25) + (Qu − g)v β η 2 dHn . Σ

Using Young’s inequality we find |2v β η∇η∇u| ≤

2 β β−1 v |∇u|2 η 2 + v β+1 |∇η|2 . 2 β

(26)

Furthermore, since v ≥ max(u, k) we have β 2

|(Qu − g)v η | ≤ v where yields

|g| k

β+1 2

η

|g| |Q| + k

,

(27)

is to be considered 0 in case k = 0. Combining (25), (26) and (27)

Z

Σ

v β−1 |∇u|2 η 2 dHn ≤

Z 4 v β+1 |∇η|2 dHn β2 Σ Z 2 |g| β+1 2 + v η |Q| + dHn . β Σ k

β+1

Hence, abbreveating w = v 2 we arrive at the estimate Z Z |∇w|2 η 2 dHn ≤ 4 w2 |∇η|2 dHn Σ Σ Z |g| 2 2 dHn . w η |Q| + +2β k Σ 10

(28)

Next, we apply the Sobolev-inequality of Michael-Simon [11] followed by H¨older’s inequality to obtain Z

2χ

n

1

χ

(ηw) dH Σ Z ≤ C (|∇η|2 w2 + η 2 |∇w|2 + η 2 w2 |H|2 ) dHn , Σ

n ˆ where χ = nˆ −2 with n ˆ = n for n ≥ 3 and 2 < n ˆ < min{q, p} for n = 2, respectively, and where C is a constant depending on n ˆ and Hn (Σ ∩ B2 ). Combining this with (28) leads to

Z

2χ

(ηw)

dH

n

Σ

1

χ

Z

w2 (η 2 + |∇η|2 ) dHn Σ Z |g| 2 2 dHn , w η |Q| + +Cβ k Σ

≤ C

(29)

the constant C now depending additionally on supΣ∩B2 |H|. Next we use interpolation inequalities for Lp -spaces, cf. [7, Section 7.1], and obtain Z w2 η 2 |Q| dHn Σ

≤

Z

(wη)

2q q−2

dH

n

Σ

q−2 Z q

q 2

Σ∩B2

|Q| dH

n

2 q

(30)

" Z 1 Z 1 #2 2χ 2 2χ n −µ 2 2 n (wη) dH ≤ ε +ε w η dH kQkLq/2 (Σ∩B2 ) Σ

Σ

n ˆ q−ˆ n

for all ε > 0 with µ = Z

> 0. Similarly, we have

|g| dHn k Σ " Z 1 1 #2 Z 2χ 2 (wη)2χ dHn w2 η 2 dHn ≤ ε + ε−˜µ w2 η 2

Σ

(31)

Σ

n ˆ with µ ˜ = p−ˆ n > 0. Hence, using (30), (31) with ε ∼ [β(kQkLq/2 (Σ∩B2 ) + −1/2 1)] in (29) we finally arrive at

Z

2χ

(ηw) Σ

dH

n

1

χ

≤ Cβ α

Z

Σ

w2 (η 2 + |∇η|2 ) dHn

(32)

with C depending on n, q, p, supΣ∩B2 |H|, Hn (Σ ∩ B2 ) and kQkLq/2 (Σ∩B2 ) , and α = α(n, p, q) > 1. 11

From here we can employ Moser’s iteration technique [12] in a manner similar to [4] and [21, Section 4]. Put γ := β + 1 ≥ 2 such that w2 = v γ . Let ρ, ρ′ be radii satisfying 1 ≤ ρ′ ≤ ρ ≤ 2 and let η ∈ Cc∞ (Σ) to be a cut-off function with 0 ≤ η ≤ 1, η = 1 in Σ ∩ Bρ′ , supp(η) ⊂ Σ ∩ Bρ , and C |∇η| ≤ ρ−ρ ′ . Then we infer from (32) the estimate Z

v χγ dHn Σ∩Bρ′

!

1 χγ

1

≤

α

Cγγγ

2

(ρ − ρ′ ) γ

!1 γ

Z

v γ dHn

Σ∩Bρ

(33)

with a constant C depending on n, q, p, supΣ∩B2 |H|, Hn (Σ ∩ B2 ) and kQkLq/2 (Σ∩B2 ) only. Now, let ρk = 1 + 2−k ,

ρ′k = ρk+1 ,

γk = 2χk

for k = 0, 1, 2, . . . .

Replacing ρ, ρ′ and γ in (33) by ρk , ρ′k and γk and iterating the resulting inequalities as k → ∞, we obtain the estimate sup v ≤ C

Σ∩B1

Z

2

v dH

n

Σ∩B2

1 2

with C depending on the same data as before. Recalling that v = u + k, this gives the desired result. Now we are ready to prove our main result. Theorem 3.3 Let Σn ⊂ Rm , 2 ≤ n ≤ 5, be a graph with flat normal bundle, and suppose that Σ ∩ B4R ⊂⊂ Σ with Hn (Σ ∩ B4R ) ≤ KRn . Then we have the estimate sup |A|2 ≤

Σ∩BR

C R2

(34)

with a constant C depending on n, K, R supΣ∩B4R |H|, R2 supΣ∩B4R (|∇H|+ K1 ) and R3 supΣ∩B4R K2 . Proof: In view of the Simons identity (7) and the estimate |Hα hαij hβjk hβki | ≤ C(n)|A|4 we infer ∆|A|2 + C(n)|A|4 ≥ −2K2 |A|. 12

Furthermore, since 2 ≤ n ≤ 5, we can apply Theorem 3.1 with a suitable cut-off function as before to obtain Z |A|q dHn ≤ CRn−q Σ∩B2R

for some q > max{n, 4}, with a constant C depending on n, K, R supΣ∩B4R |H|, R2 supΣ∩B4R (|∇H| + K1 ) and R3 supΣ∩B4R K2 . Hence, applying Lemma 3.2 with u = |A|2 , Q = C(n)|A|2 , g = −2K2 |A| and p = 2q, the desired estimate follows easily. In case H = 0 the constant in Theorem 3.3 is independent of R. Therefore, letting R → ∞ in (34) we obtain the Bernstein result of Smoczyk, Wang and Xin [16] and Wang [20]: Corollary 3.4 Suppose that Σ = {(x, ψ(x)) : x ∈ Rn } ⊂ Rm , 2 ≤ n ≤ 5, is an entire minimal graph with flat normal bundle. If Hn (Σ ∩ BR (p)) ≤ KRn for some point p ∈ Σ and some sequence R → ∞ with a constant K independent of R, then ψ is an affine linear function.

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Steffen Fr¨ ohlich Technische Universit¨ at Darmstadt FB Mathematik Schloßgartenstraße 7 64289 Darmstadt, Germany [email protected] Sven Winklmann Centro di Ricerca Matematica Ennio De Giorgi Scuola Normale Superiore di Pisa Piazza dei Cavalieri 3 56100 Pisa, Italy [email protected]

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