A Bernstein Property of Affine Maximal Hypersurfaces

Annals of Global Analysis and Geometry 23: 359–372, 2003. © 2003 Kluwer Academic Publishers. Printed in the Netherlands.

359

A Bernstein Property of Affine Maximal Hypersurfaces AN-MIN LI and FANG JIA Department of Mathematics, Sichuan University, Chengdu, Sichuan, P. R. China. e-mail: {math-li,galaxyly2000}@yahoo.com.cn (Received: 18 March 2002; accepted: 5 December 2002) Communicated by: D. Ferus (TU Berlin) Abstract. Let x: M → An+1 be a locally strongly convex hypersurface, given as a graph of a n locally strongly convex function
xn+1 = f (x1 , . . . , xn ) defined in a domain ⊂ A . We introduce a Riemannian metric G# = (∂ 2 f /∂xi ∂xj ) dxi dxj on M. In this paper, we investigate the affine maximal hypersurfaces which are complete with respect to the metric G# and prove a Bernstein property for the affine maximal hypersurfaces. Mathematics Subject Classification (2000): 53A15. Key words: Bernstein property, affine maximal hypersurface.

Introduction Affine maximal hypersurfaces are extremals of the interior variation of affinely invariant volume. The corresponding Euler–Lagrange equation is a fourth-order PDE. Originally, these hypersurfaces were called ‘affine minimal hypersurfaces’. Calabi calculated the second variation and proposed calling them ‘affine maximal’ (see [1]). For affine maximal surfaces, there are different versions of so called affine Bernstein conjectures. One is Chern’s conjecture (see [6]). Another is a problem raised by Calabi (see [2]), called Calabi’s conjecture (see [12]). The two conjectures differ in the assumptions on the completeness of the affine maximal surface considered. While Chern assumed that the surface is a convex graph over R 2 , which means that the surface is Euclidean complete, Calabi assumed that the surface is complete with respect to the Blaschke metric. In [13], the authors present a proof of Chern’s conjecture. Under an additional assumption, the first author gave a partial answer to Calabi’s conjecture (see [8]). Calabi’s conjecture was recently solved, see [9, 14]. Let x: M → An+1 be an affine maximal hypersurface given by a locally strongly convex function xn+1 = f (x1 , x2 , . . . , xn )

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defined in a domain ⊂ An . In this paper, we say that f is ‘locally strongly convex’ which means that the Hessian of the function f is positive definite. Then x (M) is an affine maximal hypersurface if and only if f satisfies the following fourth-order PDE:   2 −1/(n+2) ∂ f  det = 0, ∂xi ∂xj where  denotes the Laplacian with respect to the Blaschke metric G, which is defined by   ∂   ∂ 1 ij G det(Gkl ) . = √ ∂xi ∂xj det(Gkl ) Following Calabi [3] and Pogorelov [11], we consider the Riemannian metric G# on M, defined by  fij dxi dxj , G# = where fij = ∂ 2 f /∂xi ∂xj . This is a very natural metric for a convex graph. Our main result can be stated as follows: THEOREM. Let xn+1 = f (x1 , . . . , xn ) be a locally strongly convex function defined in a domain ⊂ An . If M = {(x1 , . . . , xn , f (x1 , . . . , xn ) | (x1 , . . . , xn ) ∈ } is an affine maximal hypersurface, and if M is complete with respect to the metric G# , then, in the case where n = 2 or n = 3, M must be an elliptic paraboloid. For the proof of the theorem, we first show that if the maximal hypersurface M is complete with respect to the metric G# and if the norm of its Ricci curvature Ric# G# is bounded, then M must be an elliptic paraboloid. Next, we use Hofer’s Lemma to prove that Ric# G# must be bounded. 1. Preliminaries Let An+1 be the unimodular affine space of dimension n + 1, M be a connected and oriented C ∞ manifold of dimension n, and x: M → An+1 a locally strongly convex hypersurface. We choose a local unimodular affine frame field x, e1 , e2 , . . . , en , en+1 on M such that e1 , . . . , en ∈ Tx M, det(e1 , . . . , en , en+1 ) = 1, Gij = δij ,

en+1 = Y,

A BERNSTEIN PROPERTY OF AFFINE MAXIMAL HYPERSURFACES

361

where we denote the Blaschke metric and the affine normal vector field by Gij and Y , respectively. Denote by U , Aij k and Bij the affine conormal vector field, the Fubini–Pick tensor and the affine Weingarten tensor with respect to the frame field x, e1 , . . . , en , and by Rij denote the Ricci curvature. We have the following local formulas (see [10]):  Aij k U,k − Bij U, (1.1) U,ij = − U = −nL1 U,  Aiik = 0, Rij =



Amli Amlj +

(1.2) (1.3) n−2 n Bij + L1 δij , 2 2

(1.4)

where L1 denotes the affine mean curvature, and ‘,’ denotes the covariant differentiation with respect to the Blaschke metric (note that with respect to the orthonormal j frame field x, e1 , . . . , en , we have Aij k = Aik ). A locally strongly convex hypersurface is called an affine maximal hypersurface if L1 = 0 everywhere. Let x: M → An+1 be given by a locally strongly convex function xn+1 = f (x1 , . . . , xn ). We choose the following unimodular affine frame field:   ∂f , e1 = 1, 0, . . . , 0, ∂x1   ∂f , e2 = 0, 1, 0, . . . , 0, ∂x2 .........   ∂f , en = 0, 0, . . . , 1, ∂xn en+1 = (0, 0, . . . , 0, 1). Then the Blaschke metric is given by (see [10])   2 −1/(n+2)  2 ∂ f ∂ f dxi dxj . G = det ∂xi ∂xj ∂xi ∂xj The affine conormal vector field U can be identified with    2 −1/(n+2)  ∂f ∂f ∂ f ,...,− ,1 . − U = det ∂xi ∂xj ∂x1 ∂xn

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The formula U = −nL1 U implies that x(M) is an affine maximal hypersurface if and only if f satisfies the following PDE:   2 −1/(n+2) ∂ f  det = 0, (1.5) ∂xi ∂xj where  denotes the Laplacian with respect to the Blaschke metric, which is defined by   ∂   ∂ 1 ij G det(Gkl ) . = √ ∂xi ∂xj det(Gkl ) Denote   2 −1/(n+2) ∂ f , ρ = det ∂xi ∂xj and =

∇ρ 2G . ρ

(1.6)

Then (1.5) gives ρ = 0.

(1.7)

Now we calculate . Let p ∈ M. We choose an orthonormal frame field around p ∈ M. Then
2 ρ,j , = ρ

ρ,i ρ,j2 2 ρ,j ρ,j i − , ,i = ρ ρ2


2 ρ,j2 i + 2 ρ,j ρ,j ii ρ,j ρ,i ρ,j i ∇ρ 4G −4 + 2 .  = ρ ρ2 ρ3 In the case where (p) = 0, it is easy to get, at p,
2 2 ρ,ij .  ≥ ρ

(1.8)

Now we assume that (p)  = 0 and choose an orthonormal frame field such that, at p ∈ M, ρ,1 = ∇ρ , ρ,i = 0, ∀i > 1. Then

2 4 2 ρ,j2 i + 2 ρ,j ρ,j ii ρ,1 ρ,1 ρ,11 −4 +2 3. (1.9)  = ρ ρ2 ρ

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A BERNSTEIN PROPERTY OF AFFINE MAXIMAL HYPERSURFACES

The Schwarz inequality gives    2 2 2 2 ≥ ρ,11 + ρ,ii +2 ρ,1i ρ,ij i>1

i>1



 1 2 ≥ ρ,11 + ρ,ii n − 1 i>1 = Similarly,  A2ml1 ≥

n ρ2 + 2 n − 1 ,11



2 +2



2 ρ,1i

i>1

2 ρ,1i .

(1.10)

i>1

n A2 . n − 1 111

(1.11)

Taking the (n + 1)-th component of U,ij = −




Aij k U,k − Bij U , we have

ρ,11 = −A111 ρ,1 − B11 ρ.

(1.12)

Using the formula (1.4) and (1.12), we get   2 2 2 =2 A2ml1 ρ,1 + (n − 2)B11 ρ,1 2 ρ,j ρ,j ii = 2R11 ρ,1 = 2



2 A2ml1 ρ,1 − (n − 2)

≥ −(n − 2)

2 ρ,11 ρ,1

ρ

−

2 ρ,11 ρ,1

ρ

− (n − 2)A111

4 (n − 2)2 (n − 1) ρ,1 . 8n ρ2

Substituting (1.10) and (1.13) into (1.9), we obtain
2 2 2n ρ,11 i>1 ρ,1i +4 −  ≥ n−1 ρ ρ  4  2 ρ,11 ρ,1 (n − 2)2 (n − 1) ρ,1 + 2− . − (n + 2) ρ2 8n ρ3 Note that

2 2 2 4 2 ρ,11 ρ,1 ρ,1 ρ,11 ,i i>1 ρ,1i =4 +4 + 3 −4 .  ρ ρ ρ ρ2 Then (1.14) and (1.15) together give us
2 ,i n2 − n − 2  ρ,i n − ,i +  ≥ 2(n − 1)  2(n − 1) ρ  4  n2 − 2 ρ,1 (n − 2)2 (n − 1) − . + 2− 8n 2(n − 1) ρ 3

3 ρ,1

ρ (1.13)

(1.14)

(1.15)

(1.16)

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For n = 2, by (1.16) we have
2 ,i 1  ≥ + 2 .  ρ

(1.17)

For n = 3, by (1.16) we have
2 ,i 2  ρ,i 3 + 16 − ,i .  ≥ 4  ρ ρ

(1.18)

Denote by # and Ric# the Laplacian and the Ricci curvature with respect to the metric G# , resp. By definition of Laplacian and a direct calculation, we get ∇ρ 2G# ρ

= ∇ρ 2G ,

# r = ρr −

(1.19)

n − 2 < ∇ρ, ∇r >G# , 2 ρ

(1.20)

where r is the geodesic distance function with respect to the metric G# on M. Denote fij k = ∂ 3 f /∂xi ∂xj ∂xk . We have the following formula (see [11, p. 38]):  (1.21) f j l f hm (fhil fmj k − fhik fmj l ), Ric#ik = 14 where (f ij ) denotes the inverse matrix of (fij ). 2. Proof of the Theorem We first prove the following lemma: LEMMA 1. Let x: M → An+1 be a locally strongly convex affine maximal hypersurface, which is given by a locally strongly convex function: xn+1 = f (x1 , . . . , xn ). If M is complete with respect to the metric G# , and if there is a constant N > 0 such that Ric# 2G# ≤ N everywhere, then in the case where n = 2 or n = 3, M must be an elliptic paraboloid. Proof. Let p0 ∈ M. By adding a linear function and taking a parameter transformation we may assume that p0 has coordinates (0, . . . , 0) and f (0) = 0,

fi (0) = 0,

fij (0) = δij .

Denote by r(p0 , p) the geodesic distance function from p0 with respect to the metric G# . For any positive number a, let Ba (p0 ) = {p ∈ M | r(p0 , p) ≤ a}. Consider the function F = (a 2 − r 2 )2 

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defined on Ba (p0 ). Obviously, F attains its supremum at some interior point p ∗ . We may assume that r 2
is a C 2 -function in a neighborhood of p ∗ , and  > 0 at ∗ ∗ p . Then at p , F,i = 0, F,ii ≤ 0, where ‘,’ denotes the covariant differentiation with respect to the Blaschke metric. We calculate both expression explicitly 2(r 2 ),i ,i − 2 = 0,  a − r2
2 ,i 2 ∇r 2 2G 2r 2  − − − ≤ 0.  2 (a 2 − r 2 )2 a 2 − r 2

(2.1) (2.2)

Inserting (2.1) into (2.2) and noting that ∇r 2 2G = 4r 2 ∇r 2G , r 2 = 2 ∇r 2G + 2rr, we get 6 ∇r 2 2G 2r 2  ≤ +  (a 2 − r 2 )2 a 2 − r 2 =

24r 2 ∇r 2G 4 ∇r 2G 4rr + + 2 . 2 2 2 2 2 (a − r ) a −r a − r2

(2.3)

Since 2(n − 2)r < ∇ρ, ∇r >G# 4r # r 4rr + 2 = 2 2 2 2 a −r a −r ρ a − r2 ρ2 ≤

4r # r + · a2 − r 2 ρ

1 24

∇ρ 2G# ρ3

+

24(n − 2)2 r 2 ∇r 2G# , (a 2 − r 2 )2 ρ

and ∇ρ 2G# = ∇ρ 2G , ρ ∇r 2G# ρ

= ∇r 2G ,

∇r 2G# = 1, we have 24r 2 ∇r 2G# 4 ∇r 2G# 4r# r  ≤ + + +  (a 2 − r 2 )2 ρ (a 2 − r 2 )ρ (a 2 − r 2 )ρ + =

1 24

 24(n − 2)2 r 2 + 2 ∇r 2G# ρ (a − r 2 )2 ρ

4 4r# r 24(1 + (n − 2)2 )r 2 + + + (a 2 − r 2 )2 ρ (a 2 − r 2 )ρ (a 2 − r 2 )ρ

1 24

 . ρ

(2.4)

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Recall that (M, G# ) is a complete Riemannian manifold with the Ricci curvature bounded from below by a constant −K, K > 0. We have √ r# r ≤ (n − 1)(1 + Kr). (2.5) Consequently, from (2.4) it follows that

√ 24(1 + (n − 2)2 )r 2 4n 4(n − 1) K · r  ≤ + 2 + +  (a 2 − r 2 )2 ρ (a − r 2 )ρ (a 2 − r 2 )ρ

1 24

 . (2.6) ρ

For n = 3, we have by (1.18)  ≥  ≥

3 4

 2,i 2

3 4

= −

+

1 6

  ,i ρi − · ρ  ρ

  2,i −3 + 2 36r 2 + (a 2 − r 2 )2 ρ

1 12

1 12

 ρ

 , ρ

(2.7)

where we used (2.1). Inserting (2.7) into (2.6) we get √ 288 192 K · r 2016r 2 + + . ≤ 2 (a − r 2 )2 a 2 − r 2 a2 − r 2

(2.8)

Multiply both sides of (2.8) by (a 2 − r 2 )2 . We obtain, at p ∗ , √ (a 2 − r 2 )2  ≤ 2304a 2 + 192 Ka 3 = b1 a 2 + b2 a 3 , (2.9) √ where b1 = 2304 and b2 = 192 K. It is obvious from (1.17) that (2.9) also holds for n = 2. Hence, at any interior point of Ba (p0 ), we have  ≤ b1

1

a2 1 −

r2 a2

1 2 + b2

a 1−

 r2 2 a2

.

Let a → ∞, then  ≡ 0. It follows that  2  ∂ f = 1, det ∂xi ∂xj and G# = G.

(2.10)

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A BERNSTEIN PROPERTY OF AFFINE MAXIMAL HYPERSURFACES

This means that M is an affine complete parabolic affine hypersphere. From a result of Calabi (see [10]) we conclude that M must be an elliptic paraboloid.
We now want to show that there is a constant N > 0 such that ||Ric# ||2G# ≤ N everywhere. To this end, we need the following lemma (see [7, p. 635, lemma 26]), which was applied several times in symplectic geometry. LEMMA 2 (Hofer [7]). Let (X, d) be a complete metric space with metric d, and Ba (p) = {x|d(p, x) ≤ a} be a ball with center p and radius a. Let H be a positive continuous function defined on B2a (p). Then there is a point q ∈ Ba (p) and a positive number - ≤ a/2 such that H (x) ≤ 2H (q)

for all x ∈ B- (q) and -H (q) ≥

a H (p). 2

Now we assume that ||Ric# ||2G# is not bounded above. Then there is a sequence of points p. ∈ M such that ||Ric# ||2G# (p. ) → ∞. Let B1 (p. ) be the geodesic ball with center p. and radius 1. Consider a family of functions /(.): B2 (p. ) → R, . ∈ N, defined by /(.) = ||Ric# ||G# +  + L,

(2.11)

where  is defined by (1.6) and  L= f il f j m f kn fij k flmn . Using Hofer’s Lemma with H = / 1/2 , we find a sequence of points q. and positive numbers -. such that / 1/2 (x) ≤ 2/ 1/2 (q. ),

∀ x ∈ B-. (q. ),

-. / 1/2 (q. ) ≥ 12 / 1/2 (p. ) → ∞.

(2.12) (2.13)

The restriction of the hypersurface x to the balls B-. (q. ) defines a family M(.) of maximal hypersurfaces. For every ., we normalize M(.) as follows: Step 1. By adding a linear function we may assume that, at ql , (x1 , . . . , xn ) = (0, . . . , 0) and f (0) = 0,

fi (0) = 0.

We take a parameter transformation:  j ai (.)xj (.), xi (.) = j

(2.14) j

where ai (.) are constants. Choosing ai (.) appropriately and using an obvious , we may assume that, for every ., we have f ij (0) = δij . Note that, notation f , /

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is invariant. under the parameter transformation (2.14), / Step 2. We take an affine transformation by xi (.),

xi (.) = a(.)

1 ≤ i ≤ n,

xn+1 (.),

xn+1 (.) = λ(.)

where λ(.) and a(.) are constants. It is easy to verify that each M(.) again is a locally strongly convex maximal hypersurface. We now choose λ(.) = a(.)2 ,

, one can see that (q. ). Using again an obvious notation f , / λ(.) = / f ij (.) = f ij (.),

(.) = /

1 (.). / λ(.)

The first equation is trivial. We calculate the second one. From (1.6), (1.21) we can easily get ,

= 1  λ ˜ ||G˜ # = ||Ric #

1 # Ric G# , λ

.

= 1L L λ Then the second equality follows.

a (q. ) = {x ∈ M(.) | r(.)(q. , x) ≤ a}, where r(.) is the geodesic We denote B

# on M(.).

(.) is defined on Then / distance function with respect to the metric G

d(.)(q. ) with the geodesic ball B d(.) = -. / 1/2 (q. ) ≥ 12 / 1/2 (p. ) → ∞. From (2.12) we have

(q. ) = 1, /

(x) ≤ 4, /

(2.15)

d(.)(q. ). ∀x ∈ B

(2.16)

We may identify the parametrization as (ξ1 , . . . , ξn ) for any index .. Then f (.) is a sequence of functions defined in a domain (.) with 0 ∈ (.). Thus we have

a sequence M(.) of maximal hypersurfaces given by f (.). We have f (.)(0) = 0,

∂ f (.) (0) = 0, ∂ξi

∂ 2 f (.) (0) = δij , ∂ξi ξj

(2.17)

A BERNSTEIN PROPERTY OF AFFINE MAXIMAL HYPERSURFACES

(.) = 1, /

369 (2.18)

(.)(x) ≤ 4, / d(.) → ∞,

d(.)(0). ∀x ∈ B

(2.19)

as . → ∞.

(2.20)

We need the following lemma: LEMMA 3. Let M be an affine maximal hypersurface defined in a neighborhood of 0 ∈ Rn . Suppose that, with the notations from above, (i)

fij (0) = δij ,

(ii)

Ric# G# +  +



f il f j m f km fij k flmn ≤ 4.


2 ξi ≤ 1/4n2 }. Then there is constant C1 > 0 such Denote D := {(ξ1 , . . . , ξn )| that, for (ξ1 , . . . , ξn ) ∈ D, the following estimates hold:
(1) fii ≤ 4n, (2) 1/C1 ≤ det(fij ) ≤ C1 ,
(3) Define do by do2 = 1/7n2 (4n)n−1 C1 then do ⊂ { ξi2 < 1/7n2 } ⊂ D, where

do is the geodesic ball with center 0 and radius do with respect to the metric G# . Proof. (1) Consider an arbitrary curve     ai2 = 1, s ≥ 0 . 4 = ξ1 = a1 s, . . . , ξn = an s  By assumption, we have  f il f j m f kn fij k flmn ≤ 4,



fii (0) = n.


Since f il f j m f kn fij k flmn is independent of the choice of coordinates ξ1 , . . . , ξn , for any point ξ(s) we may assume that fij = λi δij . Then 

il

f f

jm

f fij k flmn = kn




2 fij k 1 2 fij k ≥
3 . λi λj λk fii

It follows that
2  1 f
iik 3 ≤
fij2 k ≤ 4 ( fii ) ( fii )3

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and, hence,




 1 fii (x(s))) =
fiik (x(s))ak ds ( fii (x(s)))3/2 
2 1/2   1/2 √ f (x(s)) 2 ≤ n
iik a k ( fii (x(s)))3 √ ≤ 2 n.
Solving this differential inequality with fii (0) = n, we get 1 d(
3/2 ( fii (x(s)))

√ 1 1 . √ −s n≤
( fii (x(s)))1/2 n From the assumption we have s ≤ 1/2n , then (1) follows. (2) Again consider an arbitrary curve     ai2 = 1, s ≥ 0 . 4 = ξ1 = a1 s, . . . , ξn = an s  By assumption, we have
ij f ρi ρj ≤ 4. ρ2 It follows that
2 ρ
i 2 ≤ 4. ( fii )ρ By (1) we get √ 1 dρ(x(s)) ≤ 4 n. ρ ds Solving this differential inequality with ρ(0) = 1, we obtain √ √ −4 ns ≤ lnρ(x(s)) ≤ 4 ns. Recall that s ≤ 1/2n, then (2) follows. (3) Denote by λmin , λmax the minimal and maximal eigenvalues of (fij ). Then, from (1) and (2), we have λmax ≤ 4n and 1 n−1 ≤ det(fij ) ≤ λmin λn−1 λmin . max ≤ (4n) C1

(2.21)

Hence, by (1) and (2.21) 4n



xi2 ≥ γ 2 ≥

 1 xi2 , n−1 C1 (4n)

(2.22)

A BERNSTEIN PROPERTY OF AFFINE MAXIMAL HYPERSURFACES

371


and (3) follows.

We continue with the proof of the theorem. Since d(.) → ∞, we have D ⊂ (.) for . big enough. In fact, by (2.22), the geodesic distance from √ 0 to the boundary

# on M(.) is less than 1/ n. By Lemma 3 and of D with respect to the metric G k bootstrapping, we may
get2 a C - estimate, independent of ., for any k. It follows that there is a ball { ξi ≤ C2 } and a subsequence (still indexed by .) such that f (.) converges to f on the ball and correspondingly all derivatives, where C2 < 1/4n2 is very close to 1/4n2 . Thus, as a limit, we get a maximal hypersurface

defined on the ball, which contains a geodesic ball d0 . We now extend the M,

as follows: For every boundary point p = (ξ10, . . . , ξn0 ) of the hypersurface M geodesic ball d0 , we first make a parameter transformation  j

ai ξj ξi =

we such that, at p, ( ξ1 , . . . , ξn ) = (0, . . . , 0) and for the limit hypersurface M, have f ij (0) = δij . We have (i ) f ij (.)(p) → f ij (0) = δij ,

as . → ∞.

It is easy to see that under the conditions (i ) and (ii) in Lemma 3, the estimates (1), (2) and (3) in Lemma 3 still hold. By the same argument as above, we conclude that there is a ball around p and a subsequence .k , such that f˜(.k ) converges to f˜ on the ball and, correspondingly, all derivatives. As a limit, we get a maximal hypersurface M˜  , which contains a geodesic ball of radius do around p. Then we return to the original parameters. Note that the geodesic distance is independent of the choice of the parameters. It is obvious that M˜ and M˜  agree on the common part. We repeat this procedure to extend M˜ to be defined on 2do , etc. In this way, we may extend M˜ to be a maximal hypersurface defined in a domain ⊂ Rn ,

# . Using (2.18) and (2.19), we get which is complete with respect to the metric G ˜ # ||G˜ # ≤ 4, ||Ric

˜ /(0) = 1.

By Lemma 1, M˜ must be an elliptic paraboloid. For a paraboloid, we have ˜ = 0. Thus, we get a contradiction. So ||Ric# ||2 # must be bounded identically / G above on M. By Lemma 1, M is an elliptic paraboloid. This completes the proof of the theorem.
Before concluding this paper, we wish to state the following problem for higher dimensions: PROBLEM. Let x: M → An+1 be a locally strongly convex hypersurface, given as a graph of a locally strongly convex function xn+1 = f (x1 , . . . , xn )

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defined in a domain ⊂ An . If x(M) is an affine maximal hypersurface and if x(M) is complete with respect to the metric G# , is it an elliptic paraboloid? Acknowledgements The first author would like to thank Professors U. Simon and L. Vrancken for many valuable discussions. Both authors are partially supported by 973 project, NSFC 10271083 grant and a Chinese-German exchange project of NSFC and DFG. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.

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