- Email: [email protected]
A Bernstein theorem for affine maximal type hypersurfaces under decaying convexity
Nonlinear Analysis 187 (2019) 170–179
Contents lists available at ScienceDirect
Nonlinear Analysis www.elsevier.com/locate/na
A Bernstein theorem for affine maximal type hypersurfaces under decaying convexity Shi-Zhong Du 1 The Department of Mathematics, Shantou University, Shantou, 515063, PR China
article
info
Article history: Received 11 October 2018 Accepted 8 April 2019 Communicated by Enzo Mitidieri MSC: 53A15 53A10 35J60
abstract In this paper, we will study a fourth order fully nonlinear PDE uij Dij w = 0,
w ≡ [det D2 u]−θ ,
θ>0
of affine maximal type. By a formula of quasi-norm of third derivatives derived from pure PDE tricks, we will prove that any entire convex solution must be a quadratic function, provided
( ( Keywords: Affine maximal hypersurfaces Bernstein theorem
θ∈
0,
1 − 2
√
N −1 2(N + 8)
) ) ⋃( 1 +
2
√ +
)
N −1 , +∞ 2(N + 8)
and the minimal convex constant λ(R) decays slower than R−α(θ,N ) for some positive constant α(θ, N ) given explicitly below. © 2019 Elsevier Ltd. All rights reserved.
1. Introduction In this paper, we study a fourth order fully nonlinear PDE Dij (U ij w) = 0
(1.1)
of affine maximal type, where U ij is the co-factor matrix of uij and w ≡ [det D2 u]−θ for some θ ̸= 0. Eq. (1.1) is the Euler–Lagrange equation of the affine area functional ∫ A(u, Ω ) ≡ [det D2 u]1−θ Ω ∫ N +1 N +2 2 = K 1−θ (1 + |Du| )ϑ dVg , ϑ = − θ, 2 2 MΩ E-mail address: [email protected]. The author is partially supported by STU Scientific Research Foundation for Talents (SRFT-NTF16006), and Special Funds for Scientific and Technological Innovation Strategy in Guangdong Province. 1
https://doi.org/10.1016/j.na.2019.04.007 0362-546X/© 2019 Elsevier Ltd. All rights reserved.
S.-Z. Du / Nonlinear Analysis 187 (2019) 170–179
for θ ̸= 1, and
∫ A(u, Ω ) ≡
171
log det D2 u
Ω N for θ ={ 1, where u is a convex function } over Ω ⊂ R and K is the Gauss curvature of the graph MΩ ≡ (x, z) ∈ RN +1 | z = u(x), x ∈ Ω . Noting that
Dj U ij = 0, ∀i = 1, 2, . . . , N, (1.1) can be written by U ij Dij w = 0, or equivalent to uij Dij w = 0
(1.2)
here [uij ] denotes the inverse of the metric [uij ] of graph M = MΩ . N +1 The classical affine maximal case θ ≡ N +2 has been studied extensively in the past, see for examples [2–4,11,15,16]. If one introduces an affine metric Aij =
uij [det D2 u]1/(N +2)
on graph M and sets H ≡ [det D2 u]−1/(N +2) , it is not hard to see that (1.2) is equivalent to △M H = 0
(1.3)
for the Laplace–Beltrami operator √ 1 △M ≡ √ Di ( AAij Dj ) = HDi (H −2 uij Dj ) A with respect to this affine metric, where A is the determinant of [Aij ] and [Aij ] stands for the inverse of [Aij ]. So the hypersurface M is affine maximal if and only if H is harmonic on M. In [5], Chern conjectured that for N = 2 and θ = 43 , any entire convex solution to (1.2) must be quadratic. A first partial result was obtained by Bernstein in [2], where he showed that if w(x) = o(|x|) as x → ∞, then w is a constant. Thus, the Chern’s conjecture follows from J¨ orgens’ theorem [12] on dimension two. Later, Calabi [3] verified the Chern’s conjecture under the hypothesis that the affine metric of the graph of the solution, defined by (1.2) is complete. Different conditions were also imposed by Calabi [4]. However, these conditions represent fairly strong restrictions on the asymptotic behaviour of the second derivatives of the function u. Combining with a result by Li [13], which tells that the Euclidean completeness implies affine completeness when affine principal curvatures are bounded, we derive the validity of Chern’s conjecture if the affine Gauss curvature is bounded from below. More recently, the conjecture was fully solved by Trudinger–Wang in their celebrating paper [15]. Moreover, it was proposed there a generalized Bernstein problem by asking whether any locally N +1 convex smooth solution of (1.2) on RN must be a quadratic function in case N ≥ 3 and θ = N +2 . The validity of this generalized conjecture was verified in [15] under an additional uniform “strict convexity” assumption. Different models of (1.2) have also attracted much attention [1,6–11,13,16]. For any N ≥ 2 and |θ| being large sufficiently, Jia–Li proved in [10] that any convex affine maximal hypersurfaces must be a paraboloid. Shortly after, they [11] generalized the Chern–Trudinger–Wang theorem [15] for N = 2, θ = 3/4 to that for N = 2, θ ≥ 3/4. In case θ = 1, (1.2) is called Abreu’s equation, see e.g. [1,6–9,16]. Zhou [16] gave a different proof to Bernstein theorem for N = 2 and 3/4 < θ ≤ 1. Moreover, it was shown that if 2 ≤ N ≤ 4 and θ = 1, the Abreu hypersurface must be paraboloid under completeness of Calabi’s metric.
172
S.-Z. Du / Nonlinear Analysis 187 (2019) 170–179
In this paper, we will study the generalized Bernstein problem of (1.2) for arbitrary dimension N ≥ 2 together with a wide range of θ ̸= 0, and prove the following Bernstein type result: Main Theorem. Let N ≥ 2 and ( θ∈
1 + 2
√
) N −1 , +∞ . 2(N + 8)
Suppose that Dij u(x)ξ i ξ j ≥
C 2 |ξ| , ∀x, ξ ∈ RN , (|x| + 1)α
holds for some positive constants C and 0 ≤ α < α(θ, N ), where α(θ, N ) is a positive constant given by (3.2)–(3.3). Then any entire convex solution of the equation must be a quadratic function. We shall firstly prove a key formula for quasi-norm of third derivatives by Lemma 2.3 in Section 2, and then verify the validity of our Bernstein theorem under decaying convexity in Section 3. 2. Equations of third derivatives For any convex solution u of (1.2) with θ ̸= 0 and N ≥ 2, we use lower indexes like ui ≡ Di u to denote the usual derivatives of u, and use upper indexes Tji ≡ uik Tkj to denote the conjugate ones, where [uij ] is the inverse of [uij ]. For examples, ai bj i,j uabc ≡ uad ubcd , uab ≡ uia ujb Ba,b c ≡ u u uijc , B
etc. will be used under below. Elementary computations show that Dk uij = −uip ujq upqk = −uij k
(2.1)
wk = −θd−θ−1 U pq upqk = −θwupq upqk = −θwuaak .
(2.2)
and
Differentiating once more on both sides of (2.2), we get { } wkl = w θ2 uaak ubbl − θ(uaakl − Bk,l )
(2.3)
for Bk,l ≡ uab k uabl . Taking trace for (2.3) and using (1.2), we obtain that ij kl ij k l ijk uik = θv + C, ik = u u uijkl = θu uki ulj + uijk u
(2.4)
ijk where v ≡ uaai ubi . Hereafter, we will call v to be quasi-norm of third derivatives of u b and C ≡ uijk u comparing to the usual norm
∥D3 u∥2 ≡ uip ujq ukr uijk upqr of u Differentiating (1.2) on xp , we get uij Dij wp = uij p Dij w
(2.5)
) ( a b a − uij Dij (wuaap ) = wuij θu u − u + B i,j p ai bj aij
(2.6)
or equivalently
S.-Z. Du / Nonlinear Analysis 187 (2019) 170–179
173
by (2.2) and (2.3). On the other hand, noting that ) ( −uij Dij (wuaap ) = −uij wDij uaap + 2wi Dj uaap + wij uaap { } = −wuij Dij uaap − 2θubbi Dj uaap , we have
} ( { ) a b a θu u − u + B . − uij Dij uaap − 2θubbi Dj uaap = uij i,j p ai bj aij
(2.7)
Using v = uaap ubp b , it yields that ( ) bp bp a a − uij Dij v = −uij Dij uaap ubp b + 2Di uap Dj ub + uap Dij ub ( ) )( pc b ij bc p uaapi − uac = −uij Dij uaap ubp ubp i uacp b − 2u bj − uj ubc − uj ubc { } pq pr q b bc b −uij uaap upq Dij ubbq − 2upq i (ubqj − uj ubcq ) − (uij − 2uj uri )ubq bij bq a a i,j = −2uij Dij uaap ubp + 4ubbij upij uaap + uiipq uap a ub b − 2uaij ub + 4uaij B bj − 2Bi,j B i,j − 4Bi,j upij ubbp − 2Bi,j uai a ub
(2.8)
and ( ) pq a c θuij ubbi Dj v = θuij ubbi 2Dj uaap ucp c − uj uap ucq ai bj ck = 2θuij ubbi Dj uaap ucp c − θuijk ua ub uc .
(2.9)
As a result, it is inferred from (2.6)–(2.9) that 1 − uij Dij v + θuij ubbi Dj v 2 1 bq a i,j = −uaaij ubij + 2ubbij upij uaap + uiipq uap a ub b + 2uaij B 2 bj ai bj ck −Bi,j B i,j − 2Bi,j upij ubbp − Bi,j uai a ub − θuijk ua ub uc ( ) bj ck b pij a + θuijk uai uap + Bi,j upij ubbp a ub uc − ubij u 1 1 bj ck a i,j Dq v + uijk uai = −uaaij ubij + ubbij upij uaap + ubq a ub uc b + 2uaij B 4 b 4 1 bq bj i,j − Bi,j upij ubbp − Bi,j uai + Bp,q uap a ub − Bi,j B a ub 2 1 1 a i,j ai bj ck = −uaaij ubij + ubbij upij uaap + ubq b + 2uaij B b Dq v + uijk ua ub uc 4 4 1 bj −Bi,j B i,j − Bi,j upij ubbp − Bi,j uai a ub ≡ J1 . 2 Considering ( ) ( ) ij bl i,j − γuaaij + λBi,j + µuijk uak · γubij + λB + µu u a b b l 2 i,j bl a i,j = −γ 2 uaaij ubij − µ2 Bk,l uak a ub − 2λγuaij B b − λ Bi,j B
−2γµubbij upij uaap − 2λµBi,j upij ubbp , if one imposes that −2λγ = 2, −2λµ = −1, −2γµ = 1, or equivalently γ = 1, λ = −1, µ = −1/2,
(2.10)
S.-Z. Du / Nonlinear Analysis 187 (2019) 170–179
174
then 1 bj a i,j −uaaij ubij + ubbij upij uaap − Bi,j B i,j − Bi,j upij ubbp − Bi,j uai a ub b + 2uaij B 4 ) ( ) ( 1 1 i,j bl · ubij − uij ≤ 0, = − uaaij − Bi,j − uijk uak a b −B l ub 2 2 since ∥V ∥2 ≡ uia ujb Vij Vab = Vij V ij is the norm of 2-tensor
1 Vij ≡ uaaij − Bi,j − uijk uak a . 2
Thus, J1 can be reformed by ( )( ) 1 1 1 bj bl i,j J1 = − uaaij − Bi,j − uijk uak ubij − uij − Bi,j uai a a ub l ub b −B 2 2 4 1 1 ai bj ck + ubk b Dk v + uijk ua ub uc . 4 4
(2.11)
To explore further, let us prove two fundamental lemmas that would be helpful. Lemma 2.1.
ai b Let Bi,j ≡ uab i uabi and v ≡ ua ubi . We have
Rij ≡ Bi,j − uijp uap a ≥−
N −1 uij v 4N
(2.12)
as matrix. Proof . Taking any vector ξ ∈ RN , after scaling and rotation, we may assume that uij = δij and ξ = (1, 0, . . . , 0). Therefore, we have Rij ξ i ξ j = R11 = Σa,b u2ab1 − Σa,p u11p uaap ≥ Σa u2aa1 − Σa u111 uaa1 ≡ L. Noting that L = u2111 − u111 Σa uaa1 + Σa≥2 u2aa1 )2 1 ( ≥ u2111 − u111 Σa uaa1 + Σa≥2 uaa1 N −1 )2 1 ( 2 = u111 − u111 Σa uaa1 + Σa uaa1 − u111 N −1 )2 N N +1 1 ( 2 = u111 − u111 Σa uaa1 + Σa uaa1 N −1 N −2 N −1 )2 N − 1( ≥− Σa uaa1 , 4N we conclude that (2.12) to be true. Lemma 2.2.
□
Suppose that u be a convex solution to (1.2). Let v, C and Bi,j be defined as above, we have ( )( ) 1 1 ij bl (θ − 1/2)2 2 bij a ak i,j uaij − Bi,j − uijk ua ub − B − ul ub ≥ v . (2.13) 2 2 N
S.-Z. Du / Nonlinear Analysis 187 (2019) 170–179
175
Proof . By the elementary inequality Vij V ij ≡ uik ujl Vij Vkl ≥ for 2-tensor V = Vij dxi
1 ij (u Vij )2 N
dxj , we have
⨂ (
)( 1 ij bl ) 1 bij i,j u − B − u u uaaij − Bi,j − uijk uak a b 2 2 l b ( )2 (θ − 1/2)2 2 ≥ N −1 uai = v , ai − C − 1/2v N
where (2.4) has also been used. So, the lemma holds true. □ Combining (2.10)–(2.11) and Lemmas 2.1–2.2, we have that ( ) 1 ij b 1 ij u ubi Dj v − u Dij v + θ − 2 4 { } N −1 2 (θ − 1/2)2 ≤− − v <0 N 16N √
(2.14)
√
for θ > 2+ 4N −1 or θ < 2− 4N −1 . √ √ To prove our Bernstein theorem and handle the case 2− 4N −1 ≤ θ ≤ 2+ 4N −1 , let us take two positive constants γ, β and set ρ ≡ v γ wβ . After direct computations, we obtain that ( ) { ( ) } 1 ij b 1 1 1 ij − u Dij ρ + θ − u ubi Dj ρ = γv γ−1 wβ − uij Dij v + θ − ubj Dj v 2 4 2 4 b γ(γ − 1) γ−2 β ij v w u Di vDj v − γβv γ−1 wβ−1 uij Di vDj w 2 ( ) β(β − 1) γ β−2 ij 1 γ β−1 bj − v w u Di wDj w + β θ − v w ub Dj w. 2 4 −
Using vj = −
β v wj + γ −1 v 1−γ w−β Dj ρ, γw
(2.15)
we deduce that ( ) } 1 β(β − 1) 2 γ(γ − 1)β 2 2 2 2 − θ −β θ− θ v γ+1 wβ θ +β θ − 2γ 2 2 4 ( ) } { 1 1 bj +γv γ−1 wβ − uij Dij v + θ − u Dj v ≤ −ζ2 v γ+1 wβ 2 4 b
1 − uij Dij ρ + ζ1j Dj ρ = 2
{
(2.16)
for (
) βθ 1 bj γ − 1 ij = θ− − u + u Di log ρ, γ 4 b 2γ { } (θ − 1/2)2 N −1 β 2 θ2 θ(2θ − 1) ζ2 (θ, N, β, γ) = − γ+ − β. N 16N 2γ 4 ζ1j (θ, β, γ)
To be summarize, we conclude the following lemma:
(2.17)
176
Lemma 2.3.
S.-Z. Du / Nonlinear Analysis 187 (2019) 170–179
Let u be a solution to (1.2) with w ≡ [det D2 u]−θ . If ( ) ( ) √ √ 2− N −1 ⋃ 2+ N −1 θ ∈ −∞, , +∞ , 4 4
we have (2.16) holds for β = 0, γ > 0 together with a positive constant ζ2 (θ, N, 0, γ). If √ √ ) ( ] [ √ √ ⋃ 1 N −1 1 N −1 N −1 1 N −1 1 , θ∈ − , − + , + 2 16 2 2(N + 8) 2 2(N + 8) 2 16
(2.18)
(2.19)
we have (2.16) holds for any β > 0, √ γ ∈ (γ− , γ+ ) ∋ |θ|β
8N N − 1 − 4(2θ − 1)2
together with a positive constant ζ2 (θ, N, β, γ), where √ ⎧ 2N θ(2θ − 1) ± 2|θ| N (N + 8)(2θ − 1)2 − 2N (N − 1) ⎪ ⎪ ⎨γ± ≡ β, N − 1 − 4(2θ − 1)2 ⎪ ⎪ ⎩γ+ = +∞, γ− = 0,
√ 1 N −1 θ ̸= ± , 2 √ 16 1 N −1 θ= ± . 2 16
Proof . The lemma is an immediate consequence of (2.16). In fact, in case (2.18), if one can choose β = 0 and any γ > 0, then ζ2 (θ, N, β, γ) > 0. More precisely, there holds ( ) ( ) √ √ 2− N −1 ⋃ 2+ N −1 ζ2 (θ, N, 0, γ) > 0 ⇔ θ ∈ −∞, , +∞ 4 4 In case (2.19), the region of γ comes from the inequality { } (θ − 1/2)2 N −1 β 2 θ2 θ(2θ − 1) ζ2 (θ, N, β, γ) = − γ+ − β>0 N 16N 2γ 4 ⇔ γ ∈ (γ− , γ+ ). □
3. Bernstein theorem under decaying λ To prove our main theorem, a key observation is the right hand side of (2.16) is non-positive in case (2.18) for β = 0 and γ > 0, or in case (2.19) for β > 0 and γ ∈ (γ− , γ+ ). However, it is not enough to yield the vanishing of ρ by usual maximum principle, since we deal with the entire solution over whole Rn . So, let us take any cut-off function η ∈ C ∞ (Ω ) ∩ C01 (Ω ) whose value and first derivatives are vanishing on the boundary of Ω , a direct computation shows that ( ) 1 ij 1 ij j j − u Dij (ρη) + ζ1 Dj (ρη) = η − u Dij ρ + ζ1 Dj ρ 2 2 1 −uij Di ρDj η − uij ρDij η + ρζ1j Dj η 2 γ + 1 ij 1 ≤ −ζ2 ρvη − u Di ρDj η − ρuij Dij η + 2γ 2
(
) β 1 θ− θ− ρubj b Dj η γ 4
S.-Z. Du / Nonlinear Analysis 187 (2019) 170–179
γ + 1 ij Dj η γ + 1 ij Di ηDj η u Di (ρη) + ρu 2γ η 2γ η ) ( √ 2 1 1 ρubj − ρuij Dij η + θ − signθ − b Dj η, 2 4 4
177
= −ζ2 ρvη −
(3.1)
where β = 0, γ > 0 in case of (2.18), and β > 0, γ ∈ (γ− , γ+ ) in case of (2.19). Given any bounded domain Ω ⊂ RN , we define λ(Ω ) ≡
inf
x∈Ω,∥ξ∥2 =1
Dij u(x)ξ i ξ j
to be the “minimal convex constant” of u on Ω , and denote λ(BR ) by λ(R) for short. We have the following Bernstein type result: Theorem 3.1.
Suppose that ( ( θ∈
0,
1 + 2
√
N −1 2(N + 8)
) ) +
√ ( ) ⋃ 1 N −1 + , +∞ 2 2(N + 8)
and λ(R) ≥ δ0 R−α , ∀R > 1 holds for some positive constant δ0 and 0 ≤ α < α(θ, N ) ≡
⎧ ⎨2,
in case of (2.18),
2γ sup , ⎩ β>0,γ∈(γ− ,γ+ ) γ + βN θ
in case of (2.19).
(3.2)
Then u must be a quadratic function on RN . It is remarkable that α(θ, N ) =
2γ = β>0,γ∈(γ− ,γ+ ) γ + βN θ sup
(3.3)
√ N (N + 8)(2θ − 1)2 − 2N (N − 1) { } √ 2 N (N + 8)(2θ − 1)2 − 2N (N − 1) + N N − 1 + 2(2θ − 1) − 4(2θ − 1)2 4N (2θ − 1) + 4
under the assumption of (2.19). Although the expression of α(θ, N ) is much more complicated, one can find the key ingredient in our proof below is the condition α<
2γ . γ + βN θ
Since β > 0, γ ∈ (γ− , γ+ ) can be chosen variantly in the above inequality, if we take supremum in for different β, γ, then we can obtain a best region α < α(θ, N ) ≡
2γ . γ + βN θ β>0,γ∈(γ− ,γ+ ) sup
Proof . Taking a standard cut-off function η ∈ C0∞ (B2R ) which satisfies that ∥Dη∥RN ≤
CN CN , ∥D2 η∥RN ≤ 2 R R
2γ γ+βN θ
S.-Z. Du / Nonlinear Analysis 187 (2019) 170–179
178
and setting η = η k for some large integer k which will be chosen later, one can evaluate the function ρη at its maximum point on both sides of (3.1). Then, β
δ1 R− γ N θα ρ
γ+1 γ
η≤ρ
γ+1 γ
β
w− γ η = ρvη
⏐ ⏐ ⏐ ⏐ ≤ Cρη k−2 uij Di ηDj η + Cρη k−1 ⏐uij Dij η ⏐ ( ) ( kγ kγ kγ ) β β − γ+1 N θα γ+1 ≤ C ρη η k−2− γ+1 + η k−1− γ+1 R−2+α+ γ+1 N θα R
≤ εR
β
− γ N θα
ρ
γ+1 γ
( η + Cε η
k γ+1 −2
β
R
)γ+1
−2+α+ γ+1 N θα
holds for some positive constant δ1 and any ε > 0, where we have used |uij Dij η| + |uij Di ηDj η| ≤ CR−2+α by our assumption on minimal convex constant λ(R). Taking ε to be small sufficiently, we get ( )γ+1 k −2 −2+α+ β N θα δ2 − βγ N θα γ+1 γ+1 ρ γ η ≤ C2 η γ+1 R R 2 for some δ2 small and C2 large. As a result, for some positive constant C3 which is independent of R, ) β ( β γ+1 (ρη) γ ≤ C3 R(γ+1) −2+α+ γ+1 N θα + γ N θα → 0 as R → +∞, where we have used (
) β β (γ + 1) −2 + α + N θα + N θα < 0 γ+1 γ ( ) γ+1 2γ ⇔ γ+1+ βN θ α < 2(γ + 1) ⇔ α < . γ γ + βN θ So, w must be identical to a positive constant since ∥D log w∥2 = θ2 v ≡ 0. Now, the result follows from Pogorelov’s theorem [14] for Monge–Ampere equation. □ Acknowledgements The author would like to express his deepest gratitude to Professor Kai-Seng Chou and Professor Xu-Jia Wang for their constant encouragement and valuable suggestions. He would also like to thank Neil Trudinger for his support and interest in this problem. References [1] M. Abreu, K¨ ahler geometry of toric varieties and extremal metrics, Int. J. Math. 9 (1998) 641–651. [2] S.N. Bernstein, Sur un theoreme de geometrie et ses applications aux equations aux derivees partielles du type elliptique, ¨ Comm. Soc. Math. Kharkov (2eme ser.) 15 (1915-17) 38–45, See also: Uber ein geometrisches Theorem und seine Anwendung auf die partiellen Differential gleichungen vom elliptischen Typus, Math. Zeit., 26 (1927) 551-558. [3] E. Calabi, Hypersurfaces with maximal affinely invariant area, Amer. J. Math. 104 (1982) 91–126. [4] E. Calabi, Convex affine maximal surfaces, Results Math. 13 (1988) 199–233.
S.-Z. Du / Nonlinear Analysis 187 (2019) 170–179
179
[5] S.S. Chern, Affine minimal hypersurfaces, in: Minimal Submanifolds and Geodesics, in: Proc. Japan-United States Sem., Tokyo, 1977, pp. 17–30, see also Selected papers of S.S. Chern, Volume III, Springer, 1989, 425-438. [6] S.K. Donaldson, Interior estimates for solutions of Abreu’s equation, Collect. Math. 56 (2005) 103–142. [7] S.K. Donaldson, Extremal metrics on toric surfaces: a continuity method, J. Differential Geom. 79 (2008) 389–432. [8] S.K. Donaldson, Constant scalar curvature metrics on toric surfaces, Geom. Funct. Anal. 19 (2009) 83–136. [9] F. Jia, A.M. Li, A Bernstein property of affine maximal hypersurfacces, Ann. Glob. Anal. Geom. 23 (2003) 359–372. [10] F. Jia, A.M. Li, Interior estimates for solutions of a fourth order nonlinear partial differential equation, Differential Geom. Appl. 25 (2007) 433–451. [11] F. Jia, A.M. Li, A Bernstein property of some fourth order partial differential equations, Results Math. 56 (2009) 109–139. ¨ [12] K. J¨ orgens, Uber die L¨ osungen der Differentialgleichung rt − s2 = 1, Math. Ann. 127 (1954) 130–134. [13] A.M. Li, Affine completeness and Euclidean completeness, Lecture Notes in Math. 1481 (1991) 116–126. [14] A.V. Pogorelov, On the improper affine hyperspheres, Geom. Dedicata 1 (1972) 33–46. [15] N.S. Trudinger, X.J. Wang, The Bernstein problem for affine maximal hypersurfaces, Invent. Math. 140 (2000) 399–422. [16] B. Zhou, The Bernstein theorem for a class of fourth order equations, Calc. Var. Partial Differential Equations 43 (2012) 25–44.
Contents lists available at ScienceDirect
Nonlinear Analysis www.elsevier.com/locate/na
A Bernstein theorem for affine maximal type hypersurfaces under decaying convexity Shi-Zhong Du 1 The Department of Mathematics, Shantou University, Shantou, 515063, PR China
article
info
Article history: Received 11 October 2018 Accepted 8 April 2019 Communicated by Enzo Mitidieri MSC: 53A15 53A10 35J60
abstract In this paper, we will study a fourth order fully nonlinear PDE uij Dij w = 0,
w ≡ [det D2 u]−θ ,
θ>0
of affine maximal type. By a formula of quasi-norm of third derivatives derived from pure PDE tricks, we will prove that any entire convex solution must be a quadratic function, provided
( ( Keywords: Affine maximal hypersurfaces Bernstein theorem
θ∈
0,
1 − 2
√
N −1 2(N + 8)
) ) ⋃( 1 +
2
√ +
)
N −1 , +∞ 2(N + 8)
and the minimal convex constant λ(R) decays slower than R−α(θ,N ) for some positive constant α(θ, N ) given explicitly below. © 2019 Elsevier Ltd. All rights reserved.
1. Introduction In this paper, we study a fourth order fully nonlinear PDE Dij (U ij w) = 0
(1.1)
of affine maximal type, where U ij is the co-factor matrix of uij and w ≡ [det D2 u]−θ for some θ ̸= 0. Eq. (1.1) is the Euler–Lagrange equation of the affine area functional ∫ A(u, Ω ) ≡ [det D2 u]1−θ Ω ∫ N +1 N +2 2 = K 1−θ (1 + |Du| )ϑ dVg , ϑ = − θ, 2 2 MΩ E-mail address: [email protected]. The author is partially supported by STU Scientific Research Foundation for Talents (SRFT-NTF16006), and Special Funds for Scientific and Technological Innovation Strategy in Guangdong Province. 1
https://doi.org/10.1016/j.na.2019.04.007 0362-546X/© 2019 Elsevier Ltd. All rights reserved.
S.-Z. Du / Nonlinear Analysis 187 (2019) 170–179
for θ ̸= 1, and
∫ A(u, Ω ) ≡
171
log det D2 u
Ω N for θ ={ 1, where u is a convex function } over Ω ⊂ R and K is the Gauss curvature of the graph MΩ ≡ (x, z) ∈ RN +1 | z = u(x), x ∈ Ω . Noting that
Dj U ij = 0, ∀i = 1, 2, . . . , N, (1.1) can be written by U ij Dij w = 0, or equivalent to uij Dij w = 0
(1.2)
here [uij ] denotes the inverse of the metric [uij ] of graph M = MΩ . N +1 The classical affine maximal case θ ≡ N +2 has been studied extensively in the past, see for examples [2–4,11,15,16]. If one introduces an affine metric Aij =
uij [det D2 u]1/(N +2)
on graph M and sets H ≡ [det D2 u]−1/(N +2) , it is not hard to see that (1.2) is equivalent to △M H = 0
(1.3)
for the Laplace–Beltrami operator √ 1 △M ≡ √ Di ( AAij Dj ) = HDi (H −2 uij Dj ) A with respect to this affine metric, where A is the determinant of [Aij ] and [Aij ] stands for the inverse of [Aij ]. So the hypersurface M is affine maximal if and only if H is harmonic on M. In [5], Chern conjectured that for N = 2 and θ = 43 , any entire convex solution to (1.2) must be quadratic. A first partial result was obtained by Bernstein in [2], where he showed that if w(x) = o(|x|) as x → ∞, then w is a constant. Thus, the Chern’s conjecture follows from J¨ orgens’ theorem [12] on dimension two. Later, Calabi [3] verified the Chern’s conjecture under the hypothesis that the affine metric of the graph of the solution, defined by (1.2) is complete. Different conditions were also imposed by Calabi [4]. However, these conditions represent fairly strong restrictions on the asymptotic behaviour of the second derivatives of the function u. Combining with a result by Li [13], which tells that the Euclidean completeness implies affine completeness when affine principal curvatures are bounded, we derive the validity of Chern’s conjecture if the affine Gauss curvature is bounded from below. More recently, the conjecture was fully solved by Trudinger–Wang in their celebrating paper [15]. Moreover, it was proposed there a generalized Bernstein problem by asking whether any locally N +1 convex smooth solution of (1.2) on RN must be a quadratic function in case N ≥ 3 and θ = N +2 . The validity of this generalized conjecture was verified in [15] under an additional uniform “strict convexity” assumption. Different models of (1.2) have also attracted much attention [1,6–11,13,16]. For any N ≥ 2 and |θ| being large sufficiently, Jia–Li proved in [10] that any convex affine maximal hypersurfaces must be a paraboloid. Shortly after, they [11] generalized the Chern–Trudinger–Wang theorem [15] for N = 2, θ = 3/4 to that for N = 2, θ ≥ 3/4. In case θ = 1, (1.2) is called Abreu’s equation, see e.g. [1,6–9,16]. Zhou [16] gave a different proof to Bernstein theorem for N = 2 and 3/4 < θ ≤ 1. Moreover, it was shown that if 2 ≤ N ≤ 4 and θ = 1, the Abreu hypersurface must be paraboloid under completeness of Calabi’s metric.
172
S.-Z. Du / Nonlinear Analysis 187 (2019) 170–179
In this paper, we will study the generalized Bernstein problem of (1.2) for arbitrary dimension N ≥ 2 together with a wide range of θ ̸= 0, and prove the following Bernstein type result: Main Theorem. Let N ≥ 2 and ( θ∈
1 + 2
√
) N −1 , +∞ . 2(N + 8)
Suppose that Dij u(x)ξ i ξ j ≥
C 2 |ξ| , ∀x, ξ ∈ RN , (|x| + 1)α
holds for some positive constants C and 0 ≤ α < α(θ, N ), where α(θ, N ) is a positive constant given by (3.2)–(3.3). Then any entire convex solution of the equation must be a quadratic function. We shall firstly prove a key formula for quasi-norm of third derivatives by Lemma 2.3 in Section 2, and then verify the validity of our Bernstein theorem under decaying convexity in Section 3. 2. Equations of third derivatives For any convex solution u of (1.2) with θ ̸= 0 and N ≥ 2, we use lower indexes like ui ≡ Di u to denote the usual derivatives of u, and use upper indexes Tji ≡ uik Tkj to denote the conjugate ones, where [uij ] is the inverse of [uij ]. For examples, ai bj i,j uabc ≡ uad ubcd , uab ≡ uia ujb Ba,b c ≡ u u uijc , B
etc. will be used under below. Elementary computations show that Dk uij = −uip ujq upqk = −uij k
(2.1)
wk = −θd−θ−1 U pq upqk = −θwupq upqk = −θwuaak .
(2.2)
and
Differentiating once more on both sides of (2.2), we get { } wkl = w θ2 uaak ubbl − θ(uaakl − Bk,l )
(2.3)
for Bk,l ≡ uab k uabl . Taking trace for (2.3) and using (1.2), we obtain that ij kl ij k l ijk uik = θv + C, ik = u u uijkl = θu uki ulj + uijk u
(2.4)
ijk where v ≡ uaai ubi . Hereafter, we will call v to be quasi-norm of third derivatives of u b and C ≡ uijk u comparing to the usual norm
∥D3 u∥2 ≡ uip ujq ukr uijk upqr of u Differentiating (1.2) on xp , we get uij Dij wp = uij p Dij w
(2.5)
) ( a b a − uij Dij (wuaap ) = wuij θu u − u + B i,j p ai bj aij
(2.6)
or equivalently
S.-Z. Du / Nonlinear Analysis 187 (2019) 170–179
173
by (2.2) and (2.3). On the other hand, noting that ) ( −uij Dij (wuaap ) = −uij wDij uaap + 2wi Dj uaap + wij uaap { } = −wuij Dij uaap − 2θubbi Dj uaap , we have
} ( { ) a b a θu u − u + B . − uij Dij uaap − 2θubbi Dj uaap = uij i,j p ai bj aij
(2.7)
Using v = uaap ubp b , it yields that ( ) bp bp a a − uij Dij v = −uij Dij uaap ubp b + 2Di uap Dj ub + uap Dij ub ( ) )( pc b ij bc p uaapi − uac = −uij Dij uaap ubp ubp i uacp b − 2u bj − uj ubc − uj ubc { } pq pr q b bc b −uij uaap upq Dij ubbq − 2upq i (ubqj − uj ubcq ) − (uij − 2uj uri )ubq bij bq a a i,j = −2uij Dij uaap ubp + 4ubbij upij uaap + uiipq uap a ub b − 2uaij ub + 4uaij B bj − 2Bi,j B i,j − 4Bi,j upij ubbp − 2Bi,j uai a ub
(2.8)
and ( ) pq a c θuij ubbi Dj v = θuij ubbi 2Dj uaap ucp c − uj uap ucq ai bj ck = 2θuij ubbi Dj uaap ucp c − θuijk ua ub uc .
(2.9)
As a result, it is inferred from (2.6)–(2.9) that 1 − uij Dij v + θuij ubbi Dj v 2 1 bq a i,j = −uaaij ubij + 2ubbij upij uaap + uiipq uap a ub b + 2uaij B 2 bj ai bj ck −Bi,j B i,j − 2Bi,j upij ubbp − Bi,j uai a ub − θuijk ua ub uc ( ) bj ck b pij a + θuijk uai uap + Bi,j upij ubbp a ub uc − ubij u 1 1 bj ck a i,j Dq v + uijk uai = −uaaij ubij + ubbij upij uaap + ubq a ub uc b + 2uaij B 4 b 4 1 bq bj i,j − Bi,j upij ubbp − Bi,j uai + Bp,q uap a ub − Bi,j B a ub 2 1 1 a i,j ai bj ck = −uaaij ubij + ubbij upij uaap + ubq b + 2uaij B b Dq v + uijk ua ub uc 4 4 1 bj −Bi,j B i,j − Bi,j upij ubbp − Bi,j uai a ub ≡ J1 . 2 Considering ( ) ( ) ij bl i,j − γuaaij + λBi,j + µuijk uak · γubij + λB + µu u a b b l 2 i,j bl a i,j = −γ 2 uaaij ubij − µ2 Bk,l uak a ub − 2λγuaij B b − λ Bi,j B
−2γµubbij upij uaap − 2λµBi,j upij ubbp , if one imposes that −2λγ = 2, −2λµ = −1, −2γµ = 1, or equivalently γ = 1, λ = −1, µ = −1/2,
(2.10)
S.-Z. Du / Nonlinear Analysis 187 (2019) 170–179
174
then 1 bj a i,j −uaaij ubij + ubbij upij uaap − Bi,j B i,j − Bi,j upij ubbp − Bi,j uai a ub b + 2uaij B 4 ) ( ) ( 1 1 i,j bl · ubij − uij ≤ 0, = − uaaij − Bi,j − uijk uak a b −B l ub 2 2 since ∥V ∥2 ≡ uia ujb Vij Vab = Vij V ij is the norm of 2-tensor
1 Vij ≡ uaaij − Bi,j − uijk uak a . 2
Thus, J1 can be reformed by ( )( ) 1 1 1 bj bl i,j J1 = − uaaij − Bi,j − uijk uak ubij − uij − Bi,j uai a a ub l ub b −B 2 2 4 1 1 ai bj ck + ubk b Dk v + uijk ua ub uc . 4 4
(2.11)
To explore further, let us prove two fundamental lemmas that would be helpful. Lemma 2.1.
ai b Let Bi,j ≡ uab i uabi and v ≡ ua ubi . We have
Rij ≡ Bi,j − uijp uap a ≥−
N −1 uij v 4N
(2.12)
as matrix. Proof . Taking any vector ξ ∈ RN , after scaling and rotation, we may assume that uij = δij and ξ = (1, 0, . . . , 0). Therefore, we have Rij ξ i ξ j = R11 = Σa,b u2ab1 − Σa,p u11p uaap ≥ Σa u2aa1 − Σa u111 uaa1 ≡ L. Noting that L = u2111 − u111 Σa uaa1 + Σa≥2 u2aa1 )2 1 ( ≥ u2111 − u111 Σa uaa1 + Σa≥2 uaa1 N −1 )2 1 ( 2 = u111 − u111 Σa uaa1 + Σa uaa1 − u111 N −1 )2 N N +1 1 ( 2 = u111 − u111 Σa uaa1 + Σa uaa1 N −1 N −2 N −1 )2 N − 1( ≥− Σa uaa1 , 4N we conclude that (2.12) to be true. Lemma 2.2.
□
Suppose that u be a convex solution to (1.2). Let v, C and Bi,j be defined as above, we have ( )( ) 1 1 ij bl (θ − 1/2)2 2 bij a ak i,j uaij − Bi,j − uijk ua ub − B − ul ub ≥ v . (2.13) 2 2 N
S.-Z. Du / Nonlinear Analysis 187 (2019) 170–179
175
Proof . By the elementary inequality Vij V ij ≡ uik ujl Vij Vkl ≥ for 2-tensor V = Vij dxi
1 ij (u Vij )2 N
dxj , we have
⨂ (
)( 1 ij bl ) 1 bij i,j u − B − u u uaaij − Bi,j − uijk uak a b 2 2 l b ( )2 (θ − 1/2)2 2 ≥ N −1 uai = v , ai − C − 1/2v N
where (2.4) has also been used. So, the lemma holds true. □ Combining (2.10)–(2.11) and Lemmas 2.1–2.2, we have that ( ) 1 ij b 1 ij u ubi Dj v − u Dij v + θ − 2 4 { } N −1 2 (θ − 1/2)2 ≤− − v <0 N 16N √
(2.14)
√
for θ > 2+ 4N −1 or θ < 2− 4N −1 . √ √ To prove our Bernstein theorem and handle the case 2− 4N −1 ≤ θ ≤ 2+ 4N −1 , let us take two positive constants γ, β and set ρ ≡ v γ wβ . After direct computations, we obtain that ( ) { ( ) } 1 ij b 1 1 1 ij − u Dij ρ + θ − u ubi Dj ρ = γv γ−1 wβ − uij Dij v + θ − ubj Dj v 2 4 2 4 b γ(γ − 1) γ−2 β ij v w u Di vDj v − γβv γ−1 wβ−1 uij Di vDj w 2 ( ) β(β − 1) γ β−2 ij 1 γ β−1 bj − v w u Di wDj w + β θ − v w ub Dj w. 2 4 −
Using vj = −
β v wj + γ −1 v 1−γ w−β Dj ρ, γw
(2.15)
we deduce that ( ) } 1 β(β − 1) 2 γ(γ − 1)β 2 2 2 2 − θ −β θ− θ v γ+1 wβ θ +β θ − 2γ 2 2 4 ( ) } { 1 1 bj +γv γ−1 wβ − uij Dij v + θ − u Dj v ≤ −ζ2 v γ+1 wβ 2 4 b
1 − uij Dij ρ + ζ1j Dj ρ = 2
{
(2.16)
for (
) βθ 1 bj γ − 1 ij = θ− − u + u Di log ρ, γ 4 b 2γ { } (θ − 1/2)2 N −1 β 2 θ2 θ(2θ − 1) ζ2 (θ, N, β, γ) = − γ+ − β. N 16N 2γ 4 ζ1j (θ, β, γ)
To be summarize, we conclude the following lemma:
(2.17)
176
Lemma 2.3.
S.-Z. Du / Nonlinear Analysis 187 (2019) 170–179
Let u be a solution to (1.2) with w ≡ [det D2 u]−θ . If ( ) ( ) √ √ 2− N −1 ⋃ 2+ N −1 θ ∈ −∞, , +∞ , 4 4
we have (2.16) holds for β = 0, γ > 0 together with a positive constant ζ2 (θ, N, 0, γ). If √ √ ) ( ] [ √ √ ⋃ 1 N −1 1 N −1 N −1 1 N −1 1 , θ∈ − , − + , + 2 16 2 2(N + 8) 2 2(N + 8) 2 16
(2.18)
(2.19)
we have (2.16) holds for any β > 0, √ γ ∈ (γ− , γ+ ) ∋ |θ|β
8N N − 1 − 4(2θ − 1)2
together with a positive constant ζ2 (θ, N, β, γ), where √ ⎧ 2N θ(2θ − 1) ± 2|θ| N (N + 8)(2θ − 1)2 − 2N (N − 1) ⎪ ⎪ ⎨γ± ≡ β, N − 1 − 4(2θ − 1)2 ⎪ ⎪ ⎩γ+ = +∞, γ− = 0,
√ 1 N −1 θ ̸= ± , 2 √ 16 1 N −1 θ= ± . 2 16
Proof . The lemma is an immediate consequence of (2.16). In fact, in case (2.18), if one can choose β = 0 and any γ > 0, then ζ2 (θ, N, β, γ) > 0. More precisely, there holds ( ) ( ) √ √ 2− N −1 ⋃ 2+ N −1 ζ2 (θ, N, 0, γ) > 0 ⇔ θ ∈ −∞, , +∞ 4 4 In case (2.19), the region of γ comes from the inequality { } (θ − 1/2)2 N −1 β 2 θ2 θ(2θ − 1) ζ2 (θ, N, β, γ) = − γ+ − β>0 N 16N 2γ 4 ⇔ γ ∈ (γ− , γ+ ). □
3. Bernstein theorem under decaying λ To prove our main theorem, a key observation is the right hand side of (2.16) is non-positive in case (2.18) for β = 0 and γ > 0, or in case (2.19) for β > 0 and γ ∈ (γ− , γ+ ). However, it is not enough to yield the vanishing of ρ by usual maximum principle, since we deal with the entire solution over whole Rn . So, let us take any cut-off function η ∈ C ∞ (Ω ) ∩ C01 (Ω ) whose value and first derivatives are vanishing on the boundary of Ω , a direct computation shows that ( ) 1 ij 1 ij j j − u Dij (ρη) + ζ1 Dj (ρη) = η − u Dij ρ + ζ1 Dj ρ 2 2 1 −uij Di ρDj η − uij ρDij η + ρζ1j Dj η 2 γ + 1 ij 1 ≤ −ζ2 ρvη − u Di ρDj η − ρuij Dij η + 2γ 2
(
) β 1 θ− θ− ρubj b Dj η γ 4
S.-Z. Du / Nonlinear Analysis 187 (2019) 170–179
γ + 1 ij Dj η γ + 1 ij Di ηDj η u Di (ρη) + ρu 2γ η 2γ η ) ( √ 2 1 1 ρubj − ρuij Dij η + θ − signθ − b Dj η, 2 4 4
177
= −ζ2 ρvη −
(3.1)
where β = 0, γ > 0 in case of (2.18), and β > 0, γ ∈ (γ− , γ+ ) in case of (2.19). Given any bounded domain Ω ⊂ RN , we define λ(Ω ) ≡
inf
x∈Ω,∥ξ∥2 =1
Dij u(x)ξ i ξ j
to be the “minimal convex constant” of u on Ω , and denote λ(BR ) by λ(R) for short. We have the following Bernstein type result: Theorem 3.1.
Suppose that ( ( θ∈
0,
1 + 2
√
N −1 2(N + 8)
) ) +
√ ( ) ⋃ 1 N −1 + , +∞ 2 2(N + 8)
and λ(R) ≥ δ0 R−α , ∀R > 1 holds for some positive constant δ0 and 0 ≤ α < α(θ, N ) ≡
⎧ ⎨2,
in case of (2.18),
2γ sup , ⎩ β>0,γ∈(γ− ,γ+ ) γ + βN θ
in case of (2.19).
(3.2)
Then u must be a quadratic function on RN . It is remarkable that α(θ, N ) =
2γ = β>0,γ∈(γ− ,γ+ ) γ + βN θ sup
(3.3)
√ N (N + 8)(2θ − 1)2 − 2N (N − 1) { } √ 2 N (N + 8)(2θ − 1)2 − 2N (N − 1) + N N − 1 + 2(2θ − 1) − 4(2θ − 1)2 4N (2θ − 1) + 4
under the assumption of (2.19). Although the expression of α(θ, N ) is much more complicated, one can find the key ingredient in our proof below is the condition α<
2γ . γ + βN θ
Since β > 0, γ ∈ (γ− , γ+ ) can be chosen variantly in the above inequality, if we take supremum in for different β, γ, then we can obtain a best region α < α(θ, N ) ≡
2γ . γ + βN θ β>0,γ∈(γ− ,γ+ ) sup
Proof . Taking a standard cut-off function η ∈ C0∞ (B2R ) which satisfies that ∥Dη∥RN ≤
CN CN , ∥D2 η∥RN ≤ 2 R R
2γ γ+βN θ
S.-Z. Du / Nonlinear Analysis 187 (2019) 170–179
178
and setting η = η k for some large integer k which will be chosen later, one can evaluate the function ρη at its maximum point on both sides of (3.1). Then, β
δ1 R− γ N θα ρ
γ+1 γ
η≤ρ
γ+1 γ
β
w− γ η = ρvη
⏐ ⏐ ⏐ ⏐ ≤ Cρη k−2 uij Di ηDj η + Cρη k−1 ⏐uij Dij η ⏐ ( ) ( kγ kγ kγ ) β β − γ+1 N θα γ+1 ≤ C ρη η k−2− γ+1 + η k−1− γ+1 R−2+α+ γ+1 N θα R
≤ εR
β
− γ N θα
ρ
γ+1 γ
( η + Cε η
k γ+1 −2
β
R
)γ+1
−2+α+ γ+1 N θα
holds for some positive constant δ1 and any ε > 0, where we have used |uij Dij η| + |uij Di ηDj η| ≤ CR−2+α by our assumption on minimal convex constant λ(R). Taking ε to be small sufficiently, we get ( )γ+1 k −2 −2+α+ β N θα δ2 − βγ N θα γ+1 γ+1 ρ γ η ≤ C2 η γ+1 R R 2 for some δ2 small and C2 large. As a result, for some positive constant C3 which is independent of R, ) β ( β γ+1 (ρη) γ ≤ C3 R(γ+1) −2+α+ γ+1 N θα + γ N θα → 0 as R → +∞, where we have used (
) β β (γ + 1) −2 + α + N θα + N θα < 0 γ+1 γ ( ) γ+1 2γ ⇔ γ+1+ βN θ α < 2(γ + 1) ⇔ α < . γ γ + βN θ So, w must be identical to a positive constant since ∥D log w∥2 = θ2 v ≡ 0. Now, the result follows from Pogorelov’s theorem [14] for Monge–Ampere equation. □ Acknowledgements The author would like to express his deepest gratitude to Professor Kai-Seng Chou and Professor Xu-Jia Wang for their constant encouragement and valuable suggestions. He would also like to thank Neil Trudinger for his support and interest in this problem. References [1] M. Abreu, K¨ ahler geometry of toric varieties and extremal metrics, Int. J. Math. 9 (1998) 641–651. [2] S.N. Bernstein, Sur un theoreme de geometrie et ses applications aux equations aux derivees partielles du type elliptique, ¨ Comm. Soc. Math. Kharkov (2eme ser.) 15 (1915-17) 38–45, See also: Uber ein geometrisches Theorem und seine Anwendung auf die partiellen Differential gleichungen vom elliptischen Typus, Math. Zeit., 26 (1927) 551-558. [3] E. Calabi, Hypersurfaces with maximal affinely invariant area, Amer. J. Math. 104 (1982) 91–126. [4] E. Calabi, Convex affine maximal surfaces, Results Math. 13 (1988) 199–233.
S.-Z. Du / Nonlinear Analysis 187 (2019) 170–179
179
[5] S.S. Chern, Affine minimal hypersurfaces, in: Minimal Submanifolds and Geodesics, in: Proc. Japan-United States Sem., Tokyo, 1977, pp. 17–30, see also Selected papers of S.S. Chern, Volume III, Springer, 1989, 425-438. [6] S.K. Donaldson, Interior estimates for solutions of Abreu’s equation, Collect. Math. 56 (2005) 103–142. [7] S.K. Donaldson, Extremal metrics on toric surfaces: a continuity method, J. Differential Geom. 79 (2008) 389–432. [8] S.K. Donaldson, Constant scalar curvature metrics on toric surfaces, Geom. Funct. Anal. 19 (2009) 83–136. [9] F. Jia, A.M. Li, A Bernstein property of affine maximal hypersurfacces, Ann. Glob. Anal. Geom. 23 (2003) 359–372. [10] F. Jia, A.M. Li, Interior estimates for solutions of a fourth order nonlinear partial differential equation, Differential Geom. Appl. 25 (2007) 433–451. [11] F. Jia, A.M. Li, A Bernstein property of some fourth order partial differential equations, Results Math. 56 (2009) 109–139. ¨ [12] K. J¨ orgens, Uber die L¨ osungen der Differentialgleichung rt − s2 = 1, Math. Ann. 127 (1954) 130–134. [13] A.M. Li, Affine completeness and Euclidean completeness, Lecture Notes in Math. 1481 (1991) 116–126. [14] A.V. Pogorelov, On the improper affine hyperspheres, Geom. Dedicata 1 (1972) 33–46. [15] N.S. Trudinger, X.J. Wang, The Bernstein problem for affine maximal hypersurfaces, Invent. Math. 140 (2000) 399–422. [16] B. Zhou, The Bernstein theorem for a class of fourth order equations, Calc. Var. Partial Differential Equations 43 (2012) 25–44.