- Open Access
Bernstein properties for α-complete hypersurfaces
© Xu and Cao; licensee Springer. 2014
Received: 14 January 2014
Accepted: 16 April 2014
Published: 6 May 2014
In the first part of this paper we focus on the Bernstein property of relative surfaces with complete α-metric. As a corollary, we give a new Bernstein type theorem for affine maximal surface and relative extremal surface. In the second part, we offer a relative simple proof of the Bernstein theorem for the affine Kähler-Ricci flat graph with complete α-metric, which was proved in (Li and Xu in Results Math. 54:329-340, 2009), based on a new observation on α-Ricci curvature.
MSC:53A15, 35J60, 53C40, 53C42.
In affine differential geometry, there are two famous conjectures about complete affine maximal surfaces, stated by Calabi and Chern, respectively. Chern assumes that the maximal surface is a convex graph on the whole , while Calabi supposes it is complete with respect to the Blaschke metric. Both versions of affine Bernstein problems attracted many mathematicians, and they were solved during the last decade. For Chern’s conjecture, please see related work [1–4] and . For Calabi’s conjecture see [4, 6] and . The two conjectures differ in the assumptions on the completeness of the affine maximal surface considered. On the other hand, Li and Jia in  considered the Bernstein problem of an affine maximal hypersurface with complete Calabi metric and proved the following.
Theorem Let be a locally strongly convex function defined in a domain . If is an affine maximal hypersurface, and if M is complete with respect to the Calabi metric , then, in the case where or , M must be an elliptic paraboloid.
where β is a constant, and the Laplacian Δ and the norm are defined with respect to the Calabi metric. The first result we obtain is as follows.
Theorem 1.1 Let be a strictly convex function defined on a convex domain , which satisfies the PDE (1.1). Assume that the graph surface is complete with respect to the α-metric. Then, if , f must be a quadratic polynomial.
From Theorem 1.1 we can obtain the following three corollaries.
For affine maximal surfaces (the case in Theorem 1.1), we have the following.
Corollary 1.2 If an affine maximal surface given by a strictly convex function is complete with respect to α-metric with , then it must be an elliptic paraboloid.
This is a new result about affine maximal surface, which generalizes the above theorem in  in dimension 2.
If α-relative extremal hypersurfaces are given by a strictly convex function f, then f should satisfy the PDE (1.1) with ; see . For α-relative extremal surfaces, we have the following.
Corollary 1.3 Let be a locally strongly convex α-relative extremal surface, complete with respect to the α-metric, which is given by a locally strongly convex function. Then, if , M is an elliptic paraboloid.
In  Xu-Xiong-Sheng used an affine blow up analysis to prove the Bernstein theorems for α-relative extremal hypersurfaces with complete α-metrics. Corollary 1.3 gives a new analytic and relative simple proof of the Bernstein property for α-relative extremal surfaces with complete α-metric.
In , Jia and Li obtained a Bernstein property for the Abreu equation in dimension under the assumption of completeness with respect to the Calabi metric. Later Xiong and Sheng used a blow up analysis to prove a Bernstein property for the Abreu equation under the assumption of completeness with respect to the -metric in . By Theorem 1.1, we get the following.
Corollary 1.4 Let be a strictly convex function defined on a convex domain , which satisfies the Abreu equation ( in PDE (1.1)). Assume that the graph surface is complete with respect to the α-metric. Then, if , f must be a quadratic polynomial.
which satisfies the PDE (1.1) with . Its graph is complete with respect to the metric.
where are real constants. In  and , Li and the first author have shown that the affine Kähler-Ricci flat graph hypersurface has a rigidity property under different completeness conditions. By deriving a differential inequality for the Laplacian ΔJ of the relative Pick invariant J and combining with estimating the norm of the relative Tchebychev vector Φ and J on a geodesic ball, we proved in  the following.
If and M is complete with respect to the α-metric then M must be an elliptic paraboloid.
Here we shall give it a relative simple proof, and do not need to estimate the relative Pick invariant J based on a new observation on α-Ricci curvature (see (4.8)).
2 α-Relative geometry
3 Proof of Theorem 1.1
In the following we will use the α-metric to do the calculations. That is to say, the norms and the Laplacian operator are defined with respect to the α-metric. Firstly we will estimate ΔΦ. Using the relationship between conformal metrics and Proposition 3.1 in  (see also Proposition 4.5.2 in ), we can prove the following.
where is a positive constant depending only on α and β. □
This means that M is an affine complete parabolic hypersphere. We apply a result of Calabi (see , p.128) and conclude that M must be an elliptic paraboloid. This completes the proof of Theorem 1.1. □
4 Proof of Theorem 1.5
As before, using the relationship between conformal metrics and Proposition 3.1 in , we have the following.
where ϵ is a positive constant to be determined later, and is a positive constant depending only n, α, and ϵ.
where is a positive constant depending only on α and n. □
Using the same method as in the proof of Theorem 1.1, we complete the proof of Theorem 1.5.
The authors would like to express their gratitude to the referees for valuable comments and suggestions. Besides, the first author is supported by grant (Nos. 11101129, 11201318 and 11171091) of NSFC and the second author is supported by grant (No. U1304101) of NSFC and NSF of Henan Province (No. 132300410141).
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