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Annals of Global Analysis and Geometry

, Volume 43, Issue 2, pp 143–152 | Cite as

Bernstein theorems for complete α-relative extremal hypersurfaces

  • Ruiwei Xu
  • Min Xiong
  • Li ShengEmail author
Article

Abstract

Let \({y : M \rightarrow \mathbb{R}^{n+1}}\) be a locally strongly convex hypersurface immersion of a smooth, connected manifold into the real affine space \({\mathbb{R}^{n+1}}\), given as graph of a strictly convex function x n+1 = f(x 1, . . . ,x n ) defined on a domain \({\Omega \subset \mathbb{R}^{n}}\). Considering the Li-normalization of the graph of the convex function f, we will prove the Bernstein theorems for relative extremal hypersurfaces with complete α-metrics. As one case of main theorem, we obtain that affine complete maximal surface given by graph is an elliptic paraboloid, which is a special case of Calabi conjecture about affine maximal surfaces.

Keywords

Bernstein theorem Relative extremal hypersurfaces Blow-up analysis 

Mathematics Subject Classification (2000)

Primary 53A15 Secondary 35J60 53C40 53C42 

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Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  1. 1.Department of MathematicsHenan Normal UniversityXinxiangPeople’s Republic of China
  2. 2.College of ScienceSouthwest University of Science and TechnologyMianyangPeople’s Republic of China
  3. 3.College of MathematicsSichuan UniversityChengduPeople’s Republic of China

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