Abstract
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 Kahler-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.
Highlights
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 R, 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
The two conjectures differ in the assumptions on the completeness of the affine maximal surface considered
Summary
There are two famous conjectures about complete affine maximal surfaces, stated by Calabi and Chern, respectively. If an affine maximal surface given by a strictly convex function is complete with respect to α-metric with α > – , it must be an elliptic paraboloid. This is a new result about affine maximal surface, which generalizes the above theorem in [ ] in dimension. Let y : M → R 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. In [ ] Xu-Xiong-Sheng used an affine blow up analysis to prove the Bernstein theorems for α-relative extremal hypersurfaces with complete α-metrics. The corresponding components of the relative metric Gi(jq), the relative Pick tensor A(ijqk), and the relative Weingarten tensor B(ijq) satisfy (for details see [ ])
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