A rigidity theorem of -relative parabolic hyperspheres
A rigidity theorem of -relative parabolic hyperspheres
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$alpha$相关抛物线超球面的刚性定理
DOI：
10.1007/s00229-017-0918-7
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发表时间：
2017
影响因子：
0.6
通讯作者：
Xu RW
中科院分区：
文献类型：
--
作者：
Xu Ruiwei;Zhu Lingyun;Xu RW
设f是定义在R^nRn上的（f ^2f <$x_ i <$x_ j\right）=\left（a_ n+ 1-∑ a_i <$f <$x_ i\right）^ n+ 2 α det <$2f <$xi <$x_j = an+ 1-∑ ai <$f <$xin+ 2 α的光滑严格凸解，其中α α是非零常数，（a_1，a_2，.. a_ n+ 1）（a 1，a 2，.，an+ 1）是R^ n+ 1 R n+ 1中的常向量。则R^ n+ 1 Rn + 1中的图超曲面M={（x，f（x））\} M=（x，f（x））是Li-几何中的α α-相对抛物仿射超球面.本文将Blaschke几何中著名的Jörgens-Calabi-Pogorelov定理推广到Li-几何中。对欧氏完备的α α-相对抛物仿射超球进行了分类，证明了具有α <$n+ 2 n+ 1，n+ 2 α <$n+ 2 n+ 1，n+ 2的上述偏微分方程的任何光滑严格凸整解必是二次多项式.
Let f be a smooth strictly convex solution of (∂^ 2 f ∂ x_ i ∂ x_ j\right)=\left (a_ n+ 1-∑ a_i ∂ f ∂ x_ i\right)^ n+ 2 α det∂ 2 f∂ xi∂ xj= an+ 1-∑ ai∂ f∂ xin+ 2 α defined on R^ n R n, where α α is a nonzero constant, and (a_1, a_2, ..., a_ n+ 1)(a 1, a 2,…, an+ 1) is a constant vector in R^ n+ 1 R n+ 1. Then the graph hypersurface M={(x, f (x))\} M=(x, f (x)) in R^ n+ 1 R n+ 1 is an α α-relative parabolic affine hypersphere in Li-geometry. In this paper, we will extend a celebrated theorem of Jörgens–Calabi–Pogorelov in Blaschke geometry to Li-geometry. We classify Euclidean complete α α-relative parabolic affine hyperspheres and show that any smooth strictly convex entire solution of the above PDE with α ∉ n+ 2 n+ 1, n+ 2 α∉ n+ 2 n+ 1, n+ 2 must be a quadratic polynomial.