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- Title
A rigidity theorem of $$\alpha $$ -relative parabolic hyperspheres.
- Authors
Xu, Ruiwei; Zhu, Lingyun
- Abstract
Let f be a smooth strictly convex solution of defined on $${\mathbb {R}}^n$$ , where $$\alpha $$ is a nonzero constant, and $$(a_1,a_2,\ldots ,a_{n+1})$$ is a constant vector in $${{\mathbb {R}}}^{n+1}$$ . Then the graph hypersurface $$M=\{(x, f(x))\}$$ in $${{\mathbb {R}}}^{n+1}$$ is an $$\alpha $$ -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 $$\alpha $$ -relative parabolic affine hyperspheres and show that any smooth strictly convex entire solution of the above PDE with $$\alpha \notin [\frac{n+2}{n+1}, n+2]$$ must be a quadratic polynomial.
- Publication
Manuscripta Mathematica, 2017, Vol 154, Issue 3/4, p503
- ISSN
0025-2611
- Publication type
Article
- DOI
10.1007/s00229-017-0918-7