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A rigidity theorem of \(\alpha \)-relative parabolic hyperspheres

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Abstract

Let f be a smooth strictly convex solution of

$$\begin{aligned} \det \left( \frac{\partial ^{2}f}{\partial x_{i}\partial x_{j}}\right) =\left( a_{n+1}-\sum a_i\frac{\partial f}{\partial x_{i}} \right) ^{\frac{n+2}{\alpha }} \end{aligned}$$

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.

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Correspondence to Ruiwei Xu.

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Research partially supported by NSFC (Nos. 11471225, 11671121) and IRTSTHN (14IRTSTHN023).

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Xu, R., Zhu, L. A rigidity theorem of \(\alpha \)-relative parabolic hyperspheres. manuscripta math. 154, 503–512 (2017). https://doi.org/10.1007/s00229-017-0918-7

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  • DOI: https://doi.org/10.1007/s00229-017-0918-7

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