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arXiv · 2106.06211

Necessary and Sufficient Conditions to Bernstein Theorem of a Hessian Equation

Shi-Zhong Du
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Abstract

The Hessian quotient equatio were studied for k-th symmetric elementary function S_k(D^2u) of eigenvalues of the Hessian matrix D^2u. Two pointwise quadratic growth conditions were found by Bao-Cheng-Guan-Ji ([1], American J. Math., 2003, 125, 301-316) ensuring Bernstein properties of Hessian quotient equation or k-Hessian equation respectively. In this paper, we will drop the point wise quadratic growth condition of [1] and prove three necessary and sufficient conditions to Bernstein property of (0.1) and (0.2), using a reverse isoperimetric type inequality, volume growth or Lp-integrable respectively.Our volume growth or Lp-integrable conditions improve largely various known point wise conditions in [1,6,7,13,18] etc.

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Shi-Zhong Du. 2021-06-11. Necessary and Sufficient Conditions to Bernstein Theorem of a Hessian Equation. https://arxiv.org/abs/2106.06211

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