Necessary and Sufficient Conditions to Bernstein Theorem of a Hessian Equation

TL;DR

This paper establishes new necessary and sufficient conditions for the Bernstein property of Hessian equations, removing previous growth restrictions and using volume and integrability conditions to improve upon earlier pointwise criteria.

Contribution

It introduces three new conditions based on volume growth and Lp-integrability that replace earlier pointwise growth conditions for Bernstein properties of Hessian equations.

Findings

Proves three necessary and sufficient conditions for Bernstein property.

Replaces pointwise growth conditions with volume and Lp-integrability conditions.

Improves upon previous criteria in the literature.

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.