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On Liouville's theorem for the Hessian quotient equation $σ_2/σ_1$

Siyuan Lu, Marcin Sroka

Abstract

We prove Liouville's theorem for semi-convex entire solutions to Hessian quotient equation $σ_2/σ_1=1$ in $\mathbb{R}^n$. The proof is based on the observation that after rewriting the quotient operator as the $σ_2$ operator, acting on a new function, one can refer to the recent result of Shankar and Yuan on Liouville's theorem for $σ_2$ equation.

On Liouville's theorem for the Hessian quotient equation $σ_2/σ_1$

Abstract

We prove Liouville's theorem for semi-convex entire solutions to Hessian quotient equation in . The proof is based on the observation that after rewriting the quotient operator as the operator, acting on a new function, one can refer to the recent result of Shankar and Yuan on Liouville's theorem for equation.
Paper Structure (3 sections, 4 theorems, 22 equations)

This paper contains 3 sections, 4 theorems, 22 equations.

Table of Contents

  1. Introduction
  2. Liouville's theorem
  3. Interior estimate

Key Result

Lemma 1.1

For $\lambda \in \Gamma_2$, the following equality holds for while $\mu \in \Gamma_2$.

Theorems & Definitions (9)

  • Lemma 1.1
  • proof
  • Theorem 2.1
  • proof
  • Remark 2.2
  • Theorem 2.3
  • proof
  • Theorem 3.1
  • proof