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On the fully nonlinear Yamabe problem with constant boundary mean curvature. I

BaoZhi Chu, YanYan Li, Zongyuan Li

Abstract

In a recent paper, we established optimal Liouville-type theorems for conformally invariant second-order elliptic equations in the Euclidean space. In this work, we prove an optimal Liouville-type theorem for these equations in the half-Euclidean space.

On the fully nonlinear Yamabe problem with constant boundary mean curvature. I

Abstract

In a recent paper, we established optimal Liouville-type theorems for conformally invariant second-order elliptic equations in the Euclidean space. In this work, we prove an optimal Liouville-type theorem for these equations in the half-Euclidean space.

Paper Structure

This paper contains 13 sections, 13 theorems, 159 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Examples
  4. More general equations.
  5. Notations
  6. Organization of the paper
  7. Proof of Theorem \ref{['nondegenerateliouville-critical']}
  8. Proof of Theorem \ref{['nondegenerateliouville']}
  9. Solutions on one variable
  10. Nonexistence of entire solutions
  11. Counterexamples in Remark \ref{['remark-optimal-range']}
  12. Counterexamples in Remark \ref{['nondegoptimalremark']}
  13. Equivalent theorems in terms of Ricci tensor

Key Result

Theorem 1.1

For $n\geq 2$ and $c\in\mathbb{R}$, let $(f,\Gamma)$ satisfy eqn-230331-0110--natural-assumption and $\lambda^*\notin \overline\Gamma$, and $v\in C^2(\overline\mathbb{R}^n_+)$ satisfy halfspace-equ-v-critical. Then where $a,b>0$ and $\bar{x}=(\bar{x}', \bar{x}_n)\in\mathbb{R}^n$ satisfy $f(2a^{-2}b\bm{e})=1$ and $2 a^{-1}b \bar{x}_n=c$ with $\bm{e}=(1,1,\dots,1)$.

Theorems & Definitions (34)

  • Theorem 1.1
  • Remark 1.1
  • Example 1.1
  • Example 1.2
  • Example 1.3
  • Theorem 1.2
  • Proposition 2.1
  • Remark 2.1
  • Proposition 2.2
  • proof : Proof of Proposition \ref{['ms_starter']}
  • ...and 24 more