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Liouville Theorem for $k-$curvature equation in half space with fully nonlinear boundary condition

Wei Wei

TL;DR

The paper establishes a Liouville-type rigidity result for metrics in the half-space $\\mathbb{R}_{+}^{n}$ with constant $\sigma_{k}$-curvature and a positive constant boundary curvature $\\mathcal{B}_{k}^{g}$, where $\\mathcal{B}_{k}^{g}$ is derived from Chen's variational functional for $\\sigma_{k}(A_{g})$. It develops the boundary linearization, proves ellipticity within the $\\Gamma_{k}^{+}$ cone, and applies the method of moving spheres to obtain an explicit Kelvin-type profile $u(y)=\left(\frac{\sqrt{b}}{1+b|y-p|^{2}}\right)^{\frac{n-2}{2}}$ with corresponding geometric data, plus a corollary in the unit ball. The second part shows that, under a Li-Li-type lifting lemma, the stronger asymptotic assumption holds, enabling the full half-space Liouville theorem. The appendix provides a Corner Hopf Lemma and other technical lemmas to support the boundary analysis. Overall, the work advances the understanding of fully nonlinear boundary conditions for $k$-curvature equations and provides tools for half-space and boundary-constrained problems in conformal geometry.

Abstract

We establish the Liouville theorem for positive constant $σ_{k}$-curvature equation in $\mathbb{R}_{+}^{n}$ and positive constant boundary $\mathcal{B}_{k}^{g}$ curvature equation, where the boundary curvature $\mathcal{B}_{k}^{g}$ is discovered by Sophie Chen \cite{Chen} from the natural variational functional for $σ_{k}(A_{g})$.

Liouville Theorem for $k-$curvature equation in half space with fully nonlinear boundary condition

TL;DR

The paper establishes a Liouville-type rigidity result for metrics in the half-space with constant -curvature and a positive constant boundary curvature , where is derived from Chen's variational functional for . It develops the boundary linearization, proves ellipticity within the cone, and applies the method of moving spheres to obtain an explicit Kelvin-type profile with corresponding geometric data, plus a corollary in the unit ball. The second part shows that, under a Li-Li-type lifting lemma, the stronger asymptotic assumption holds, enabling the full half-space Liouville theorem. The appendix provides a Corner Hopf Lemma and other technical lemmas to support the boundary analysis. Overall, the work advances the understanding of fully nonlinear boundary conditions for -curvature equations and provides tools for half-space and boundary-constrained problems in conformal geometry.

Abstract

We establish the Liouville theorem for positive constant -curvature equation in and positive constant boundary curvature equation, where the boundary curvature is discovered by Sophie Chen \cite{Chen} from the natural variational functional for .
Paper Structure (8 sections, 14 theorems, 125 equations)

This paper contains 8 sections, 14 theorems, 125 equations.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Linearization of $\mathcal{B}_{k}^{g}$ on $\partial\mathbb{R}_{+}^{n}$
  4. Liouville Theorem under the Stronger condition
  5. Proof of Theorem \ref{['thm:half space liouville']}
  6. Appendix
  7. Corner Hopf Lemma
  8. Useful Lemmas

Key Result

Theorem 1

Given a positive constant $c_{0}$, let $g_{u}=u^{\frac{4}{n-2}}|dx|^{2}$ in $\mathbb{R}_{+}^{n}$ satisfy where $\sigma_{k}(A_{g_{u}})$ and $\mathcal{B}_{k}^{g_{u}}$ are defined by (expression of sigmak) and (eq:boundary equation) respectively. Assume that $\lim_{x\rightarrow0}u_{0,1}$ exists, where $u_{0,1}(x):=|x|^{2-n}u\left(\frac{x}{|x|^{2}}\right)$. Then there exist a positive constant $b\in\

Theorems & Definitions (22)

  • Theorem 1
  • Corollary 1.1
  • Lemma 2.1
  • proof
  • Theorem 2
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • ...and 12 more