Liouville theorem for <i>k</i> -curvature equation in half space with fully nonlinear boundary condition

Abstract We establish the Liouville theorem for positive constant <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>σ</m:mi> <m:mi>k</m:mi> </m:msub> </m:math> {\sigma_{k}} -curvature equation in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msubsup> <m:mi>ℝ</m:mi> <m:mo>+</m:mo> <m:mi>n</m:mi> </m:msubsup> </m:math> {\mathbb{R}_{+}^{n}} and positive constant boundary <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msubsup> <m:mi mathvariant="script">ℬ</m:mi> <m:mi>k</m:mi> <m:mi>g</m:mi> </m:msubsup> </m:math> {\mathcal{B}_{k}^{g}} curvature equation, where the boundary curvature <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msubsup> <m:mi mathvariant="script">ℬ</m:mi> <m:mi>k</m:mi> <m:mi>g</m:mi> </m:msubsup> </m:math> {\mathcal{B}_{k}^{g}} is discovered by Sophie Chen in [S.-Y. S. Chen, Conformal deformation on manifolds with boundary, Geom. Funct. Anal. 19 2009, 4, 1029–1064] from the natural variational functional for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>σ</m:mi> <m:mi>k</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi>A</m:mi> <m:mi>g</m:mi> </m:msub> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {\sigma_{k}(A_{g})} .