Exterior Serrin Rigidity for the Homogeneous{k}-Hessian Equation

In this paper, we study the exterior overdetermined problems for the homogeneous k-Hessian equations $${\sigma }_k(D^{2}u)=0\text{in }{\mathbb{R}}^n\setminus \bar{\mathrm{\Omega }}$$ in three dimensional regimes. For $2\le k<n$ and smooth strictly star-shaped domain $\mathrm{\Omega }⋐{\mathbb{R}}^n$, we establish ball rigidity results in all three dimensional regimes. If $2\le k<\frac{n}{2}$, the solution with boundary value $-1$ and limit zero at infinity is treated under strict $(k-1)$-convexity of $\mathrm{\partial }\mathrm{\Omega }$. If $k=\frac{n}{2}$, the solution with boundary value $0$ and logarithmic growth at infinity is considered under strict $(k-1)$-convexity of $\mathrm{\partial }\mathrm{\Omega }$; if $k>\frac{n}{2}$, the solution with boundary $1$ and fundamental power-growth at infinity is considered under strict $(k-1)$-convexity of $\mathrm{\partial }\mathrm{\Omega }$. In each case, for a positive constant $c$, $\mathrm{\mid }Du\mathrm{\mid }=c>0$ on $\mathrm{\partial }\mathrm{\Omega }$ forces $\mathrm{\Omega }$ to be a Euclidean ball and determines the solution explicitly. The three arguments are organized by the single parameter $q=(n-k)\mathrm{/}k$ and the scale function $$F_q(t)=\frac{t^{1-q}-1}{1-q},F_1(t)=\log t\mathrm{.}$$ After normalization, $U=F_q(v)$ and $M_q[v]=v{D}^{2}v-qDv\otimes Dv$ satisfies ${\sigma }_k(M_q[v])=0$. A unified contact-point calculation gives the sharp bound $\mathrm{\mid }Dv\mathrm{\mid }\le b$;a boundary viscosity contact then yields $H_k\ge qb{H}_{k-1}$. The integral mechanism that closes the argument changes at the critical exponent: Rellich–Pohozaev identities are used for $q>1$, a scale-invariant Wronskian–Newton current for $q=1$, and a renormalized Newton–Jacobi mass for $q<1$. When $q\le 1,$ we also obtain a global strict k-convex defining function by Minkowski gauge, this help us extend the exterior construction from convex domains to strictly star-shaped domains.