Isolated Singularities of k-Hessian Equations

We classify isolated singularities of nonpositive, $k$-admissible classical solutions to $k$-Hessian equations without radial symmetry assumptions. For $1\le k<n\mathrm{/}2$, we consider $F_k[u]=(-u{)}^p$ with $1<p\le nk\mathrm{/}(n-2k)$. Below the Serrin exponent, we obtain fundamental-solution asymptotics and characterize removability by the vanishing of the leading coefficient. At the Serrin exponent, we determine the exact logarithmic correction and its universal coefficient for every nonremovable singularity. In the critical dimension $k=n\mathrm{/}2$, we establish logarithmic asymptotics and a removability criterion for $F_k[u]=f(x,u,\mathrm{\nabla }u)$ with continuous sources satisfying $0\le f(x,s,\xi )\le C\mathrm{\mid }\xi {\mathrm{\mid }}^{\mathrm{\ell }}{e}^{-\alpha s}$, where $\alpha >0$ and $0\le \mathrm{\ell }<n$. The proofs combine Hessian-measure compactness, Wolff-potential estimates, a global Bôcher theorem, and a bounded-variation analysis of spherical extrema.