A Bernstein Theorem for the Self-Shrinking $J$-Equation and Some Generalizations

We prove that every entire smooth plurisubharmonic solution of the self-shrinking $J$-equation on ${\mathbb{C}}^n$ is a quadratic polynomial. This removes the asymptotic lower bound assumption on the complex Hessian in \cite[Theorem 4]{HJ}. The result also recovers the corresponding real rigidity theorem in \cite[Theorem 1.1]{HOW} as a special case. More generally, our method applies to a broad class of fully nonlinear elliptic operators satisfying suitable structural conditions, including the inverse complex Hessian quotient operators $-{\sigma }_{k-1}\mathrm{/}{\sigma }_k$ for $1\le k\le n$.