Interior Hessian estimates for $k$-convex solutions of Hessian quotient equations

Abstract

We study interior Hessian estimates for $k$-convex solutions of Hessian quotient equations $\frac{{\sigma }_k(D^{2}u)}{{\sigma }_{\ell }(D^{2}u)}=1$ with $2\le k\le n$, $0\le \ell <k$, and $k-\ell \in \{1,2\}$. We first adopt the unified quantitative separation-propagation method of Li--Wu [LW26b] to derive a doubling inequality, and then use the compactness argument of Shankar--Yuan [SY25] to obtain the estimates. This approach also applies to a class of general Hessian equations.

arXiv comment16 pages