Interior $C^2$ estimate for Hessian quotient equation in general dimension

Abstract: In this paper, we study the interior $C^2$ regularity problem for the Hessian quotient equation $\left(\frac{\sigma_n}{\sigma_k}\right)(D^2u)=f$. We give a complete answer to this longstanding problem: for $k=n-1,n-2$, we establish an interior $C^2$ estimate; for $k\leq n-3$, we show that interior $C^2$ estimate fails by finding a singular solution.