Geometric Viewpoint of $L^{\mathrm{\infty }}$ Estimates for Linearized Monge-Ampère Equations

We prove a Harnack inequality on Monge–Ampère gradient graphs under suitable geometric bounds, giving us interior $L^{\mathrm{\infty }}$ estimates for the linearized Monge–Ampère equation. The proof combines Moser iteration on extrensic balls with a gradient estimate for the extrinsic distance function. The argument avoids the Caffarelli–Gutiérrez Harnack inequality and the engulfing and covering properties of Monge–Ampère sections.