A strict upper volume bound for minimal graphs in the unit ball

Abstract

Let $u$ be a solution of the minimal surface equation on a domain containing the closed unit ball $\overline{B^n}\subset\mathbb R^n$. A classical calibration argument gives $|Graph_u\cap B^{n+1}| \leq \frac12 |\mathbb S^n|.$ A basic question is whether this half-sphere bound is sharp for minimal graphs. We show that it is not. More precisely, for every $n\geq2$, there exists an explicit constant $\delta_n>0$, depending only on $n$, such that $ | Graph_u\cap B^{n+1}|\leq \frac12 |\mathbb S^n|-\delta_n$. The proof combines calibration with a canonical spherical filling associated with the subgraph and a quantitative incompatibility between near equality in the calibration estimate and the divergence-free structure of the minimal surface equation. We also give an improved explicit gap and formulate the corresponding sharp extremal problem.