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 $|Grap{h}_u\cap B^{n+1}|\le \frac{1}{2}|{\mathbb{S}}^n|.$ A basic question is whether this half-sphere bound is sharp for minimal graphs. We show that it is not. More precisely, for every $n\ge 2$, there exists an explicit constant ${\delta }_n>0$, depending only on $n$, such that $|Grap{h}_u\cap B^{n+1}|\le \frac{1}{2}|{\mathbb{S}}^n|-{\delta }_n$. 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.

11 pages, no figure. This version replaces the withdrawn previous version. The previous sharpness claim was incorrect. The present version proves instead a uniform positive gap below the half-sphere bound. All comments are welcome!