The Nitsche--Hopf conjecture for minimal graphs

We prove the Nitsche–Hopf conjecture for non-parametric minimal graphs over disks. If $S$ is a minimal graph over a disk of radius $R$, and if $\xi$ is the point above the center, then $$W(\xi {)}^2\mathrm{\mid }K(\xi )\mathrm{\mid }<\frac{{\pi }^2}{2R^2}\mathrm{.}$$ Here $K$ is the Gaussian curvature and $$W=\sqrt{1+\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^2}=\frac{1}{n_3}$$ is the reciprocal of the vertical component of the upward unit normal. The constant is sharp, as shown by the horizontal tangent-plane extremal sequence of Finn and Osserman. The main difficulty is that the bicentric-quadrilateral comparison theorem for Gaussian curvature controls $\mathrm{\mid }K\mathrm{\mid }$, but it does not by itself control the normalized quantity $W^2\mathrm{\mid }K\mathrm{\mid }$: the slope factor $W$ can be arbitrarily large. We show that the missing information is recovered inside the Scherk-type comparison family from the zero equation for the horizontal harmonic projection. More precisely, in the fixed-arc normalization the point corresponding to the center of the physical disk is a distinguished zero $z_{\circ }$ of the harmonic projection. The equation $f(z_{\circ })=0$, written in harmonic-measure coordinates, reduces the sharp Hopf estimate to a scalar derivative inequality at the admissible zero of a monotone function $G_{A,B}$. We prove this scalar inequality on the full admissible parameter domain by a barrier argument and two explicit Bernstein-polynomial positivity certificates. Combined with the bicentric-quadrilateral comparison theorem of the first author and Melentijević, the Scherk-family estimate gives the sharp normalized Hopf estimate for arbitrary minimal graphs over disks. As a byproduct, we obtain the two-sided bound $$\frac{{\pi }^2}{4}\le W^2\mathrm{\mid }K\mathrm{\mid }\le \frac{{\pi }^2}{2}$$ throughout the normalized Scherk-type comparison family, evaluated at the distinguished point corresponding to the center.