标题： 关于极小图的曲率猜想的一个精确形式 显示英文标题

标题： On a sharp form of curvature conjecture for minimal graphs

摘要： 最近，作者与Melentijević通过证明在单位圆盘上的极小图在位于原点正上方的点处的尖锐不等式\[ |\mathcal{K}| < c_0 = \frac{\pi^2}{2} \]，解决了长期存在的高斯曲率问题。 常数\( c_0 \)被称为\emph{海因茨常数}。 在此结果的基础上，我们得到了对霍普常数\( c_1 \)的改进估计。 此外，我们证明了对于任何给定的单位法向量\( \mathbf{n} \)，存在一个在单位圆盘上的极小图——沿着坐标方向弯曲——其在原点正上方的高斯曲率严格小于但可以任意接近于具有相同法向量且位于双心四边形上方的相关Scherk型曲面的曲率。 这个尖锐不等式加强了Finn和Osserman的经典结果，该结果在单位法向量为\( (0,0,1) \)的特殊情况下适用。 显示英文摘要

摘要： Recently, the author and Melentijevi\'c resolved the longstanding Gaussian curvature problem by proving the sharp inequality \[ |\mathcal{K}| < c_0 = \frac{\pi^2}{2} \] for minimal graphs over the unit disk, evaluated at the point of the graph lying directly above the origin. The constant \( c_0 \) is known as the \emph{Heinz constant}. Building on this result, we obtain an improved estimate for the Hopf constant \( c_1 \). In addition, we show that for any prescribed unit normal vector \( \mathbf{n} \), there exists a minimal graph over the unit disk -- bending in the coordinate directions -- whose Gaussian curvature at the point above the origin is strictly smaller, yet arbitrarily close to, the curvature of the associated Scherk-type surface with the same normal, situated above a bicentric quadrilateral. This sharp inequality strengthens the classical result of Finn and Osserman, which applies in the special case when the unit normal is \( (0,0,1) \).