The classical four-vertex theorem describes a fundamental property of simple closed planar curves. It has been extended to space curves, namely a smooth, simple closed curve in $\mathbb{R}^3$ has at least four points with vanishing torsion if it lies on a convex surface. More recently, Ghomi [6] extended this property to curves lying on locally convex surfaces. In this paper we provide an alternative approach to the result via the theory of Monge–Ampère equations.

中文翻译：

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关于局部凸曲面上的曲线的四顶点定理

经典的四顶点定理描述了简单闭合平面曲线的基本性质。它已扩展到空间曲线，即$ \ mathbb {R} ^ 3 $中的平滑简单闭合曲线至少有四个点，如果它位于凸面上则具有消失的扭转。最近，Ghomi [6]将此特性扩展到位于局部凸面上的曲线。在本文中，我们通过Monge–Ampère方程理论为结果提供了一种替代方法。