In this paper we prove an extrinsic one-sided curvature estimate for disks embedded in $\mathbb{R}^3$ with constant mean curvature, which is independent of the value of the constant mean curvature. We apply this extrinsic one-sided curvature estimate in [26] to prove a weak chord arc result for these disks. In Section 4 we apply this weak chord arc result to obtain an intrinsic version of the one-sided curvature estimate for disks embedded in $\mathbb{R}^3$ with constant mean curvature. In a natural sense, these one-sided curvature estimates generalize respectively, the extrinsic and intrinsic one-sided curvature estimates for minimal disks embedded in $\mathbb{R}^3$ given by Colding and Minicozzi in Theorem 0.2 of [8] and in Corollary 0.8 of [9].

中文翻译：

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$ H $磁盘的单侧曲率估计

在本文中，我们证明了嵌入具有恒定平均曲率的$ \ mathbb {R} ^ 3 $中的磁盘的外在一侧曲率估计，该估计与恒定平均曲率的值无关。我们在[26]中应用了这种外在的单侧曲率估计，以证明这些圆盘的弱弦弧结果。在第4节中，我们应用此弱弦弧结果来获得嵌入具有恒定平均曲率的$ \ mathbb {R} ^ 3 $中的磁盘的单侧曲率估计的本征形式。从自然意义上讲，这些单侧曲率估计分别概括，由Colding和Minicozzi在[8]定理0.2中给出的$ \ mathbb {R} ^ 3 $中嵌入的最小圆盘的外部和固有单侧曲率估计，以及在[9]的推论0.8中。