It is known that the surface of a cone over the unit disc with large height has smaller distortion than the standard embedding of the 2-sphere in |$\mathbb{R}^3$|⁠. In this note we show that distortion minimizers exist among convex embedded 2-spheres and have uniformly bounded eccentricity. Moreover, we prove that |$\pi /2$| is a sharp lower bound on the distortion of embedded closed surfaces of positive genus.

中文翻译：

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空间中球和表面的变形

众所周知，单位圆盘上高度较大的圆锥体的表面变形要比2球体在| $ \ mathbb {R} ^ 3 $ |⁠中的标准嵌入小。在此注释中，我们表明，扭曲最小化器存在于凸形嵌入2个球体之间，并且具有均匀边界的偏心率。此外，我们证明| $ \ pi / 2 $ | 是正型嵌入的闭合曲面的变形的下限。