论文标题
涉及封闭表面的平均曲率的几何不平等现象
Geometric inequalities involving mean curvature for closed surfaces
论文作者
论文摘要
在本文中，我们证明了欧几里得三个空间中封闭表面的一些几何不平等现象。由Gage的凸曲线不等式的动机，我们首先验证了凸面表面Willmore Energy以某些规模不变的数量为界。特别是，我们在凸度下获得了Willmore能量与等值比之间的最佳缩放定律。此外，我们还针对连接的封闭表面的topping猜想相关的直径和平均曲率。我们在简单连接的轴对称表面中证明了这种猜想，此外，获得了一个鲜明的剩余术语，该术语确保了最佳形状即使没有凸度的最佳形状也必须是笔直的第一个证据。
In this paper we prove some geometric inequalities for closed surfaces in Euclidean three-space. Motivated by Gage's inequality for convex curves, we first verify that for convex surfaces the Willmore energy is bounded below by some scale-invariant quantities. In particular, we obtain an optimal scaling law between the Willmore energy and the isoperimetric ratio under convexity. In addition, we also address Topping's conjecture relating diameter and mean curvature for connected closed surfaces. We prove this conjecture in the class of simply-connected axisymmetric surfaces, and moreover obtain a sharp remainder term which ensures the first evidence that optimal shapes are necessarily straight even without convexity.
