论文标题
定量估计值几乎恒定的平均曲率超曲面
Quantitative estimates for almost constant mean curvature hypersurfaces
论文作者
论文摘要
Alexandrov的肥皂泡定理断言，球形是欧几里得空间中唯一连接的闭合嵌入式式超曲面，其平均曲率恒定。该定理可以扩展到空间形式，并且具有主要曲线的更通用功能。 在这篇简短的综述中，我们讨论了近年来作者获得的有关亚历山德罗夫定理的定量稳定性结果。特别是，我们认为具有接近常数的平均曲率的高度曲面，并定量描述了与单个球体的接近度或根据平均曲率的振荡而与切线球的集合。此外，我们还考虑了非本地环境中的问题，并且我们表明，非局部效应给问题带来了更强的刚性，并防止了冒泡的出现。
Alexandrov's soap bubble theorem asserts that spheres are the only connected closed embedded hypersurfaces in the Euclidean space with constant mean curvature. The theorem can be extended to space forms and it holds for more general functions of the principal curvatures. In this short review, we discuss quantitative stability results regarding Alexandrov's theorem which have been obtained by the author in recent years. In particular, we consider hypersurfaces having mean curvature close to a constant and we quantitatively describe the proximity to a single sphere or to a collection of tangent spheres in terms of the oscillation of the mean curvature. Moreover, we also consider the problem in a non local setting, and we show that the non local effect gives a stronger rigidity to the problem and prevents the appearance of bubbling.
