论文标题
通过凹形曲率函数移动的超曲面的凸度估计值
Convexity estimates for hypersurfaces moving by concave curvature functions
论文作者
论文摘要
我们研究了完全非线性的几何流量，这些几何流量严格地形成了欧几里得空间中$ k $ convex Hypersurfaces，其主要弯曲功能给出了正常速度。具体而言，我们考虑的速度是通过在主要曲线的平均值和$ k $ harmonic平均值之间进行非线性插值来获得的。我们的主要结果是凸度估计值表明，在紧凑型溶液上，高曲率的区域大约是凸。与平均曲率流相反，此处考虑的完全非线性流在里曼尼亚背景中保留$ k $ convexity，我们表明，只要环境曲率满足自然捏合条件，凸度的估计值就会延续到这种情况下。
We study fully nonlinear geometric flows that deform strictly $k$-convex hypersurfaces in Euclidean space with pointwise normal speed given by a concave function of the principal curvatures. Specifically, the speeds we consider are obtained by performing a nonlinear interpolation between the mean and the $k$-harmonic mean of the principal curvatures. Our main result is a convexity estimate showing that, on compact solutions, regions of high curvature are approximately convex. In contrast to the mean curvature flow, the fully nonlinear flows considered here preserve $k$-convexity in a Riemannian background, and we show that the convexity estimate carries over to this setting as long as the ambient curvature satisfies a natural pinching condition.
