We consider the flow of closed convex hypersurfaces in Euclidean space $\mathbb{R}^{n+1}$ with speed given by a power of the $k$‑th mean curvature $E_k$ plus a global term chosen to impose a constraint involving the enclosed volume $V_{n+1}$ and the mixed volume $V_{n+1-k}$ of the evolving hypersurface. We prove that if the initial hypersurface is strictly convex, then the solution of the flow exists for all time and converges to a round sphere smoothly. No curvature pinching assumption is required on the initial hypersurface.

中文翻译：

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第k个平均曲率的幂保持体积的流量

我们考虑欧几里德空间$ \ mathbb {R} ^ {n + 1} $中的闭合凸超曲面的流动，其速度由$ k $次平均曲率$ E_k $的幂加上选择用来施加约束涉及封闭的体积$ V_ {n + 1} $和演化的超曲面的混合体积$ V_ {n + 1-k} $。我们证明，如果初始超曲面是严格凸的，则流的解一直存在，并且平滑地收敛到圆球。在初始超曲面上不需要曲率收缩假设。