In an ambient space with rotational symmetry around an axis (which include the Hyperbolic and Euclidean spaces), we study the evolution under the volume-preserving mean curvature flow of a revolution hypersurface M generated by a graph over the axis of revolution and with boundary in two totally geodesic hypersurfaces (tgh for short). Requiring that, for each time t ≥ 0, the evolving hypersurface M t meets such tgh orthogonally, we prove that: (a) the flow exists while M t does not touch the axis of rotation; (b) throughout the time interval of existence, (b1) the generating curve of M t remains a graph, and (b2) the averaged mean curvature is double side bounded by positive constants; (c) the singularity set (if non-empty) is finite and lies on the axis; (d) under a suitable hypothesis relating the enclosed volume to the n-volume of M, we achieve long time existence and convergence to a revolution hypersurface of constant mean curvature.
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机译：在围绕轴具有旋转对称性的环境空间（包括双曲线和欧几里得空间）中，我们研究了旋转超曲面M的体积保持平均曲率流在体积保留平均曲率流下的演化，该曲面由旋转轴上的图生成并在边界上存在两个完全测地的超曲面（简称tgh）。要求，对于每当t≥0时，演化的超曲面M t 都正交地满足这样的tgh，我们证明：（a）存在流动，而M t 不接触旋转轴； （b）在整个存在时间间隔内，（b1）M t 的生成曲线仍为图，（b2）平均平均曲率是由正常数界定的双侧； （c）奇异集（如果为非空）是有限的并且位于轴上； （d）在将封闭体积与M的n体积相关的合适假设下，我们实现了长时间存在并收敛到具有恒定平均曲率的旋转超曲面。
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