The mean curvature flow of submanifolds of high codimension
The mean curvature flow of submanifolds of high codimension
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DOI：
10.25911/5d5155ff2112e
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发表时间：
2011-04
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影响因子：
--
通讯作者：
C. Baker
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文献类型：
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作者：
C. Baker
从偏微分方程组的角度研究平均曲率流始于1984年Gerhard Huisken的开创性工作。自那时以来，超曲面的平均曲率流一直是一个活跃的研究领域。虽然Huisken的开创性论文现在只有25年多一点的历史，但对高余维子流形的平均曲率流的研究直到最近才开始受到关注。子流形的平均曲率流是本文的主要研究对象，我们所得到的中心结果可以看作是一些早期超曲面定理的高余维类比。Huisken 1984年的论文的结果大致表明，凸超曲面在有限时间内在平均曲率流下演化为圆点。这里我们得到的结果是，如果第二基本形式的长度与平均曲率向量的长度之比是有界的(由依赖于维度而不是余维的显式常数决定)，则子流形将在有限时间内在平均曲率流下演化到圆点。我们研究了在平坦和弯曲背景下的演化，并探索了当第一个奇异时间逼近时流动的奇异行为。
The study of the mean curvature flow from the perspective of partial differential equations began with Gerhard Huisken's pioneering work in 1984. Since that time, the mean curvature flow of hypersurfaces has been a lively area of study. Although Huisken's seminal paper is now just over twenty-five years old, the study of the mean curvature flow of submanifolds of higher codimension has only recently started to receive attention. The mean curvature flow of submanifolds is the main object of investigation in this thesis, and indeed, the central results we obtain can be considered as high codimension analogues of some early hypersurface theorems. The result of Huisken's 1984 paper roughly says that convex hypersurfaces evolve under the mean curvature flow to round points in finite time. Here we obtain the result that if the ratio of the length of the second fundamental form to the length of the mean curvature vector is bounded (by some explicit constant depending on dimension but not codimension), then the submanifold will evolve under the mean curvature flow to a round point in finite time. We investigate evolutions in flat and curved backgrounds, and explore the singular behaviour of the flows as the first singular time is approached.