| Evolution of hypersurfaces by the mean curvature minus an external force field |
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| 引用本文： | Yan-nan LIU & Huai-yu JIAN Department of Mathematical Sciences,Tsinghua University,Beijing 100084,China. Evolution of hypersurfaces by the mean curvature minus an external force field[J]. 中国科学A辑(英文版), 2007, 50(2): 231-239. DOI: 10.1007/s11425-007-2077-x |
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| 作者姓名： | Yan-nan LIU & Huai-yu JIAN Department of Mathematical Sciences Tsinghua University Beijing 100084 China |
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| 作者单位： | Yan-nan LIU & Huai-yu JIAN Department of Mathematical Sciences，Tsinghua University，Beijing 100084，China |
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| 基金项目： | 国家自然科学基金;北京市科委科研项目 |
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| 摘 要： | In this paper, we study the evolution of hypersurface moving by the mean curvature minus an external force field. It is shown that the flow will blow up in a finite time if the mean curvature of the initial surface is larger than some constant depending on the boundness of derivatives of the external force field. For a linear force, we prove that the convexity of the hypersurface is preserved during the evolution and the flow has a unique smooth solution in any finite time and expands to infinity as the time tends to infinity if the initial curvature is smaller than the slope of the force.
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| 收稿时间： | 2006-01-23 |
| 修稿时间： | 2006-09-04 |
Evolution of hypersurfaces by the mean curvature minus an external force field |
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| Yan-nan Liu,Huai-yu Jian. Evolution of hypersurfaces by the mean curvature minus an external force field[J]. Science in China(Mathematics), 2007, 50(2): 231-239. DOI: 10.1007/s11425-007-2077-x |
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| Authors: | Yan-nan Liu Huai-yu Jian |
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| Affiliation: | Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China |
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| Abstract: | In this paper, we study the evolution of hypersurface moving by the mean curvature minus an external force field. It is shown that the flow will blow up in a finite time if the mean curvature of the initial surface is larger than some constant depending on the boundness of derivatives of the external force field. For a linear force, we prove that the convexity of the hypersurface is preserved during the evolution and the flow has a unique smooth solution in any finite time and expands to infinity as the time tends to infinity if the initial curvature is smaller than the slope of the force. |
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| Keywords: | parabolic equation mean curvature flow maximum principle (for tensor) |
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