Mean Curvature Flow and Isoperimetric Inequalities
Mean Curvature Flow and Isoperimetric Inequalities
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DOI：
10.1007/978-3-0346-0213-6
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发表时间：
2009-11
期刊：
影响因子：
--
通讯作者：
M. Ritoré;C. Sinestrari
中科院分区：
文献类型：
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作者：
M. Ritoré;C. Sinestrari
几何流在物理学和几何学中有许多应用。在某些物理模型中，在描述界面演化过程时会出现平均曲率流。这与这样的流动是面积泛函的梯度流动的性质有关，因此在表面能最小化的问题中自然出现。平均曲率流也有许多几何应用，类似于抽象黎曼流形上的度量的里奇流。人们可以使用这个流作为一种工具，以获得满足一定曲率条件的曲面的分类结果，以及构造极小曲面。由几何抛物方程的解得到的几何流可以被看作是证明等周不等式的另一种工具。另一方面，等周不等式可以帮助处理这些流的收敛的几个方面。等周不等式在其他几何领域有许多应用，如双曲流形。
Geometric flows have many applications in physics and geometry. The mean curvature flow occurs in the description of the interface evolution in certain physical models. This is related to the property that such a flow is the gradient flow of the area functional and therefore appears naturally in problems where a surface energy is minimized. The mean curvature flow also has many geometric applications, in analogy with the Ricci flow of metrics on abstract riemannian manifolds. One can use this flow as a tool to obtain classification results for surfaces satisfying certain curvature conditions, as well as to construct minimal surfaces. Geometric flows, obtained from solutions of geometric parabolic equations, can be considered as an alternative tool to prove isoperimetric inequalities. On the other hand, isoperimetric inequalities can help in treating several aspects of convergence of these flows. Isoperimetric inequalities have many applications in other fields of geometry, like hyperbolic manifolds.