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In the field of differential geometry in mathematics, inverse mean curvature flow (IMCF) is an example of a geometric flow of hypersurfaces of a Riemannian manifold (for example, smooth surfaces in 3-dimensional Euclidean space). Intuitively, a family of surfaces evolves under IMCF if the outward normal speed at which a point on the surface moves is given by the reciprocal of the mean curvature of the surface. For example, a round sphere evolves under IMCF by expanding outward uniformly at an exponentially growing rate (see below). In general, this flow does not exist (for example, if a point on the surface has zero mean curvature), and even if it does, it generally develops singularities. Nevertheless, it has recently been an important tool in differential geometry and mathematical problems in general relativity. == Example: a round sphere == Consider a two-dimensional sphere of radius evolving under IMCF in 3-dimensional Euclidean space, where is the time parameter of the flow. (By symmetry considerations, a round sphere will remain round under this flow, so that the radius at time determines the surface at time .) The outward speed under the flow is the derivative, , and the mean curvature equals . (This may be computed from the first variation of area formula.) Setting the speed equal to the reciprocal of the mean curvature, we have the ordinary differential equation : which possesses a unique, smooth solution given by : where is the radius of the sphere at time . Thus, in this case we see that a round sphere evolves under IMCF by uniformly expanding outward with an exponentially increasing radius. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「inverse mean curvature flow」の詳細全文を読む スポンサード リンク
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