|
In mathematics, in Riemannian geometry, Mikhail Gromov's filling area conjecture asserts that among all possible fillings of the Riemannian circle of length 2 by a surface with the strongly isometric property, the round hemisphere has the least area. Here the ''Riemannian circle'' refers to the unique closed 1-dimensional Riemannian manifold of total 1-volume 2 and Riemannian diameter . ==Explanation== To explain the conjecture, we start with the observation that the equatorial circle of the unit 2-sphere : is a Riemannian circle ''S''1 of length 2 and diameter . More precisely, the Riemannian distance function of ''S''1 is the restriction of the ambient Riemannian distance on the sphere. This property is ''not'' satisfied by the standard imbedding of the unit circle in the Euclidean plane, where a pair of opposite points are at distance 2, not . We consider all fillings of ''S''1 by a surface, such that the restricted metric defined by the inclusion of the circle as the boundary of the surface is the Riemannian metric of a circle of length 2. The inclusion of the circle as the boundary is then called a strongly isometric imbedding of the circle. In 1983 Gromov conjectured that the round hemisphere gives the "best" way of filling the circle among all filling surfaces. 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「Filling area conjecture」の詳細全文を読む スポンサード リンク
|