Filling Area Conjecture
An alternative formulation of Pu's inequality is the following. Of all possible fillings of the Riemannian circle of length by a -dimensional disk with the strongly isometric property, the round hemisphere has the least area.
To explain this formulation, we start with the observation that the equatorial circle of the unit -sphere is a Riemannian circle of length . More precisely, the Riemannian distance function of is induced from the ambient Riemannian distance on the sphere. Note that this property is not satisfied by the standard imbedding of the unit circle in the Euclidean plane. Indeed, the Euclidean distance between a pair of opposite points of the circle is only, whereas in the Riemannian circle it is .
We consider all fillings of by a -dimensional disk, such that the metric induced by the inclusion of the circle as the boundary of the disk is the Riemannian metric of a circle of length . 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 even when the filling surface is allowed to have positive genus.
Read more about this topic: Pu's Inequality
Famous quotes containing the words filling, area and/or conjecture:
“To play is nothing but the imitative substitution of a pleasurable, superfluous and voluntary action for a serious, necessary, imperative and difficult one. At the cradle of play as well as of artistic activity there stood leisure, tedium entailed by increased spiritual mobility, a horror vacui, the need of letting forms no longer imprisoned move freely, of filling empty time with sequences of notes, empty space with sequences of form.”
—Max J. Friedländer (1867–1958)
“... nothing is more human than substituting the quantity of words and actions for their character. But using imprecise words is very similar to using lots of words, for the more imprecise a word is, the greater the area it covers.”
—Robert Musil (1880–1942)
“There is something fascinating about science. One gets such wholesale returns of conjecture out of such a trifling investment of fact.”
—Mark Twain [Samuel Langhorne Clemens] (1835–1910)