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:
“I am filling the room
with the words from my pen.
Words leak out of it like a miscarriage.
I am zinging words out into the air
and they come back like squash balls.
Yet there is silence.”
—Anne Sexton (1928–1974)
“During the Civil War the area became a refuge for service- dodging Texans, and gangs of bushwhackers, as they were called, hid in its fastnesses. Conscript details of the Confederate Army hunted the fugitives and occasional skirmishes resulted.”
—Administration in the State of Texa, U.S. public relief program (1935-1943)
“What these perplexities of my uncle Toby were,—’tis impossible for you to guess;Mif you could,—I should blush ... as an author; inasmuch as I set no small store by myself upon this very account, that my reader has never yet been able to guess at any thing. And ... if I thought you was able to form the least ... conjecture to yourself, of what was to come in the next page,—I would tear it out of my book.”
—Laurence Sterne (1713–1768)