Pu's Inequality - Filling Area Conjecture

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.