Filling area conjecture
In differential geometry, Mikhail Gromov's filling area conjecture asserts that among all possible Riemannian surfaces that isometrically fill a Riemannian circle of given length, the hemisphere has the least area.
Every Riemannian manifold M (including any curve or surface in Euclidean space) is a metric space, in which the (intrinsic) distance dM(x,y) between two points x and y of M is the infimum of the lengths of the curves that go along M from x to y. In particular, any Riemannian circle of total length 2π is a metric space with diameter π, in fact any point x of the circle has a unique opposite point y in the circle, at the maximum distance π from x.
A compact Riemannian surface is said to fill a Riemannian circle if the boundary of the surface is the circle, and the filling is said isometric if for any pair of points on the circle, the intrinsic distance between them along the surface is the same, and not less, than the intrinsic distance along the circle. In other words, an isometric filling is a filling that does not introduce shortcuts between the points of the circle.
For example, in three-dimensional Euclidean space, the circle
is filled both by the flat disk
and by the hemisphere
Of these two fillings, only the hemisphere is isometric. The flat disk is not an isometric filling because, in it, the intrinsic distance between any two opposite points is 2, which is less than π.
In 1983, Mikhail Gromov conjectured that the hemisphere has least area among the orientable compact Riemannian surfaces that fill isometrically a circle of given length. In fact he conjectured an analogous statement for higher dimensions as well.
Gromov proved his conjecture restricted to fillings homeomorphic to a disk. He did so by applying Pu's inequality to the surface, homeomorphic to the real projective plane, obtained from the filling surface by identifying opposite points of its boundary.
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