To explain the conjecture, we start with the observation that the equatorial circle of the unit 2-sphere S2 in R3 is a Riemannian circle S1 of length 2π. More precisely, the Riemannian distance function of S1 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 2, whereas in the Riemannian circle it is π.
We consider all fillings of S1 by a surface, such that the metric induced by the inclusion of the circle as the boundary of the surface is the Riemannian metric of a circle of length 2π. The inclusion of the circle as the boundary is then called a strongly isometric imbedding of the circle.
In '83 Gromov conjectured that the round hemisphere gives the "best" way of filling the circle among all filling surfaces.
The case of simply connected fillings is equivalent to Pu's inequality for the real projective plane.
The case of genus 1 fillings has recently been settled in the
affirmative, see
Bangert, V; Croke, C.; Ivanov, S.; Katz, M.: Filling area conjecture
and ovalless real hyperelliptic surfaces, Geometric and Functional
Analysis (GAFA) 15 ('05), no. 3, 577-597. See
arXiv:math.DG/0405583.
A $50 prize is offered for the solution of the filling area conjecture
for genus 2 fillings of the circle.