Filling area conjecture

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In mathematics, in Riemannian geometry, Mikhail Gromov's filling area conjecture asserts that among all possible fillings of the Riemannian circle of length 2π by a surface with the strongly isometric property, the round hemisphere has the least area. Here the Riemannian circle refers to the unique closed 1-dimensional Riemannian manifold of total 1-volume 2π and Riemannian diameter π.

Contents 1 Explanation 2 Relation to Pu's inequality 3 See also 4 References

[edit] Explanation

To explain the conjecture, we start with the observation that the equatorial circle of the unit 2-sphere

is a Riemannian circle S¹ of length 2π and diameter π.

More precisely, the Riemannian distance function of S¹ is the restriction of the ambient Riemannian distance on the sphere. This property is not satisfied by the standard imbedding of the unit circle in the Euclidean plane, where a pair of opposite points are at distance 2, not π.

We consider all fillings of S¹ by a surface, such that the restricted metric defined 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 1983 Gromov conjectured that the round hemisphere gives the "best" way of filling the circle among all filling surfaces.

[edit] Relation to Pu's inequality

The case of simply-connected fillings is equivalent to Pu's inequality for the real projective plane. Recently the case of genus-1 fillings was settled affirmatively, as well. Namely, it turns out that one can exploit a half-century old formula by J. Hersch from integral geometry. Namely, consider the family of figure-8 loops on a football, with the self-intersection point at the equator (see figure at the beginning of the article). Hersch's formula expresses the area of a metric in the conformal class of the football, as an average of the energies of the figure-8 loops from the family. An application of Hersch's formula to the hyperelliptic quotient of the Riemann surface proves the filling area conjecture in this case.

[edit] See also

• Filling radius
• Pu's inequality for the real projective plane
• Systolic geometry

[edit] References

• Gromov, M.: Filling Riemannian manifolds, J. Diff. Geom. 18 (1983), 1-147.
• Bangert, V; Croke, C.; Ivanov, S.; Katz, M.: Filling area conjecture and ovalless real hyperelliptic surfaces, Geometric and Functional Analysis (GAFA) 15 (2005) no. 3, 577-597. See arXiv:math.DG/0405583