Filling area conjecture

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.

Explanation

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

:$S^2\; subset\; R^3\; ,!$

is a Riemannian circle "S"¹ of length 2π and diameter π. More precisely, the Riemannian distance function of "S"¹ 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 "S"¹ 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.

Relation to Pu's inequality

[
Roman Surface representing RP² in R³] 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.

ee also

*Filling radius
*Pu's inequality
*Systolic geometry

References

* 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

*math-citation|authors=Gromov, M.|title=Filling Riemannian manifolds|journal=J. Diff. Geom.|volume=18|year=1983|pages=1-147

*Citation | last1=Katz | first1=Mikhail G. | title=Systolic geometry and topology| publisher=American Mathematical Society | location=Providence, R.I. | series=Mathematical Surveys and Monographs | isbn=978-0-8218-4177-8 | year=2007 | volume=137