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| In mathematics, in [[Riemannian geometry]], [[Mikhail Gromov (mathematician)|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.
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| ==Explanation==
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| To explain the conjecture, we start with the observation that the equatorial circle of the unit 2-sphere
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| :<math>S^2 \subset \R^3 \,\!</math>
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| is a [[Riemannian circle]] ''S''<sup>1</sup> of length 2π and diameter π. More precisely, the Riemannian distance function of ''S''<sup>1</sup> 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. | |
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| We consider all fillings of ''S''<sup>1</sup> 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.
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| ==Relation to Pu's inequality==
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| [[Image:Steiner's Roman Surface.gif|thumb|An animation of the [[Roman Surface]] representing RP<sup>2</sup> in R<sup>3</sup>]]
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| The case of simply-connected fillings is equivalent to [[Pu's inequality|Pu's inequality for the real projective plane]] RP<sup>2</sup>. Recently the case of [[genus (mathematics)|genus]]-1 fillings was settled affirmatively, as well (see Bangert ''et al''). 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.
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| ==See also==
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| *[[Filling radius]]
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| *[[Pu's inequality]]
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| *[[Systolic geometry]]
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| ==References==
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| * [[Victor Bangert|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}}
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| *{{math-citation|authors=Gromov, M.|title=Filling Riemannian manifolds|journal=J. Diff. Geom.|volume=18|year=1983|pages=1–147}}
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| *{{Citation | last1=[[Mikhail Katz|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}}
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