Filling area conjecture

Gromov's proof for the case of Riemannian disks

In the same paper where Gromov stated the conjecture, he proved that

the hemisphere has least area among the Riemannian surfaces that isometrically fill a circle of given length, and are homeomorphic to a disk.

Proof: Let ${\displaystyle M}$ be a Riemannian disk that isometrically fills its boundary of length ${\displaystyle 2L}$. Glue each point ${\displaystyle x\in \mathrm{\partial }M}$ with its antipodal point ${\displaystyle -x}$, defined as the unique point of ${\displaystyle \mathrm{\partial }M}$ that is at the maximum possible distance ${\displaystyle L}$ from ${\displaystyle x}$. Gluing in this way we obtain a closed Riemannian surface ${\displaystyle M^{\prime }}$ that is homeomorphic to the real projective plane and whose systole (the length of the shortest non-contractible curve) is equal to ${\displaystyle L}$. (And reciprocally, if we cut open a projective plane along a shortest noncontractible loop of length ${\displaystyle L}$, we obtain a disk that fills isometrically its boundary of length ${\displaystyle 2L}$.) Thus the minimum area that the isometric filling ${\displaystyle M}$ can have is equal to the minimum area that a Riemannian projective plane of systole ${\displaystyle L}$ can have. But then Pu's systolic inequality asserts precisely that a Riemannian projective plane of given systole has minimum area if and only if it is round (that is, obtained from a Euclidean sphere by identifying each point with its opposite). The area of this round projective plane equals the area of the hemisphere (because each of them has half the area of the sphere).

The proof of Pu's inequality relies, in turn, on the uniformization theorem.

Fillings with Finsler metrics

In 2001, Sergei Ivanov presented another way to prove that the hemisphere has smallest area among isometric fillings homeomorphic to a disk. His argument does not employ the uniformization theorem and is based instead on the topological fact that two curves on a disk must cross if their four endpoints are on the boundary and interlaced. Moreover, Ivanov's proof applies more generally to disks with Finsler metrics, which differ from Riemannian metrics in that they need not satisfy the Pythagorean equation at the infinitesimal level. The area of a Finsler surface can be defined in various inequivalent ways, and the one employed here is the Holmes–Thompson area, which coincides with the usual area when the metric is Riemannian. What Ivanov proved is that

The hemisphere has minimum Holmes–Thompson area among Finsler disks that isometrically fill a closed curve of given length.

Unlike the Riemannian case, there is a great variety of Finsler disks that isometrically fill a closed curve and have the same Holmes–Thompson area as the hemisphere. If the Hausdorff area is used instead, then the minimality of the hemisphere still holds, but the hemisphere becomes the unique minimizer. This follows from Ivanov's theorem since the Hausdorff area of a Finsler manifold is never less than the Holmes–Thompson area, and the two areas are equal if and only if the metric is Riemannian.

Non-minimality of the hemisphere among rational fillings with Finsler metrics

A Euclidean disk that fills a circle can be replaced, without decreasing the distances between boundary points, by a Finsler disk that fills the same circle N=10 times (in the sense that its boundary wraps around the circle N times), but whose Holmes–Thompson area is less than N times the area of the disk. For the hemisphere, a similar replacement can be found. In other words, the filling area conjecture is false if Finsler 2-chains with rational coefficients are allowed as fillings, instead of orientable surfaces (which can be considered as 2-chains with integer coefficients).

Riemannian fillings of genus one and hyperellipticity

An orientable Riemannian surface of genus one that isometrically fills the circle cannot have less area than the hemisphere. The proof in this case again starts by gluing antipodal points of the boundary. The non-orientable closed surface obtained in this way has an orientable double cover of genus two, and is therefore hyperelliptic. The proof then exploits a 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. 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.

Almost flat manifolds are minimal fillings of their boundary distances

If a Riemannian manifold M (of any dimension) is almost flat (more precisely, M is a region of ${\displaystyle {\mathbb{R}}^n}$ with a Riemannian metric that is ${\displaystyle C^2}$-near the standard Euclidean metric), then M is a volume minimizer: it cannot be replaced by an orientable Riemannian manifold that fills the same boundary and has less volume without reducing the distance between some boundary points. This implies that if a piece of sphere is sufficiently small (and therefore, nearly flat), then it is a volume minimizer. If this theorem can be extended to large regions (namely, to the whole hemisphere), then the filling area conjecture is true. It has been conjectured that all simple Riemannian manifolds (those that are convex at their boundary, and where every two points are joined by a unique geodesic) are volume minimizers.

The proof that each almost flat manifold M is a volume minimizer involves embedding M in ${\displaystyle L^{\mathrm{\infty }}(\mathrm{\partial }M)}$, and then showing that any isometric replacement of M can also be mapped into the same space ${\displaystyle L^{\mathrm{\infty }}(\mathrm{\partial }M)}$, and projected onto M, without increasing its volume. This implies that the replacement has not less volume than the original manifold M.