In
differential geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...
,
Mikhail Gromov's filling area conjecture asserts that the
hemisphere
Hemisphere refers to:
* A half of a sphere
As half of the Earth
* A hemisphere of Earth
** Northern Hemisphere
** Southern Hemisphere
** Eastern Hemisphere
** Western Hemisphere
** Land and water hemispheres
* A half of the (geocentric) celes ...
has minimum area among the
orientable
In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclockwise". A space i ...
surfaces that fill a closed curve of given length without introducing shortcuts between its points.
Definitions and statement of the conjecture
Every smooth surface or curve in
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
is a
metric space
In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...
, in which the
(intrinsic) distance between two points of is defined as the infimum of the lengths of the curves that go from to ''along'' . For example, on a closed curve
of length , for each point of the curve there is a unique other point of the curve (called the antipodal of ) at distance from .
A
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ...
surface fills a closed curve if its border (also called
boundary
Boundary or Boundaries may refer to:
* Border, in political geography
Entertainment
* ''Boundaries'' (2016 film), a 2016 Canadian film
* ''Boundaries'' (2018 film), a 2018 American-Canadian road trip film
*Boundary (cricket), the edge of the pla ...
, denoted ) is the curve . The filling is said isometric if for any two points of the boundary curve , the distance between them along is the same (not less) than the distance along the boundary. In other words, to fill a curve isometrically is to fill it without introducing shortcuts.
Question: ''How small can be the area of a surface that isometrically fills its boundary curve, of given length?''
For example, in three-dimensional Euclidean space, the circle
:
(of length 2) is filled by the flat disk
:
which is not an isometric filling, because any straight chord along it is a shortcut. In contrast, the hemisphere
:
is an isometric filling of the same circle , which has
twice the area of the flat disk. Is this the minimum possible area?
The surface can be imagined as made of a flexible but non-stretchable material, that allows it to be moved around and bended in Euclidean space. None of these transformations modifies the area of the surface nor the length of the curves drawn on it, which are the magnitudes relevant to the problem. The surface can be removed from Euclidean space altogether, obtaining a
Riemannian surface, which is an abstract
smooth surface
In mathematics, the differential geometry of surfaces deals with the differential geometry of smooth surfaces with various additional structures, most often, a Riemannian metric.
Surfaces have been extensively studied from various perspective ...
with a
Riemannian metric
In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space '' ...
that encodes the lengths and area. Reciprocally, according to the
Nash-Kuiper theorem
The Nash embedding theorems (or imbedding theorems), named after John Forbes Nash Jr., state that every Riemannian manifold can be isometrically embedded into some Euclidean space. Isometric means preserving the length of every path. For instan ...
, any Riemannian surface with boundary can be embedded in Euclidean space preserving the lengths and area specified by the Riemannian metric. Thus the filling problem can be stated equivalently as a question about
Riemannian surfaces, that are not placed in Euclidean space in any particular way.
:Conjecture (Gromov's filling area conjecture, 1983): ''The hemisphere has minimum area among the
orientable
In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclockwise". A space i ...
compact Riemannian surfaces that fill isometrically their boundary curve, of given length.''
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
Disk or disc may refer to:
* Disk (mathematics), a geometric shape
* Disk storage
Music
* Disc (band), an American experimental music band
* ''Disk'' (album), a 1995 EP by Moby
Other uses
* Disk (functional analysis), a subset of a vector sp ...
.''
[
Proof: Let be a Riemannian disk that isometrically fills its boundary of length . Glue each point with its antipodal point , defined as the unique point of that is at the maximum possible distance from . Gluing in this way we obtain a closed Riemannian surface that is homeomorphic to the ]real projective plane
In mathematics, the real projective plane is an example of a compact non-orientable two-dimensional manifold; in other words, a one-sided surface. It cannot be embedded in standard three-dimensional space without intersecting itself. It has b ...
and whose systole
Systole ( ) is the part of the cardiac cycle during which some chambers of the heart contract after refilling with blood. The term originates, via New Latin, from Ancient Greek (''sustolē''), from (''sustéllein'' 'to contract'; from ''sun ...
(the length of the shortest non-contractible curve) is equal to . (And reciprocally, if we cut open a projective plane along a shortest noncontractible loop of length , we obtain a disk that fills isometrically its boundary of length .) Thus the minimum area that the isometric filling can have is equal to the minimum area that a Riemannian projective plane of systole 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
In mathematics, the uniformization theorem says that every simply connected Riemann surface is conformally equivalent to one of three Riemann surfaces: the open unit disk, the complex plane, or the Riemann sphere. The theorem is a generalization ...
.
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
In mathematics, the uniformization theorem says that every simply connected Riemann surface is conformally equivalent to one of three Riemann surfaces: the open unit disk, the complex plane, or the Riemann sphere. The theorem is a generalization ...
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
In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite ...
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.''
Let be a Finsler disk that isometrically fills its boundary of length . We may assume that is the standard round disk in , and the Finsler metric
In mathematics, particularly differential geometry, a Finsler manifold is a differentiable manifold where a (possibly asymmetric) Minkowski functional is provided on each tangent space , that enables one to define the length of any smooth curve ...
is smooth and strongly convex. The Holmes–Thompson area of the filling can be computed by the formula
:
where for each point , the set is the dual unit ball of the norm (the unit ball of the dual norm
In functional analysis, the dual norm is a measure of size for a continuous linear function defined on a normed vector space.
Definition
Let X be a normed vector space with norm \, \cdot\, and let X^* denote its continuous dual space. The dual ...
), and is its usual area as a subset of .
Choose a collection of boundary points, listed in counterclockwise order. For each point , we define on ''M'' the scalar function . These functions have the following properties:
* Each function is Lipschitz on ''M'' and therefore (by Rademacher's theorem In mathematical analysis, Rademacher's theorem, named after Hans Rademacher, states the following: If is an open subset of and is Lipschitz continuous, then is differentiable almost everywhere in ; that is, the points in at which is ''not'' d ...
) differentiable at almost every
In measure theory (a branch of mathematical analysis), a property holds almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities. The notion of "almost everywhere" is a companion notion t ...
point .
* If is differentiable at an interior point , then there is a unique shortest curve from to ''x'' (parametrized with unit speed), that arrives at ''x'' with a speed . The differential has norm 1 and is the unique covector such that .
* In each point where all the functions are differentiable, the covectors are distinct and placed in counterclockwise order on the dual unit sphere . Indeed, they must be distinct because different geodesics cannot arrive at with the same speed. Also, if three of these covectors (for some