Pu's inequality
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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 ...
, Pu's inequality, proved by
Pao Ming Pu Pao Ming Pu (the form of his name he used in Western languages, although the Wade-Giles transliteration would be Pu Baoming; ; August 1910 – February 22, 1988), was a mathematician born in Jintang County, Sichuan, China.. He was a student ...
, relates the
area Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape A shape or figure is a graphics, graphical representation of an obje ...
of an arbitrary Riemannian surface 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 ...
with the
lengths Length is a measure of distance. In the International System of Quantities, length is a quantity with dimension distance. In most systems of measurement a base unit for length is chosen, from which all other units are derived. In the Intern ...
of the closed curves contained in it.


Statement

A student of
Charles Loewner Charles Loewner (29 May 1893 – 8 January 1968) was an American mathematician. His name was Karel Löwner in Czech and Karl Löwner in German. Karl Loewner was born into a Jewish family in Lany, about 30 km from Prague, where his father Sig ...
, Pu proved in his 1950 thesis that every Riemannian surface M 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 ...
satisfies the inequality : \operatorname(M) \geq \frac \operatorname(M)^2 , where \operatorname(M) is the
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 ...
of M . The equality is attained precisely when the metric has constant
Gaussian curvature In differential geometry, the Gaussian curvature or Gauss curvature of a surface at a point is the product of the principal curvatures, and , at the given point: K = \kappa_1 \kappa_2. The Gaussian radius of curvature is the reciprocal of . F ...
. In other words, if all noncontractible loops in M have length at least L , then \operatorname(M) \geq \frac L^2, and the equality holds if and only if M is obtained from a Euclidean sphere of radius r=L/\pi by identifying each point with its antipodal. Pu's paper also stated for the first time Loewner's inequality, a similar result for Riemannian metrics on the
torus In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle. If the axis of revolution does not tou ...
.


Proof

Pu's original proof relies 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 ...
and employs an averaging argument, as follows. By uniformization, the Riemannian surface (M,g) is conformally diffeomorphic to a round projective plane. This means that we may assume that the surface M is obtained from the Euclidean unit sphere S^2 by identifying antipodal points, and the Riemannian length element at each point x is : \mathrm = f(x) \mathrm_, where \mathrm_ is the Euclidean length element and the function f: S^2\to(0,+\infty) , called the conformal factor, satisfies f(-x)=f(x) . More precisely, the universal cover of M is S^2 , a loop \gamma\subseteq M is noncontractible if and only if its lift \widetilde\gamma\subseteq S^2 goes from one point to its opposite, and the length of each curve \gamma is : \operatorname(\gamma)=\int_ f \, \mathrm_. Subject to the restriction that each of these lengths is at least L , we want to find an f that minimizes the : \operatorname(M,g)=\int_ f(x)^2\,\mathrm_(x), where S^2_+ is the upper half of the sphere. A key observation is that if we average several different f_i that satisfy the length restriction and have the same area A , then we obtain a better conformal factor f_ = \frac \sum_ f_i, that also satisfies the length restriction and has : \operatorname(M,g_) = \int_\left(\frac 1n\sum_i f_i(x)\right)^2\mathrm_(x) : \qquad\qquad \leq \frac\sum_i\left(\int_ f_i(x)^2\mathrm_(x)\right) = A, and the inequality is strict unless the functions f_i are equal. A way to improve any non-constant f is to obtain the different functions f_i from f using rotations of the sphere R_i\in SO^3 , defining f_i(x)=f(R_i(x)). If we average over all possible rotations, then we get an f_ that is constant over all the sphere. We can further reduce this constant to minimum value r=\frac L\pi allowed by the length restriction. Then we obtain the obtain the unique metric that attains the minimum area 2\pi r^2 = \frac 2\pi L^2 .


Reformulation

Alternatively, every metric on the sphere S^2 invariant under the antipodal map admits a pair of opposite points p,q\in S^2 at Riemannian distance d=d(p,q) satisfying d^2 \leq \frac \operatorname (S^2). A more detailed explanation of this viewpoint may be found at the page
Introduction to systolic geometry Systolic geometry is a branch of differential geometry, a field within mathematics, studying problems such as the relationship between the area inside a closed curve ''C'', and the length or perimeter of ''C''. Since the area ''A'' may be smal ...
.


Filling area conjecture

An alternative formulation of Pu's inequality is the following. Of all possible fillings of the
Riemannian circle In metric space theory and Riemannian geometry, the Riemannian circle is a great circle with a characteristic length. It is the circle equipped with the ''intrinsic'' Riemannian metric of a compact one-dimensional manifold of total length 2, or ...
of length 2\pi by a 2-dimensional disk with the strongly isometric property, the round
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 the least area. To explain this formulation, we start with the observation that the equatorial circle of the unit 2-sphere S^2 \subset \mathbb R^3 is a
Riemannian circle In metric space theory and Riemannian geometry, the Riemannian circle is a great circle with a characteristic length. It is the circle equipped with the ''intrinsic'' Riemannian metric of a compact one-dimensional manifold of total length 2, or ...
S^1 of length 2\pi. More precisely, the Riemannian distance function of S^1 is induced from the ambient Riemannian distance on the sphere. Note that this property is not satisfied by the standard imbedding of the unit circle in the Euclidean plane. Indeed, the Euclidean distance between a pair of opposite points of the circle is only 2, whereas in the Riemannian circle it is \pi. We consider all fillings of S^1 by a 2-dimensional disk, such that the metric induced by the inclusion of the circle as the boundary of the disk is the Riemannian metric of a circle of length 2\pi. The inclusion of the circle as the boundary is then called a strongly isometric imbedding of the circle. Gromov
conjectured In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis (still a conjecture) or Fermat's Last Theorem (a conjecture until proven in 199 ...
that the round hemisphere gives the "best" way of filling the circle even when the filling surface is allowed to have positive genus .


Isoperimetric inequality

Pu's inequality bears a curious resemblance to the classical
isoperimetric inequality In mathematics, the isoperimetric inequality is a geometric inequality involving the perimeter of a set and its volume. In n-dimensional space \R^n the inequality lower bounds the surface area or perimeter \operatorname(S) of a set S\subset\R^n ...
: L^2 \geq 4\pi A for Jordan curves in the plane, where L is the length of the curve while A is the area of the region it bounds. Namely, in both cases a 2-dimensional quantity (area) is bounded by (the square of) a 1-dimensional quantity (length). However, the inequality goes in the opposite direction. Thus, Pu's inequality can be thought of as an "opposite" isoperimetric inequality.


See also

*
Filling area conjecture In differential geometry, Mikhail Gromov's filling area conjecture asserts that the hemisphere has minimum area among the orientable surfaces that fill a closed curve of given length without introducing shortcuts between its points. Definitions ...
*
Gromov's systolic inequality for essential manifolds In the mathematical field of Riemannian geometry, M. Gromov's systolic inequality bounds the length of the shortest non-contractible loop on a Riemannian manifold in terms of the volume of the manifold. Gromov's systolic inequality was proved in 1 ...
*
Gromov's inequality for complex projective space In Riemannian geometry, Gromov's optimal stable 2- systolic inequality is the inequality : \mathrm_2^n \leq n! \;\mathrm_(\mathbb^n), valid for an arbitrary Riemannian metric on the complex projective space, where the optimal bound is attained b ...
*
Loewner's torus inequality In differential geometry, Loewner's torus inequality is an inequality due to Charles Loewner. It relates the systole and the area of an arbitrary Riemannian metric on the 2-torus. Statement In 1949 Charles Loewner proved that every metric on ...
* Systolic geometry * Systoles of surfaces


References

* * * * * {{Systolic geometry navbox Riemannian geometry Geometric inequalities Differential geometry of surfaces Systolic geometry