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In
differential geometry Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of Calculus, single variable calculus, vector calculus, lin ...
, Pu's inequality, proved by Pao Ming Pu, relates the
area Area is the measure of a region's size on a surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an open surface or the boundary of a three-di ...
of an arbitrary Riemannian surface homeomorphic to the
real projective plane In mathematics, the real projective plane, denoted or , is a two-dimensional projective space, similar to the familiar Euclidean plane in many respects but without the concepts of distance, circles, angle measure, or parallelism. It is the sett ...
with the lengths 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. Early life and career Karl Loewner was born into a Jewish family in Lany, about 30 km from Prag ...
, Pu proved in his 1950 thesis that every Riemannian surface M homeomorphic to the
real projective plane In mathematics, the real projective plane, denoted or , is a two-dimensional projective space, similar to the familiar Euclidean plane in many respects but without the concepts of distance, circles, angle measure, or parallelism. It is the sett ...
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. Its contrasting phase is diastole, the relaxed phase of the cardiac cycle when the chambers of the heart are refilling ...
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 smooth Surface (topology), surface in three-dimensional space at a point is the product of the principal curvatures, and , at the given point: K = \kappa_1 \kappa_2. For ...
. 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 (: tori or toruses) is a surface of revolution generated by revolving a circle in three-dimensional space one full revolution about an axis that is coplanarity, coplanar with the circle. The main types of toruses inclu ...
.


Proof

Pu's original proof relies on the
uniformization theorem In mathematics, the uniformization theorem states 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 generali ...
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.


Filling area conjecture

An alternative formulation of Pu's inequality is the following. Of all possible fillings of the
Riemannian circle In mathematics, a metric circle is the metric space of arc length on a circle, or equivalently on any rectifiable simple closed curve of bounded length. The metric spaces that can be embedded into metric circles can be characterized by a four-p ...
of length 2\pi by a 2-dimensional disk with the strongly isometric property, the round
hemisphere Hemisphere may refer to: In geometry * Hemisphere (geometry), a half of a sphere As half of Earth or any spherical astronomical object * A hemisphere of Earth ** Northern Hemisphere ** Southern Hemisphere ** Eastern Hemisphere ** Western Hemi ...
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 mathematics, a metric circle is the metric space of arc length on a circle, or equivalently on any rectifiable simple closed curve of bounded length. The metric spaces that can be embedded into metric circles can be characterized by a four-p ...
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 or Fermat's conjecture (now a theorem, proven in 1995 by Andrew Wiles), have sh ...
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 square of the circumference of a closed curve in the plane and the area of a plane region it encloses, as well as its various generalizations. '' Isoperimetric'' ...
: 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 * Gromov's inequality for complex projective space * Loewner's torus inequality *
Systolic geometry In mathematics, systolic geometry is the study of systolic invariants of manifolds and polyhedra, as initially conceived by Charles Loewner and developed by Mikhail Gromov, Michael Freedman, Peter Sarnak, Mikhail Katz, Larry Guth, and ...
* Systoles of surfaces


References

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