Pu's inequality

Summary

In differential geometry, Pu's inequality, proved by Pao Ming Pu, relates the area of an arbitrary Riemannian surface homeomorphic to the real projective plane with the lengths of the closed curves contained in it.

An animation of the Roman surface representing RP2 in R3

Statement

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A student of Charles Loewner, Pu proved in his 1950 thesis (Pu 1952) that every Riemannian surface   homeomorphic to the real projective plane satisfies the inequality

 

where   is the systole of  . The equality is attained precisely when the metric has constant Gaussian curvature.

In other words, if all noncontractible loops in   have length at least  , then   and the equality holds if and only if   is obtained from a Euclidean sphere of radius   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.

Proof

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Pu's original proof relies on the uniformization theorem and employs an averaging argument, as follows.

By uniformization, the Riemannian surface   is conformally diffeomorphic to a round projective plane. This means that we may assume that the surface   is obtained from the Euclidean unit sphere   by identifying antipodal points, and the Riemannian length element at each point   is

 

where   is the Euclidean length element and the function  , called the conformal factor, satisfies  .

More precisely, the universal cover of   is  , a loop   is noncontractible if and only if its lift   goes from one point to its opposite, and the length of each curve   is

 

Subject to the restriction that each of these lengths is at least  , we want to find an   that minimizes the

 

where   is the upper half of the sphere.

A key observation is that if we average several different   that satisfy the length restriction and have the same area  , then we obtain a better conformal factor  , that also satisfies the length restriction and has

 
 

and the inequality is strict unless the functions   are equal.

A way to improve any non-constant   is to obtain the different functions   from   using rotations of the sphere  , defining  . If we average over all possible rotations, then we get an   that is constant over all the sphere. We can further reduce this constant to minimum value   allowed by the length restriction. Then we obtain the obtain the unique metric that attains the minimum area  .

Reformulation

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Alternatively, every metric on the sphere   invariant under the antipodal map admits a pair of opposite points   at Riemannian distance   satisfying  

A more detailed explanation of this viewpoint may be found at the page Introduction to systolic geometry.

Filling area conjecture

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An alternative formulation of Pu's inequality is the following. Of all possible fillings of the Riemannian circle of length   by a  -dimensional disk with the strongly isometric property, the round hemisphere has the least area.

To explain this formulation, we start with the observation that the equatorial circle of the unit  -sphere   is a Riemannian circle   of length  . More precisely, the Riemannian distance function of   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  , whereas in the Riemannian circle it is  .

We consider all fillings of   by a  -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  . The inclusion of the circle as the boundary is then called a strongly isometric imbedding of the circle.

Gromov conjectured that the round hemisphere gives the "best" way of filling the circle even when the filling surface is allowed to have positive genus (Gromov 1983).

Isoperimetric inequality

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Pu's inequality bears a curious resemblance to the classical isoperimetric inequality

 

for Jordan curves in the plane, where   is the length of the curve while   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

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References

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  • Gromov, Mikhael (1983). "Filling Riemannian manifolds". J. Differential Geom. 18 (1): 1–147. doi:10.4310/jdg/1214509283. MR 0697984.
  • Gromov, Mikhael (1996). "Systoles and intersystolic inequalities". In Besse, Arthur L. (ed.). Actes de la Table Ronde de Géométrie Différentielle (Luminy, 1992) [Proceedings of the Roundtable on Differential Geometry]. Séminaires et Congrès. Vol. 1. Paris: Soc. Math. France. pp. 291–362. ISBN 2-85629-047-7. MR 1427752.
  • Gromov, Misha (1999) [1981]. Metric structures for Riemannian and non-Riemannian spaces. Progress in Mathematics. Vol. 152. With appendices by M. Katz, P. Pansu and S. Semmes. Translated from the French by Sean Michael Bates. Boston, MA: Birkhäuser Boston, Inc. ISBN 0-8176-3898-9. MR 1699320.
  • Katz, Mikhail G. (2007). Systolic geometry and topology. Mathematical Surveys and Monographs. Vol. 137. With an appendix by J. Solomon. Providence, RI: American Mathematical Society. doi:10.1090/surv/137. ISBN 978-0-8218-4177-8. MR 2292367. S2CID 118039315.
  • Pu, Pao Ming (1952). "Some inequalities in certain nonorientable Riemannian manifolds". Pacific J. Math. 2 (1): 55–71. doi:10.2140/pjm.1952.2.55. MR 0048886.