Pu's inequality

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.

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

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

is obtained from a Euclidean sphere of radius

Pu's paper also stated for the first time Loewner's inequality, a similar result for Riemannian metrics on the torus.

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

is conformally diffeomorphic to a round projective plane.

is obtained from the Euclidean unit sphere

is the Euclidean length element and the function

, called the conformal factor, satisfies

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

that satisfy the length restriction and have the same area

, that also satisfies the length restriction and has and the inequality is strict unless the functions

allowed by the length restriction.

invariant under the antipodal map admits a pair of opposite points

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

An alternative formulation of Pu's inequality is the following.

Of all possible fillings of the Riemannian circle of length

-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

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

-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).

Pu's inequality bears a curious resemblance to the classical isoperimetric inequality for Jordan curves in the plane, where