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 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. 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.