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.
Statement A student of Charles Loewner, Pu proved in his 1950 thesis (Pu 1952) that every Riemannian surface M {\displaystyle M} homeomorphic to the real projective plane satisfies the inequality
Area ( M ) ≥ 2 π Systole ( M ) 2 , {\displaystyle \operatorname {Area} (M)\geq {\frac {2}{\pi }}\operatorname {Systole} (M)^{2},}
where Systole ( M ) {\displaystyle \operatorname {Systole} (M)} , the systole of M {\displaystyle M} , is the length of the shortest loop in M that cannot be contracted to a point in the ambient space X. The equality is attained precisely when the metric has constant Gaussian curvature. In other words, if all noncontractible loops in M {\displaystyle M} have length at least L {\displaystyle L} , then Area ( M ) ≥ 2 π L 2 , {\displaystyle \operatorname {Area} (M)\geq {\frac {2}{\pi }}L^{2},} and the equality holds if and only if M {\displaystyle M} is obtained from a Euclidean sphere of radius r = L / π {\displaystyle 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.
Proof Pu's original proof relies on the uniformization theorem and employs an averaging argument, as follows. By uniformization, the Riemannian surface ( M , g ) {\displaystyle (M,g)} is conformally diffeomorphic to a round projective plane. This means that we may assume that the surface M {\displaystyle M} is obtained from the Euclidean unit sphere S 2 {\displaystyle S^{2}} by identifying antipodal points, and the Riemannian length element at each point x {\displaystyle x} is
d L e n g t h = f ( x ) d L e n g t h Euclidean , {\displaystyle \mathrm {dLength} =f(x)\mathrm {dLength} _{\text{Euclidean}},}
where d L e n g t h Euclidean {\displaystyle \mathrm {dLength} _{\text{Euclidean}}} is the Euclidean length element and the function f : S 2 → ( 0 , + ∞ ) {\displaystyle f:S^{2}\to (0,+\infty )} , called the conformal factor, satisfies f ( − x ) = f ( x ) {\displaystyle f(-x)=f(x)} . More precisely, the universal cover of M {\displaystyle M} is S 2 {\displaystyle S^{2}} , a loop γ ⊆ M {\displaystyle \gamma \subseteq M} is noncontractible if and only if its lift γ ~ ⊆ S 2 {\displaystyle {\widetilde {\gamma }}\subseteq S^{2}} goes from one point to its opposite, and the length of each curve γ {\displaystyle \gamma } is
Length ( γ ) = ∫ γ ~ f d L e n g t h Euclidean . {\displaystyle \operatorname {Length} (\gamma )=\int _{\widetilde {\gamma }}f\,\mathrm {dLength} _{\text{Euclidean}}.}
Subject to the restriction that each of these lengths is at least L {\displaystyle L} , we want to find an f {\displaystyle f} that minimizes the
Area ( M , g ) = ∫ S + 2 f ( x ) 2 d A r e a Euclidean ( x ) , {\displaystyle \operatorname {Area} (M,g)=\int _{S_{+}^{2}}f(x)^{2}\,\mathrm {dArea} _{\text{Euclidean}}(x),}
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