Area‐minimizing projective planes in 3‐manifolds

Abstract

Abstract Let ( M,g ) be a compact Riemannian manifold of dimension 3, and let ℱ denote the collection of all embedded surfaces homeomorphic to \input amssym ${\Bbb R}{ \Bbb P}^2$ . We study the infimum of the areas of all surfaces in ℱ. This quantity is related to the systole of ( M,g ). It makes sense whenever ℱ is nonempty. In this paper, we give an upper bound for this quantity in terms of the minimum of the scalar curvature of ( M,g ). Moreover, we show that equality holds if and only if ( M,g ) is isometric to \input amssym ${\Bbb R}{ \Bbb P}^3$ up to scaling. The proof uses the formula for the second variation of area and Hamilton's Ricci flow. © 2010 Wiley Periodicals, Inc.