Area-Minimizing Projective Planes in 3-Manifolds

Let (M, g) be a compact Riemannian manifold of dimension 3, and let F denote the collection of all embedded surfaces homeomorphic to RP2. 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 RP3 up to scaling. The proof uses the formula for the second variation of area and Hamilton's Ricci flow. (C) 2010 Wiley Periodicals, Inc.