Area-minimizing projective planes in three-manifolds

Details Area-minimizing projective planes in three-manifolds Area-minimizing projective planes in three-manifolds

2009-09-09

arxiv.org/abs/0909.1665v2

Mathematics

Differential Geometry

To appear in Comm. Pure Appl. Math

Scientific paper

Let (M,g) be a compact Riemannian manifold of dimension 3, and let \mathscr{F} denote the collection of all embedded surfaces homeomorphic to \mathbb{RP}^2. We study the infimum of the areas of all surfaces in \mathscr{F}. This quantity is related to the systole of (M,g). It makes sense whenever \mathscr{F} is non-empty. 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 \mathbb{RP}^3 up to scaling. %The proof uses the formula for the second variation of area, and Hamilton's Ricci flow.

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