Area-minimizing projective planes in three-manifolds
Differential Geometry
2010-01-04 v2 Analysis of PDEs
Abstract
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.
Cite
@article{arxiv.0909.1665,
title = {Area-minimizing projective planes in three-manifolds},
author = {H. Bray and S. Brendle and M. Eichmair and A. Neves},
journal= {arXiv preprint arXiv:0909.1665},
year = {2010}
}
Comments
To appear in Comm. Pure Appl. Math