Area-Minimizing Projective Planes in 3-Manifolds

Journal Article (Journal Article)

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

Full Text

Duke Authors

Cited Authors

  • Bray, H; Brendle, S; Eichmair, M; Neves, A

Published Date

  • September 1, 2010

Published In

Volume / Issue

  • 63 / 9

Start / End Page

  • 1237 - 1247

Electronic International Standard Serial Number (EISSN)

  • 0010-3640

International Standard Serial Number (ISSN)

  • 0010-3640

Digital Object Identifier (DOI)

  • 10.1002/cpa.20319

Citation Source

  • Scopus