Area-minimizing projective planes in three-manifolds

2009/09/09 by Hubert L. Bray, Simon Brendle, Bray, H. +5 · 2 citations
Mathematics · #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Point processes and geometric inequalities

paper · pdf · doi:10.48550/arxiv.0909.1665

openalex publication_date 2009/09/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let (M,g) be a compact Riemannian manifold of dimension 3, and let \mathscrF denote the collection of all embedded surfaces homeomorphic to \mathbbRP². We study the infimum of the areas of all surfaces in \mathscrF. This quantity is related to the systole of (M,g). It makes sense whenever \mathscrF 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 \mathbbRP³ up to scaling. %The proof uses the formula for the second variation of area, and Hamilton's Ricci flow.

Citations

• A new conformal invariant and its applications to the Willmore conjecture and the first eigenvalue of compact surfaces

Cited by

• A 2-systolic inequality on non-rational compact Kähler surfaces with positive scalar curvature
• Stability and rigidity of extremal surfaces in Riemannian geometry and General Relativity

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